[
{
"id": "authors:bxqj9-jy145",
"collection": "authors",
"collection_id": "bxqj9-jy145",
"cite_using_url": "https://authors.library.caltech.edu/records/bxqj9-jy145",
"type": "article",
"title": "Double-exponential susceptibility growth in Dyson's hierarchical model with |x\u00a0\u2212\u00a0y|\u207b\u00b2 interaction",
"author": [
{
"family_name": "Easo",
"given_name": "Philip",
"orcid": "0000-0002-5606-3727"
},
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
},
{
"family_name": "Kurrek",
"given_name": "Jana",
"orcid": "0009-0002-8449-8899"
}
],
"abstract": "We study long-range percolation on the *d*-dimensional hierarchical lattice, in which each possible edge {*x*, *y*} is included independently at random with inclusion probability 1 − exp(−*β* \u2016*x* − *y*\u2016^{−d−α}), where *α* > 0 is fixed and *β* ≥ 0 is a parameter. This model is known to have a phase transition at some *β*_{c} < ∞ if and only if *α* < *d*. We study the model in the regime *α* ≥ *d*, in which *β*_{c} = ∞, and prove that the susceptibility *χ*(*β*) (i.e., the expected volume of the cluster at the origin) satisfies *χ*(*β*) = *β(*d/1−d)^(−o(1)) as *β*↑∞ if *α* > *d* and *χ*(*β*) = e^(e^e(Θ(*β*) as *β*↑∞ if *α* = *d*. This resolves a problem raised by Georgakopoulos and Haslegrave (2020), who showed that *χ*(*β*) grows between exponentially and double-exponentially when *α* = *d*. Our results imply that analogous results hold for a number of related models including Dyson’s hierarchical Ising model, for which the double-exponential susceptibility growth we establish appears to be a new phenomenon even at the heuristic level.

",
"doi": "10.1063/5.0147340",
"issn": "0022-2488",
"publisher": "AIP Publishing",
"publication": "Journal of Mathematical Physics",
"publication_date": "2024-02",
"series_number": "2",
"volume": "65",
"issue": "2",
"pages": "023301"
},
{
"id": "authors:y9141-cjk45",
"collection": "authors",
"collection_id": "y9141-cjk45",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20230420-614686900.16",
"type": "article",
"title": "Logarithmic Corrections to Scaling in the Four-dimensional Uniform Spanning Tree",
"author": [
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
},
{
"family_name": "Sousi",
"given_name": "Perla",
"clpid": "Sousi-Perla"
}
],
"abstract": "We compute the precise logarithmic corrections to mean-field scaling for various quantities describing the uniform spanning tree of the four-dimensional hypercubic lattice Z\u2074. We are particularly interested in the distribution of the past of the origin, that is, the finite piece of the tree that is separated from infinity by the origin. We prove that the probability that the past contains a path of length n is of order (log n)^(1/3)n\u207b\u00b9, that the probability that the past containsat least n vertices is of order (log n)^(1/6)n^(\u22121/2), and that the probability that the past reaches the boundary of the box [\u2212n, n]\u2074 is of order (log n)^(2/3+o(1))n\u207b\u00b2. An important part of our proof is to prove concentration estimates for the capacity of the four-dimensional loop-erased random walk which may be of independent interest. Our results imply that the Abelian sandpile model also exhibits non-trivial polylogarithmic corrections to mean-field scaling in four dimensions, although it remains open to compute the precise order of these corrections.",
"doi": "10.1007/s00220-023-04686-w",
"issn": "0010-3616",
"publisher": "Springer",
"publication": "Communications in Mathematical Physics",
"publication_date": "2023-04-27"
},
{
"id": "authors:em9gs-e7c18",
"collection": "authors",
"collection_id": "em9gs-e7c18",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20230411-695015900.4",
"type": "article",
"title": "High-dimensional near-critical percolation and the torus plateau",
"author": [
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
},
{
"family_name": "Michta",
"given_name": "Emmanuel",
"orcid": "0000-0001-7222-0422",
"clpid": "Michta-Emmanuel"
},
{
"family_name": "Slade",
"given_name": "Gordon",
"orcid": "0000-0001-9389-9497",
"clpid": "Slade-Gordon"
}
],
"abstract": "We consider percolation on Z\u1d48 and on the d-dimensional discrete torus, in dimensions d \u2265 11 for the nearest-neighbour model and in dimensions d > 6 for spread-out models. For Z\u1d48 we employ a wide range of techniques and previous results to prove that there exist positive constants c and C such that the slightly subcritical two-point function and one-arm probabilities satisfy \nP_(p_c \u2212 \u03b5)(0 \u2194 x) \u2264 C/(\u2225x\u2225\u1d48\u207b\u00b2)e^(\u2212c\u03b5^(1/2)\u2225x\u2225), \n(c/r\u00b2)e^(\u2212C\u03b5^((1/2)r)) \u2264 P_(pc\u2212\u03b5)(0 \u2194 \u2202[\u2212r,r]\u1d48) \u2264 C/(r\u00b2)e^(\u2212c\u03b5(1/2)r). \n\nUsing this, we prove that throughout the critical window the torus two-point function has a \"plateau,\" meaning that it decays for small x as \u2225x\u2225\u207b\u207d\u1d48\u207b\u00b2\u207e but for large x is essentially constant and of order V^(\u22122/3) where V is the volume of the torus. The plateau for the two-point function leads immediately to a proof of the torus triangle condition, which is known to have many implications for the critical behaviour on the torus, and also leads to a proof that the critical values on the torus and on Z\u1d48 are separated by a multiple of V^(\u22121/3). The torus triangle condition and the size of the separation of critical points have been proved previously, but our proofs are different and are direct consequences of the bound on the Z\u1d48 two-point function. In particular, we use results derived from the lace expansion on Z\u1d48, but in contrast to previous work on high-dimensional torus percolation, we do not need or use a separate torus lace expansion.",
"doi": "10.1214/22-aop1608",
"issn": "0091-1798",
"publisher": "Institute of Mathematical Statistics",
"publication": "Annals of Probability",
"publication_date": "2023-03",
"series_number": "2",
"volume": "51",
"issue": "2",
"pages": "580-625"
},
{
"id": "authors:a7hr1-srd50",
"collection": "authors",
"collection_id": "a7hr1-srd50",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20230103-818063100.57",
"type": "article",
"title": "The bunkbed conjecture holds in the p \u2191 1 limit",
"author": [
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
},
{
"family_name": "Kent",
"given_name": "Alexander",
"clpid": "Kent-Alexander"
},
{
"family_name": "Nizi\u0107-Nikolac",
"given_name": "Petar",
"clpid": "Nizi\u0107-Nikolac-Petar"
}
],
"abstract": "Let G = (V, E) be a countable graph. The Bunkbed graph of G is the product graph G x K\u2082, which has vertex set V x {0,1} with \"horizontal\" edges inherited from G and additional \"vertical\" edges connecting (w,0) and (w,1) for each w \u03f5 V. Kasteleyn's Bunkbed conjecture states that for each u, v \u03f5 V and p \u03f5 [0,1], the vertex (u,0) is at least as likely to be connected to (v,0) as to (v,1) under Bernoulli-p bond percolation on the bunkbed graph. We prove that the conjecture holds in the p \u2191 1 limit in the sense that for each finite graph G there exists \u03b5(G) > 0 such that the bunkbed conjecture holds for p \u2a7e 1 - \u03b5(G).",
"doi": "10.1017/s096354832200027x",
"issn": "0963-5483",
"publisher": "Cambridge University Press",
"publication": "Combinatorics, Probability and Computing",
"publication_date": "2023-02-07",
"pages": "1-7"
},
{
"id": "authors:95j8r-pwq77",
"collection": "authors",
"collection_id": "95j8r-pwq77",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20230725-745306000.2",
"type": "article",
"title": "Transience and anchored isoperimetric dimension of supercritical percolation clusters",
"author": [
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
}
],
"abstract": "We establish several equivalent characterisations of the anchored isoperimetric dimension of supercritical clusters in Bernoulli bond percolation on transitive graphs. We deduce from these characterisations together with a theorem of Duminil-Copin, Goswami, Raoufi, Severo, and Yadin (Duke Math. J. 2020) that if G is a transient transitive graph then the infinite clusters of Bernoulli percolation on G are transient for p sufficiently close to 1. It remains open to extend this result down to the critical probability. Along the way we establish two new cluster repulsion inequalities that are of independent interest.",
"doi": "10.1214/23-ejp905",
"issn": "1083-6489",
"publisher": "Institute of Mathematical Statistics",
"publication": "Electronic Journal of Probability",
"publication_date": "2023-01-20",
"volume": "28",
"pages": "1-15"
},
{
"id": "authors:c1dgm-xs860",
"collection": "authors",
"collection_id": "c1dgm-xs860",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20221129-370786800.2",
"type": "article",
"title": "Sharp hierarchical upper bounds on the critical two-point function for long-range percolation on \u2124\u1d48",
"author": [
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
}
],
"abstract": "Consider long-range Bernoulli percolation on \u2124\u1d48 in which we connect each pair of distinct points x and y by an edge with probability 1 \u2212 exp(\u2212\u03b2\u2016x \u2212 y\u2016^(\u2212d\u2212\u03b1)), where \u03b1 > 0 is fixed and \u03b2 \u2a7e 0 is a parameter. We prove that if 0 < \u03b1 < d, then the critical two-point function satisfies (1/|\u039b_r|)\u2211_(x\u03f5\u039b_(r))P_(\u03b2_(c))(0 \u2194 x) \u2264 r^(\u2212d+a) for every r \u2a7e 1, where \u039b_r = [\u2212r,r]\u1d48 \u2229 \u2124\u1d48. In other words, the critical two-point function on \u2124\u1d48 is always bounded above on average by the critical two-point function on the hierarchical lattice. This upper bound is believed to be sharp for values of \u03b1 strictly below the crossover value \u03b1_(c)(d), where the values of several critical exponents for long-range percolation on \u2124\u1d48 and the hierarchical lattice are believed to be equal.",
"doi": "10.1063/5.0088450",
"issn": "0022-2488",
"publisher": "American Institute of Physics",
"publication": "Journal of Mathematical Physics",
"publication_date": "2022-11",
"series_number": "11",
"volume": "63",
"issue": "11",
"pages": "Art. No. 113301"
},
{
"id": "authors:0pmve-wtk48",
"collection": "authors",
"collection_id": "0pmve-wtk48",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20220907-386218000",
"type": "article",
"title": "Slightly supercritical percolation on non-amenable graphs I: The distribution of finite clusters",
"author": [
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
}
],
"abstract": "We study the distribution of finite clusters in slightly supercritical (p\u2193pc) Bernoulli bond percolation on transitive nonamenable graphs, proving in particular that if G is a transitive nonamenable graph satisfying the L2 boundedness condition (pc0 such that Pp(n\u2264|K|<\u221e)\u224dn\u22121/2exp[\u2212\u0398(|p\u2212pc|2n)] and Pp(r\u2264Rad(K)<\u221e)\u224dr\u22121exp[\u2212\u0398(|p\u2212pc|r)] for every p\u2208(pc\u2212\u03b4,pc+\u03b4) and n,r\u22651, where all implicit constants depend only on G. We deduce in particular that the critical exponents \u03b3\u2032 and \u0394\u2032 describing the rate of growth of the moments of a finite cluster as p\u2193pc take their mean-field values of 1 and 2 respectively. These results apply in particular to Cayley graphs of nonelementary hyperbolic groups, to products with trees, and to transitive graphs of spectral radius \u03c1<1/2. In particular, every finitely generated nonamenable group has a Cayley graph to which these results apply. They are new for graphs that are not trees. The corresponding facts are yet to be understood on \u2124d even for d very large. In a second paper in this series, we will apply these results to study the geometric and spectral properties of infinite slightly supercritical clusters in the same setting.",
"doi": "10.1112/plms.12474",
"issn": "0024-6115",
"publisher": "London Mathematical Society",
"publication": "Proceedings of the London Mathematical Society",
"publication_date": "2022-10",
"series_number": "4",
"volume": "125",
"issue": "4",
"pages": "968-1013"
},
{
"id": "authors:zwvhy-s1238",
"collection": "authors",
"collection_id": "zwvhy-s1238",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20220823-628154700.756",
"type": "article",
"title": "On the Derivation of Mean-Field Percolation Critical Exponents from the Triangle Condition",
"author": [
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
}
],
"abstract": "We give a new derivation of mean-field percolation critical behaviour from the triangle condition that is quantitatively much better than previous proofs when the triangle diagram \u2207pc is large. In contrast to earlier methods, our approach continues to yield bounds of reasonable order when the triangle diagram \u2207p is unbounded but diverges slowly as p\u2191pc, as is expected to occur in percolation on Zd at the upper-critical dimension d=6. Indeed, we show in particular that if the triangle diagram diverges polylogarithmically as p\u2191pc then mean-field critical behaviour holds to within a polylogarithmic factor. We apply the methods we develop to deduce that for long-range percolation on the hierarchical lattice, mean-field critical behaviour holds to within polylogarithmic factors at the upper-critical dimension. As part of the proof, we introduce a new method for comparing diagrammatic sums on general transitive graphs that may be of independent interest.",
"doi": "10.1007/s10955-022-02967-7",
"issn": "0022-4715",
"publisher": "Springer",
"publication": "Journal of Statistical Physics",
"publication_date": "2022-10",
"series_number": "1",
"volume": "189",
"issue": "1",
"pages": "Art. No. 6"
},
{
"id": "authors:dv0h8-wnc25",
"collection": "authors",
"collection_id": "dv0h8-wnc25",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-203748960",
"type": "article",
"title": "What are the limits of universality?",
"author": [
{
"family_name": "Halberstam",
"given_name": "Noah",
"clpid": "Halberstam-Noah"
},
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
}
],
"abstract": "It is a central prediction of renormalization group theory that the critical behaviours of many statistical mechanics models on Euclidean lattices depend only on the dimension and not on the specific choice of lattice. We investigate the extent to which this universality continues to hold beyond the Euclidean setting, taking as case studies Bernoulli bond percolation and lattice trees. We present strong numerical evidence that the critical exponents governing these models on transitive graphs of polynomial volume growth depend only on the volume-growth dimension of the graph and not on any other large-scale features of the geometry. For example, our results strongly suggest that percolation, which has upper-critical dimension 6, has the same critical exponents on Z\u2074 and the Heisenberg group despite the distinct large-scale geometries of these two lattices preventing the relevant percolation models from sharing a common scaling limit. On the other hand, we also show that no such universality should be expected to hold on fractals, even if one allows the exponents to depend on a large number of standard fractal dimensions. Indeed, we give natural examples of two fractals which share Hausdorff, spectral, topological and topological Hausdorff dimensions but exhibit distinct numerical values of the percolation Fisher exponent \u03c4. This gives strong evidence against a conjecture of Balankin et al. (2018 Phys. Lett. A382, 12\u201319 (doi:10.1016/j.physleta.2017.10.035)).",
"doi": "10.1098/rspa.2021.0857",
"issn": "1364-5021",
"publisher": "Royal Society",
"publication": "Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences",
"publication_date": "2022-03",
"series_number": "2259",
"volume": "478",
"issue": "2259",
"pages": "Art. No. 20210857"
},
{
"id": "authors:39xfc-ekz38",
"collection": "authors",
"collection_id": "39xfc-ekz38",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-202129801",
"type": "article",
"title": "Collisions of random walks in dynamic random environments",
"author": [
{
"family_name": "Halberstam",
"given_name": "Noah",
"clpid": "Halberstam-Noah"
},
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
}
],
"abstract": "We study dynamic random conductance models on \u2124\u00b2 in which the environment evolves as a reversible Markov process that is stationary under space-time shifts. We prove under a second moment assumption that two conditionally independent random walks in the same environment collide infinitely often almost surely. These results apply in particular to random walks on dynamical percolation.",
"doi": "10.1214/21-EJP738",
"issn": "1083-6489",
"publisher": "Institute of Mathematical Statistics",
"publication": "Electronic Journal of Probability",
"publication_date": "2022-01-17",
"volume": "27",
"pages": "Art. No. 8"
},
{
"id": "authors:b2avf-5rb22",
"collection": "authors",
"collection_id": "b2avf-5rb22",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-202109319",
"type": "article",
"title": "No Percolation at Criticality on Certain Groups of Intermediate Growth",
"author": [
{
"family_name": "Hermon",
"given_name": "Jonathan",
"orcid": "0000-0002-2935-3999",
"clpid": "Hermon-Jonathan"
},
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
}
],
"abstract": "We prove that critical percolation has no infinite clusters almost surely on any unimodular quasi-transitive graph satisfying a return probability upper bound of the form p_n(v,v) \u2264 exp[\u2212\u03a9(n^\u03b3)] for some \u03b3 > \u00bd\u2060. The result is new in the case that the graph is of intermediate volume growth.",
"doi": "10.1093/imrn/rnz265",
"issn": "1073-7928",
"publisher": "Oxford University Press",
"publication": "International Mathematics Research Notices",
"publication_date": "2021-11",
"series_number": "22",
"volume": "2021",
"issue": "22",
"pages": "17433-17455"
},
{
"id": "authors:h9k7j-fzm15",
"collection": "authors",
"collection_id": "h9k7j-fzm15",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-202253573",
"type": "monograph",
"title": "High-dimensional near-critical percolation and the torus plateau",
"author": [
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
},
{
"family_name": "Michta",
"given_name": "Emmanuel",
"orcid": "0000-0001-7222-0422",
"clpid": "Michta-Emmanuel"
},
{
"family_name": "Slade",
"given_name": "Gordon",
"orcid": "0000-0001-9389-9497",
"clpid": "Slade-Gordon"
}
],
"abstract": "We consider percolation on Z^d and on the d-dimensional discrete torus, in dimensions d \u2265 11 for the nearest-neighbour model and in dimensions d > 6 for spread-out models. For \u2124^d, we employ a wide range of techniques and previous results to prove that there exist positive constants c and C such that the slightly subcritical two-point function and one-arm probabilities satisfy\n\u2119_(p_c \u2212 \u03b5) (0\u2194x) \u2264 C/(\u2016x\u2016^(d\u22122)) e^(\u2212c\u03b5^(1/2)\u2016x\u2016) and c/r^2 e^(\u2212C\u03b5^((1/2)r( \u2264 \u2119_(p_c \u2212 \u03b5)(0\u2194\u2202[\u2212r,r]^d) \u2264 C/r^2 e^(\u2212c\u03b5^((1/2)r)). \n\nUsing this, we prove that throughout the critical window the torus two-point function has a \"plateau,\" meaning that it decays for small x as \u2016x\u2016^(\u2212(d\u22122)) but for large x is essentially constant and of order V^(\u22122/3) where V is the volume of the torus. The plateau for the two-point function leads immediately to a proof of the torus triangle condition, which is known to have many implications for the critical behaviour on the torus, and also leads to a proof that the critical values on the torus and on \u2124^d are separated by a multiple of V^(\u22121/3). The torus triangle condition and the size of the separation of critical points have been proved previously, but our proofs are different and are direct consequences of the bound on the \u2124^d two-point function. In particular, we use results derived from the lace expansion on \u2124^d, but in contrast to previous work on high-dimensional torus percolation we do not need or use a separate torus lace expansion.",
"doi": "10.48550/arXiv.2107.12971",
"publisher": "arXiv",
"publication_date": "2021-09-27"
},
{
"id": "authors:8bzn7-k1747",
"collection": "authors",
"collection_id": "8bzn7-k1747",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-202116146",
"type": "monograph",
"title": "Slightly supercritical percolation on nonamenable graphs I: The distribution of finite clusters",
"author": [
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
}
],
"abstract": "We study the distribution of finite clusters in slightly supercritical (p\u2193pc) Bernoulli bond percolation on transitive nonamenable graphs, proving in particular that if G is a transitive nonamenable graph satisfying the L2 boundedness condition (pc0 such that\nPp(n\u2264|K|<\u221e)\u224dn\u22121/2exp[\u2212\u0398(|p\u2212pc|2n)]\nand\nPp(r\u2264Rad(K)<\u221e)\u224dr\u22121exp[\u2212\u0398(|p\u2212pc|r)]\nfor every p\u2208(pc\u2212\u03b4,pc+\u03b4) and n,r\u22651, where all implicit constants depend only on G. We deduce in particular that the critical exponents \u03b3\u2032 and \u0394\u2032 describing the rate of growth of the moments of a finite cluster as p\u2193pc take their mean-field values of 1 and 2 respectively.\nThese results apply in particular to Cayley graphs of nonelementary hyperbolic groups, to products with trees, and to transitive graphs of spectral radius \u03c1<1/2. In particular, every finitely generated nonamenable group has a Cayley graph to which these results apply. They are new for graphs that are not trees. The corresponding facts are yet to be understood on \u2124d even for d very large. In a second paper in this series, we will apply these results to study the geometric and spectral properties of infinite slightly supercritical clusters in the same setting.",
"doi": "10.48550/arXiv.2002.02916",
"publisher": "arXiv",
"publication_date": "2021-09-27"
},
{
"id": "authors:rsb7g-z9w39",
"collection": "authors",
"collection_id": "rsb7g-z9w39",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-202126385",
"type": "monograph",
"title": "Power-law bounds for critical long-range percolation below the upper-critical dimension",
"author": [
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
}
],
"abstract": "We study long-range Bernoulli percolation on \u2124d in which each two vertices x and y are connected by an edge with probability 1\u2212exp(\u2212\u03b2\u2016x\u2212y\u2016\u2212d\u2212\u03b1). It is a theorem of Noam Berger (CMP, 2002) that if 0<\u03b1",
"doi": "10.48550/arXiv.2008.11197",
"publisher": "arXiv",
"publication_date": "2021-09-27"
},
{
"id": "authors:r0bnm-b8563",
"collection": "authors",
"collection_id": "r0bnm-b8563",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-202143857",
"type": "monograph",
"title": "Non-triviality of the phase transition for percolation on finite transitive graphs",
"author": [
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
},
{
"family_name": "Tointon",
"given_name": "Matthew",
"orcid": "0000-0001-8086-9280",
"clpid": "Tointon-Matthew"
}
],
"abstract": "We prove that if (G_n)_(n \u2265 1) = ((V_n,E_n))_(n \u2265 1) is a sequence of\nfinite, vertex-transitive graphs with bounded degrees and |V_n|\u2192\u221e that\nis at least (1+\u03f5)-dimensional for some \u03f5 > 0 in the sense that\ndiam(G_n)=O(|V_n|^(1/(1+\u03f5) as n \u2192 \u221e then this sequence of graphs has a non-trivial phase transition\nfor Bernoulli bond percolation. More precisely, we prove under these conditions\nthat for each 0<\u03b1<1 there exists p_c(\u03b1) < 1 such that for each\np \u2265 p_c(\u03b1), Bernoulli-p bond percolation on G_n has a cluster of\nsize at least \u03b1|V_n| with probability tending to 1 as n \u2192 \u221e.\nIn fact, we prove more generally that there exists a universal constant a\nsuch that the same conclusion holds whenever diam(G_n) = 0(|V_n|/(log|V_n|\u03b1) as n \u2192 \u221e.\nThis verifies a conjecture of Benjamini up to the value of the constant a,\nwhich he suggested should be 1.\nWe also prove a generalization of this result to quasitransitive graph\nsequences with a bounded number of vertex orbits and prove that one may indeed\ntake a = 1 when the graphs G_n are all Cayley graphs of Abelian groups. A key\nstep in our proof is to adapt the methods of Duminil-Copin, Goswami, Raoufi,\nSevero, and Yadin from infinite graphs to finite graphs. This adaptation also\nleads to an isoperimetric criterion for infinite graphs to have a nontrivial\nuniqueness phase (i.e., to have p_u < 1) which is of independent interest. We\nalso prove that the set of possible values of the critical probability of an\ninfinite quasitransitive graph has a gap at 1 in the sense that for every\nk,n < \u221e there exists \u03f5 > 0 such that every infinite graph G of\ndegree at most k whose vertex set has at most n orbits under Aut(G)\neither has p_c = 1 or p_c \u2264 1 - \u03f5.",
"doi": "10.48550/arXiv.2104.05607",
"publisher": "arXiv",
"publication_date": "2021-09-27"
},
{
"id": "authors:b7n9w-z9n48",
"collection": "authors",
"collection_id": "b7n9w-z9n48",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-202119558",
"type": "monograph",
"title": "On the tail of the branching random walk local time",
"author": [
{
"family_name": "Angel",
"given_name": "Omer",
"orcid": "0000-0002-6451-8242",
"clpid": "Angel-Omer"
},
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
},
{
"family_name": "J\u00e1rai",
"given_name": "Antal A.",
"orcid": "0000-0003-3522-498X",
"clpid": "J\u00e1rai-Antal-A"
}
],
"abstract": "Consider a critical branching random walk on \u2124^d, d\u22651, started with a single particle at the origin, and let L(x) be the total number of particles that ever visit a vertex x. We study the tail of L(x) under suitable conditions on the offspring distribution. In particular, our results hold if the offspring distribution has an exponential moment.",
"doi": "10.48550/arXiv.2002.12188",
"publisher": "arXiv",
"publication_date": "2021-09-27"
},
{
"id": "authors:vw7fw-2em04",
"collection": "authors",
"collection_id": "vw7fw-2em04",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-202133236",
"type": "monograph",
"title": "Logarithmic corrections to scaling in the four-dimensional uniform spanning tree",
"author": [
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
},
{
"family_name": "Sousi",
"given_name": "Perla",
"clpid": "Sousi-Perla"
}
],
"abstract": "We compute the precise logarithmic corrections to mean-field scaling for various quantities describing the uniform spanning tree of the four-dimensional hypercubic lattice \u2124\u2074. We are particularly interested in the distribution of the past of the origin, that is, the finite piece of the tree that is separated from infinity by the origin. We prove that the probability that the past contains a path of length n is of order (log n)^(1/3)n\u207b\u00b9, that the probability that the past contains at least n vertices is of order (log n)^(1/6)n^(\u22121/2), and that the probability that the past reaches the boundary of the box [\u2212n,n]\u2074 is of order (log n)^(2/3+o)(1))n\u207b\u00b2. An important part of our proof is to prove concentration estimates for the capacity of the four-dimensional loop-erased random walk which may be of independent interest. \n\nOur results imply that the Abelian sandpile model also exhibits non-trivial polylogarithmic corrections to mean-field scaling in four dimensions, although it remains open to compute the precise order of these corrections.",
"doi": "10.48550/arXiv.2010.15830",
"publisher": "arXiv",
"publication_date": "2021-09-27"
},
{
"id": "authors:qstde-m2w06",
"collection": "authors",
"collection_id": "qstde-m2w06",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-202140311",
"type": "monograph",
"title": "The critical two-point function for long-range percolation on the hierarchical lattice",
"author": [
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
}
],
"abstract": "We prove up-to-constants bounds on the two-point function (i.e.,\npoint-to-point connection probabilities) for critical long-range percolation on\nthe d-dimensional hierarchical lattice. More precisely, we prove that if we\nconnect each pair of points x and y by an edge with probability\n1-exp(-\u03b2||x-y||^(-d-\u03b1)), where 0 < \u03b1 < d is fixed and \u03b2 \u2265 0 is a parameter, then the critical two-point function satisfies P_(\u03b2_c)(x \u2194 y)||x-y||^(-d+\u03b1) for\nevery pair of distinct points x and y. We deduce in particular that the\nmodel has mean-field critical behaviour when \u03b1 < d/3 and does not have\nmean-field critical behaviour when \u03b1 > d/3.",
"doi": "10.48550/arXiv.2103.17013",
"publisher": "arXiv",
"publication_date": "2021-09-27"
},
{
"id": "authors:k4eb1-c9n17",
"collection": "authors",
"collection_id": "k4eb1-c9n17",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-202112742",
"type": "monograph",
"title": "Non-intersection of transient branching random walks",
"author": [
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
}
],
"abstract": "Let G be a Cayley graph of a nonamenable group with spectral radius \u03c1<1. It is known that branching random walk on G with offspring distribution \u03bc is transient, i.e., visits the origin at most finitely often almost surely, if and only if the expected number of offspring \u03bc\u00af satisfies \u03bc\u00af\u2264\u03c1\u22121. Benjamini and M\u00fcller (2010) conjectured that throughout the transient supercritical phase 1<\u03bc\u00af\u2264\u03c1\u22121, and in particular at the recurrence threshold \u03bc\u00af=\u03c1\u22121, the trace of the branching random walk is tree-like in the sense that it is infinitely-ended almost surely on the event that the walk survives forever. This is essentially equivalent to the assertion that two independent copies of the branching random walk intersect at most finitely often almost surely. We prove this conjecture, along with several other related conjectures made by the same authors.\nA central contribution of this work is the introduction of the notion of local unimodularity, which we expect to have several further applications in the future.",
"doi": "10.48550/arXiv.1910.01018",
"publisher": "arXiv",
"publication_date": "2021-09-27"
},
{
"id": "authors:9ak9w-nca68",
"collection": "authors",
"collection_id": "9ak9w-nca68",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-202147400",
"type": "monograph",
"title": "On the derivation of mean-field percolation critical exponents from the triangle condition",
"author": [
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
}
],
"abstract": "We give a new derivation of mean-field percolation critical behaviour from the triangle condition that is quantitatively much better than previous proofs when the triangle diagram \u2207_(p_c) is large. In contrast to earlier methods, our approach continues to yield bounds of reasonable order when the triangle diagram \u2207^p is unbounded but diverges slowly as p \u2191 p_c, as is expected to occur in percolation on \u2124^d at the upper-critical dimension d=6. Indeed, we show in particular that if the triangle diagram diverges polylogarithmically as p\u2191pc then mean-field critical behaviour holds to within a polylogarithmic factor. We apply the methods we develop to deduce that for long-range percolation on the hierarchical lattice, mean-field critical behaviour holds to within polylogarithmic factors at the upper-critical dimension. \n\nAs part of the proof, we introduce a new method for comparing diagrammatic sums on general transitive graphs that may be of independent interest.",
"doi": "10.48550/arXiv.2106.06400v2",
"publisher": "arXiv",
"publication_date": "2021-06-11"
},
{
"id": "authors:1j2jy-xp413",
"collection": "authors",
"collection_id": "1j2jy-xp413",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210922-193306644",
"type": "article",
"title": "Supercritical percolation on nonamenable graphs: isoperimetry, analyticity, and exponential decay of the cluster size distribution",
"author": [
{
"family_name": "Hermon",
"given_name": "Jonathan",
"orcid": "0000-0002-2935-3999",
"clpid": "Hermon-Jonathan"
},
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
}
],
"abstract": "Let G be a connected, locally finite, transitive graph, and consider Bernoulli bond percolation on G. We prove that if G is nonamenable and p > p_c(G) then there exists a positive constant c_p such that \n\nP_p(n \u2264 |K| < \u221e) \u2264 e^(\u2212c_p)n) \n\nfor every n \u2265 1, where K is the cluster of the origin. We deduce the following two corollaries: \n\n1. Every infinite cluster in supercritical percolation on a transitive nonamenable graph has anchored expansion almost surely. This answers positively a question of Benjamini, Lyons, and Schramm (1997). \n\n2. For transitive nonamenable graphs, various observables including the percolation probability, the truncated susceptibility, and the truncated two-point function are analytic functions of p throughout the supercritical phase.",
"doi": "10.1007/s00222-020-01011-3",
"issn": "0020-9910",
"publisher": "Springer",
"publication": "Inventiones Mathematicae",
"publication_date": "2021-05",
"series_number": "2",
"volume": "224",
"issue": "2",
"pages": "445-486"
},
{
"id": "authors:1nt9d-yhc77",
"collection": "authors",
"collection_id": "1nt9d-yhc77",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210922-193307537",
"type": "article",
"title": "Large, lengthy graphs look locally like lines",
"author": [
{
"family_name": "Benjamini",
"given_name": "Itai",
"clpid": "Benjamini-Itai"
},
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
}
],
"abstract": "We apply the theory of unimodular random rooted graphs to study the metric geometry of large, finite, bounded degree graphs whose diameter is proportional to their volume. We prove that for a positive proportion of the vertices of such a graph, there exists a mesoscopic scale on which the graph looks like R in the sense that the rescaled ball is close to a line segment in the Gromov\u2013Hausdorff metric.",
"doi": "10.1112/blms.12436",
"issn": "1469-2120",
"publisher": "Wiley",
"publication": "Bulletin of the Lindon Mathematical Society",
"publication_date": "2021-04",
"series_number": "2",
"volume": "53",
"issue": "2",
"pages": "482-492"
},
{
"id": "authors:ty32r-s8w85",
"collection": "authors",
"collection_id": "ty32r-s8w85",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210922-193307624",
"type": "article",
"title": "Geometric and spectral properties of causal maps",
"author": [
{
"family_name": "Curien",
"given_name": "Nicolas",
"clpid": "Curien-Nicolas"
},
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
},
{
"family_name": "Nachmias",
"given_name": "Asaf",
"orcid": "0000-0002-4852-5645",
"clpid": "Nachmias-Asaf"
}
],
"abstract": "We study the random planar map obtained from a critical, finite variance, Galton\u2013Watson plane tree by adding the horizontal connections between successive vertices at each level. This random graph is closely related to the well-known causal dynamical triangulation that was introduced by Ambj\u00f8rn and Loll and has been studied extensively by physicists. We prove that the horizontal distances in the graph are smaller than the vertical distances, but only by a subpolynomial factor: The diameter of the set of vertices at level n is both o(n) and n^(1\u2212o(1)). This enables us to prove that the spectral dimension of the infinite version of the graph is almost surely equal to 2, and consequently the random walk is diffusive almost surely. We also initiate an investigation of the case in which the offspring distribution is critical and belongs to the domain of attraction of an \u03b1-stable law for \u03b1 \u2208 (1,2), for which our understanding is much less complete.",
"doi": "10.4171/jems/1001",
"issn": "1435-9855",
"publisher": "European Mathematical Society",
"publication": "Journal of the European Mathematical Society",
"publication_date": "2020-12",
"series_number": "12",
"volume": "22",
"issue": "12",
"pages": "3997-4024"
},
{
"id": "authors:ave15-h5f50",
"collection": "authors",
"collection_id": "ave15-h5f50",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-201102398",
"type": "article",
"title": "New critical exponent inequalities for percolation and the random cluster model",
"author": [
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
}
],
"abstract": "We apply a variation on the methods of Duminil-Copin, Raoufi, and Tassion (Ann. of Math. (2) 189:1 (2019), 75\u201399) to establish a new differential inequality applying to both Bernoulli percolation and the Fortuin\u2013Kasteleyn random cluster model. This differential inequality has a similar form to that derived for Bernoulli percolation by Menshikov (Dokl. Akad. Nauk 288:6 (1986), 1308\u20131311) but with the important difference that it describes the distribution of the volume of a cluster rather than of its radius. We apply this differential inequality to prove the following: \n\n1. The critical exponent inequalities \u03b3 \u2264 \u03b4 \u2212 1 and \u0394 \u2264 \u03b3 + 1 hold for percolation and the random cluster model on any transitive graph. These inequalities are new even in the context of Bernoulli percolation on Z^d, and are saturated in mean-field for Bernoulli percolation and for the random cluster model with q \u2208 [1,2). \n\n2. The volume of a cluster has an exponential tail in the entire subcritical phase of the random cluster model on any transitive graph. This proof also applies to infinite-range models, where the result is new even in the Euclidean setting.",
"doi": "10.2140/pmp.2020.1.147",
"issn": "2690-1005",
"publisher": "Mathematical Sciences Publishers",
"publication": "Probability and Mathematical Physics",
"publication_date": "2020-11-19",
"series_number": "1",
"volume": "1",
"issue": "1",
"pages": "147-165"
},
{
"id": "authors:ksvt4-kdg91",
"collection": "authors",
"collection_id": "ksvt4-kdg91",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-202136797",
"type": "monograph",
"title": "Transience and recurrence of sets for branching random walk via non-standard stochastic orders",
"author": [
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
}
],
"abstract": "We study how the recurrence and transience of space-time sets for a branching random walk on a graph depends on the offspring distribution. Here, we say that a space-time set A is recurrent if it is visited infinitely often almost surely on the event that the branching random walk survives forever, and say that A is transient if it is visited at most finitely often almost surely. We prove that if \u03bc and \u03bd are supercritical offspring distributions with means \u03bc\u00af<\u03bd\u00af then every space-time set that is recurrent with respect to the offspring distribution \u03bc is also recurrent with respect to the offspring distribution \u03bd and similarly that every space-time set that is transient with respect to the offspring distribution \u03bd is also transient with respect to the offspring distribution \u03bc. To prove this, we introduce a new order on probability measures that we call the germ order and prove more generally that the same result holds whenever \u03bc is smaller than \u03bd in the germ order. Our work is inspired by the work of Johnson and Junge (AIHP 2018), who used related stochastic orders to study the frog model.",
"doi": "10.48550/arXiv.2011.06402",
"publisher": "arXiv",
"publication_date": "2020-11-12"
},
{
"id": "authors:s777e-ew214",
"collection": "authors",
"collection_id": "s777e-ew214",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210922-193307690",
"type": "article",
"title": "The L\u00b2 boundedness condition in nonamenable percolation",
"author": [
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
}
],
"abstract": "Let G = (V,E) be a connected, locally finite, transitive graph, and consider Bernoulli bond percolation on G. In recent work, we conjectured that if G is nonamenable then the matrix of critical connection probabilities T_(p_c) (u,v) = \u2119_(p_c) (u\u2194v) is bounded as an operator T_(p_c) : L\u00b2(V)\u2192L\u00b2(V) and proved that this conjecture holds for several classes of graphs. We also noted in that work that the conjecture implies two older conjectures, namely that percolation on transitive nonamenable graphs always has a nontrivial nonuniqueness phase, and that critical percolation on the same class of graphs has mean-field critical behaviour. \n\nIn this paper we further investigate the consequences of the L\u00b2 boundedness conjecture. In particular, we prove that the following hold for all transitive graphs: i) The two-point function decays exponentially in the distance for all p < p_(2\u21922); ii) If p_c < p_(2\u21922), then the critical exponent governing the extrinsic diameter of a critical cluster is 1; iii) Below p_(2\u21922), percolation is \"ballistic\" in the sense that the intrinsic distance between two points is exponentially unlikely to be much larger than their extrinsic distance; iv) If p_c < p_(2\u21922), then \u2016T_(p_c) \u2016_(q\u2192q) \u224d (q\u22121)\u22121 and p_(q\u2192q) \u2212 p_c \u224d q \u2212 1 as q\u21931. v) If p_c < p_(2\u21922), then various 'multiple-arm' events have probabilities comparable to the upper bound given by the BK inequality. In particular, the probability that the origin is a trifurcation point is of order (p \u2212 p_c)\u00b3 as p \u2193 p_c. All of these results are new even in the Gromov hyperbolic case. \n\nFinally, we apply these results together with duality arguments to compute the critical exponents governing the geometry of intrinsic geodesics at the uniqueness threshold of percolation in the hyperbolic plane.",
"doi": "10.1214/20-ejp525",
"issn": "1083-6489",
"publisher": "Electronic Journal of Probability",
"publication": "Electronic Journal of Probability",
"publication_date": "2020-10-16",
"volume": "25",
"pages": "Art. No. 127"
},
{
"id": "authors:m8n9v-j1t44",
"collection": "authors",
"collection_id": "m8n9v-j1t44",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210922-193307881",
"type": "article",
"title": "Non-intersection of transient branching random walks",
"author": [
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
}
],
"abstract": "Let G be a Cayley graph of a nonamenable group with spectral radius \u03c1 < 1. It is known that branching random walk on G with offspring distribution \u03bc is transient, i.e., visits the origin at most finitely often almost surely, if and only if the expected number of offspring \u03bc[bar] satisfies \u03bc[bar] \u2264 \u03c1 \u2212 1. Benjamini and M\u00fcller (2010) conjectured that throughout the transient supercritical phase 1< \u03bc[bar] \u2264 \u03c1 \u2212 1, and in particular at the recurrence threshold \u03bc[bar] = \u03c1 \u2212 1, the trace of the branching random walk is tree-like in the sense that it is infinitely-ended almost surely on the event that the walk survives forever. This is essentially equivalent to the assertion that two independent copies of the branching random walk intersect at most finitely often almost surely. We prove this conjecture, along with several other related conjectures made by the same authors. \n\nA central contribution of this work is the introduction of the notion of local unimodularity, which we expect to have several further applications in the future.",
"doi": "10.1007/s00440-020-00964-z",
"issn": "0178-8051",
"publisher": "Springer",
"publication": "Probability Theory and Related Fields",
"publication_date": "2020-10",
"series_number": "1-2",
"volume": "178",
"issue": "1-2",
"pages": "1-23"
},
{
"id": "authors:0an9f-d4t30",
"collection": "authors",
"collection_id": "0an9f-d4t30",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210922-193307816",
"type": "article",
"title": "Anomalous diffusion of random walk on random planar maps",
"author": [
{
"family_name": "Gwynne",
"given_name": "Ewain",
"clpid": "Gwynne-Ewain"
},
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
}
],
"abstract": "We prove that the simple random walk on the uniform infinite planar triangulation (UIPT) typically travels graph distance at most n^(1/4 + o_n(1)) in n units of time. Together with the complementary lower bound proven by Gwynne and Miller (2017) this shows that the typical graph distance displacement of the walk after n steps is n^(1/4 + o_n(1)), as conjectured by Benjamini and Curien (2013). More generally, we show that the simple random walks on a certain family of random planar maps in the \u03b3-Liouville quantum gravity (LQG) universality class for \u03b3\u2208(0,2)---including spanning tree-weighted maps, bipolar-oriented maps, and mated-CRT maps---typically travels graph distance n^(1/d_\u03b3 + o_n(1)) in n units of time, where d\u03b3 is the growth exponent for the volume of a metric ball on the map, which was shown to exist and depend only on \u03b3 by Ding and Gwynne (2018). Since d_\u03b3 > 2, this shows that the simple random walk on each of these maps is subdiffusive. \n\nOur proofs are based on an embedding of the random planar maps under consideration into C wherein graph distance balls can be compared to Euclidean balls modulo subpolynomial errors. This embedding arises from a coupling of the given random planar map with a mated-CRT map together with the relationship of the latter map to SLE-decorated LQG.",
"doi": "10.1007/s00440-020-00986-7",
"issn": "0178-8051",
"publisher": "Springer",
"publication": "Probability Theory and Related Fields",
"publication_date": "2020-10",
"series_number": "1-2",
"volume": "178",
"issue": "1-2",
"pages": "567-611"
},
{
"id": "authors:w8j7r-3fk10",
"collection": "authors",
"collection_id": "w8j7r-3fk10",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210922-193307757",
"type": "article",
"title": "Nonuniqueness and mean-field criticality for percolation on nonunimodular transitive graphs",
"author": [
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
}
],
"abstract": "We study Bernoulli bond percolation on nonunimodular quasi-transitive graphs, and more generally graphs whose automorphism group has a nonunimodular quasi-transitive subgroup. We prove that percolation on any such graph has a nonempty phase in which there are infinite light clusters, which implies the existence of a nonempty phase in which there are infinitely many infinite clusters. That is, we show that p_c < p_h < P_u for any such graph. This answers a question of H\u00e4ggstr\u00f6m, Peres, and Schonmann (1999), and verifies the nonunimodular case of a well-known conjecture of Benjamini and Schramm (1996). We also prove that the triangle condition holds at criticality on any such graph, which implies that various critical exponents exist and take their mean-field values. \n\n\nAll our results apply, for example, to the product T_k x Z^d of a k-regular tree with Z^d for k \u2265 3 and d \u2265 1, for which these results were previously known only for large k. Furthermore, our methods also enable us to establish the basic topological features of the phase diagram for anisotropic percolation on such products, in which tree edges and Z^d edges are given different retention probabilities. These features had only previously been established for d = 1, k large.",
"doi": "10.1090/jams/953",
"issn": "0894-0347",
"publisher": "American Mathematical Society",
"publication": "Journal of the American Mathematical Society",
"publication_date": "2020-10",
"series_number": "4",
"volume": "33",
"issue": "4",
"pages": "1101-1165"
},
{
"id": "authors:ba2tq-frh31",
"collection": "authors",
"collection_id": "ba2tq-frh31",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210922-193307950",
"type": "article",
"title": "Kazhdan groups have cost 1",
"author": [
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
},
{
"family_name": "Pete",
"given_name": "G\u00e1bor",
"clpid": "Pete-G\u00e1bor"
}
],
"abstract": "We prove that every countably infinite group with Kazhdan's property (T) has cost 1, answering a well-known question of Gaboriau. It remains open if they have fixed price 1.",
"doi": "10.1007/s00222-020-00967-6",
"issn": "0020-9910",
"publisher": "Springer",
"publication": "Inventiones Mathematicae",
"publication_date": "2020-09",
"series_number": "3",
"volume": "221",
"issue": "3",
"pages": "873-891"
},
{
"id": "authors:2xfrr-8xy91",
"collection": "authors",
"collection_id": "2xfrr-8xy91",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-202122977",
"type": "monograph",
"title": "Continuity of the Ising phase transition on nonamenable groups",
"author": [
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
}
],
"abstract": "We prove rigorously that the ferromagnetic Ising model on any nonamenable Cayley graph undergoes a continuous (second-order) phase transition in the sense that there is a unique Gibbs measure at the critical temperature. The proof of this theorem is quantitative and also yields power-law bounds on the magnetization at and near criticality. Indeed, we prove more generally that the magnetization \u27e8\u03c3o\u27e9+\u03b2,h is a locally H\u00f6lder-continuous function of the inverse temperature \u03b2 and external field h throughout the non-negative quadrant (\u03b2,h)\u2208[0,\u221e)2. As a second application of the methods we develop, we also prove that the free energy of Bernoulli percolation is twice differentiable at pc on any transitive nonamenable graph.",
"doi": "10.48550/arXiv.2007.15625",
"publisher": "arXiv",
"publication_date": "2020-07-30"
},
{
"id": "authors:d2qn7-4t233",
"collection": "authors",
"collection_id": "d2qn7-4t233",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210922-193308018",
"type": "article",
"title": "Locality of the critical probability for transitive graphs of exponential growth",
"author": [
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
}
],
"abstract": "Around 2008, Schramm conjectured that the critical probabilities for Bernoulli bond percolation satisfy the following continuity property: If (G_n)_(n \u2265 1) is a sequence of transitive graphs converging locally to a transitive graph G and limsup_(n \u2192 \u221e)p_c(G_n) < 1, then p_c(G_n) \u2192 p_c(G) as n \u2192 \u221e. We verify this conjecture under the additional hypothesis that there is a uniform exponential lower bound on the volume growth of the graphs in question. The result is new even in the case that the sequence of graphs is uniformly nonamenable. \n\nIn the unimodular case, this result is obtained as a corollary to the following theorem of independent interest: For every g > 1 and M < \u221e, there exist positive constants C = C(g,M) and \u03b4 = \u03b4(g,M) such that if G is a transitive unimodular graph with degree at most M and growth gr(G):= inf_(r \u2265 1)|B(o,r)|^(1/r) \u2265 g, then P_(p_c)(|K_o| \u2265 n) \u2264 C_n^(\u2212\u03b4) for every n \u2265 1, where K_o is the cluster of the root vertex o. The proof of this inequality makes use of new universal bounds on the probabilities of certain two-arm events, which hold for every unimodular transitive graph.",
"doi": "10.1214/19-AOP1395",
"issn": "0091-1798",
"publisher": "Institute of Mathematical Statistics",
"publication": "Annals of Probability",
"publication_date": "2020-05",
"series_number": "3",
"volume": "48",
"issue": "3",
"pages": "1352-1371"
},
{
"id": "authors:cqd0v-w8754",
"collection": "authors",
"collection_id": "cqd0v-w8754",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210922-193308155",
"type": "article",
"title": "Indistinguishability of collections of trees in the uniform spanning forest",
"author": [
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
}
],
"abstract": "We prove the following indistinguishability theorem for k-tuples of trees in the uniform spanning forest of Z^d: Suppose that A is a property of a k-tuple of components that is stable under finite modifications of the forest. Then either every k-tuple of distinct trees has property A almost surely, or no k-tuple of distinct trees has property A almost surely. This generalizes the indistinguishability theorem of the author and Nachmias (2016), which applied to individual trees. Our results apply more generally to any graph that has the Liouville property and for which every component of the USF is one-ended.",
"doi": "10.1214/19-AIHP988",
"issn": "0246-0203",
"publisher": "Institut Henri Poincar\u00e9",
"publication": "Annales de l'Institut Henri Poincar\u00e9, Probabilit\u00e9s et Statistiques",
"publication_date": "2020-05",
"series_number": "2",
"volume": "56",
"issue": "2",
"pages": "917-927"
},
{
"id": "authors:7dw4y-4c316",
"collection": "authors",
"collection_id": "7dw4y-4c316",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210923-213625943",
"type": "article",
"title": "Universality of high-dimensional spanning forests and sandpiles",
"author": [
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
}
],
"abstract": "We prove that the wired uniform spanning forest exhibits mean-field behaviour on a very large class of graphs, including every transitive graph of at least quintic volume growth and every bounded degree nonamenable graph. Several of our results are new even in the case of Z^d, d \u2265 5. In particular, we prove that every tree in the forest has spectral dimension 4/3 and walk dimension 3 almost surely, and that the critical exponents governing the intrinsic diameter and volume of the past of a vertex in the forest are 1 and 1/2 respectively. (The past of a vertex in the uniform spanning forest is the union of the vertex and the finite components that are disconnected from infinity when that vertex is deleted from the forest.) We obtain as a corollary that the critical exponent governing the extrinsic diameter of the past is 2 on any transitive graph of at least five dimensional polynomial growth, and is 1 on any bounded degree nonamenable graph. We deduce that the critical exponents describing the diameter and total number of topplings in an avalanche in the Abelian sandpile model are 2 and 1/2 respectively for any transitive graph with polynomial growth of dimension at least five, and are 1 and 1/2 respectively for any bounded degree nonamenable graph. In the case of Z^d, d \u2265 5, some of our results regarding critical exponents recover earlier results of Bhupatiraju et al. (Electron J Probab 22(85):51, 2017). In this case, we improve upon their results by showing that the tail probabilities in question are described by the appropriate power laws to within constant-order multiplicative errors, rather than the polylogarithmic-order multiplicative errors present in that work.",
"doi": "10.1007/s00440-019-00923-3",
"issn": "0178-8051",
"publisher": "Springer",
"publication": "Probability Theory and Related Fields",
"publication_date": "2020-02",
"series_number": "1-2",
"volume": "176",
"issue": "1-2",
"pages": "533-597"
},
{
"id": "authors:reagx-64650",
"collection": "authors",
"collection_id": "reagx-64650",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210922-193308087",
"type": "article",
"title": "Mallows permutations and finite dependence",
"author": [
{
"family_name": "Holroyd",
"given_name": "Alexander E.",
"clpid": "Holroyd-Alexander-E"
},
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
},
{
"family_name": "Levy",
"given_name": "Avi",
"clpid": "Levy-Avi"
}
],
"abstract": "We use the Mallows permutation model to construct a new family of stationary finitely dependent proper colorings of the integers. We prove that these colorings can be expressed as finitary factors of i.i.d. processes with finite mean coding radii. They are the first colorings known to have these properties. Moreover, we prove that the coding radii have exponential tails, and that the colorings can also be expressed as functions of countable-state Markov chains. We deduce analogous existence statements concerning shifts of finite type and higher-dimensional colorings.",
"doi": "10.1214/19-AOP1363",
"issn": "0091-1798",
"publisher": "Institute of Mathematical Statistics",
"publication": "Annals of Probability",
"publication_date": "2020-01",
"series_number": "1",
"volume": "48",
"issue": "1",
"pages": "343-379"
},
{
"id": "authors:mcd2a-8fa05",
"collection": "authors",
"collection_id": "mcd2a-8fa05",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210922-193309372",
"type": "article",
"title": "The component graph of the uniform spanning forest: transitions in dimensions 9,10,11, ...",
"author": [
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
},
{
"family_name": "Peres",
"given_name": "Yuval",
"orcid": "0000-0001-5456-6323",
"clpid": "Peres-Yuval"
}
],
"abstract": "We prove that the uniform spanning forests of Z^d and Z^\u2113 have qualitatively different connectivity properties whenever \u2113 > d \u2265 4. In particular, we consider the graph formed by contracting each tree of the uniform spanning forest down to a single vertex, which we call the component graph. We introduce the notion of ubiquitous subgraphs and show that the set of ubiquitous subgraphs of the component graph changes whenever the dimension changes and is above 8. To separate dimensions 5, 6, 7, and 8, we prove a similar result concerning ubiquitous subhypergraphs in the component hypergraph. Our result sharpens a theorem of Benjamini, Kesten, Peres, and Schramm, who proved that the diameter of the component graph increases by one every time the dimension increases by four.",
"doi": "10.1007/s00440-018-0884-3",
"issn": "0178-8051",
"publisher": "Springer",
"publication": "Probability Theory and Related Fields",
"publication_date": "2019-10",
"series_number": "1-2",
"volume": "175",
"issue": "1-2",
"pages": "141-208"
},
{
"id": "authors:4264h-6jw70",
"collection": "authors",
"collection_id": "4264h-6jw70",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210922-193309305",
"type": "article",
"title": "Uniform spanning forests of planar graphs",
"author": [
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
},
{
"family_name": "Nachmias",
"given_name": "Asaf",
"orcid": "0000-0002-4852-5645",
"clpid": "Nachmias-Asaf"
}
],
"abstract": "We prove that the free uniform spanning forest of any bounded degree proper plane graph is connected almost surely, answering a question of Benjamini, Lyons, Peres and Schramm. We provide a quantitative form of this result, calculating the critical exponents governing the geometry of the uniform spanning forests of transient proper plane graphs with bounded degrees and codegrees. We find that the same exponents hold universally over this entire class of graphs provided that measurements are made using the hyperbolic geometry of their circle packings rather than their usual combinatorial geometry.",
"doi": "10.1017/fms.2019.14",
"issn": "2050-5094",
"publisher": "Cambridge University Press",
"publication": "Forum Mathetmatics, Sigma",
"publication_date": "2019-09-13",
"volume": "7",
"pages": "Art. No. e29"
},
{
"id": "authors:m6mec-31e56",
"collection": "authors",
"collection_id": "m6mec-31e56",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210922-193309236",
"type": "article",
"title": "Self-avoiding walk on nonunimodular transitive graphs",
"author": [
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
}
],
"abstract": "We study self-avoiding walk on graphs whose automorphism group has a transitive nonunimodular subgroup. We prove that self-avoiding walk is ballistic, that the bubble diagram converges at criticality, and that the critical two-point function decays exponentially in the distance from the origin. This implies that the critical exponent governing the susceptibility takes its mean-field value, and hence that the number of self-avoiding walks of length n is comparable to the nth power of the connective constant. We also prove that the same results hold for a large class of repulsive walk models with a self-intersection based interaction, including the weakly self-avoiding walk. All of these results apply in particular to the product T_k \u00d7 Z^d of a k-regular tree (k \u2265 3) with Z^d, for which these results were previously only known for large k.",
"doi": "10.1214/18-AOP1322",
"issn": "0091-1798",
"publisher": "Institute of Mathematical Statistics",
"publication": "Annals of Probability",
"publication_date": "2019-09",
"series_number": "5",
"volume": "47",
"issue": "5",
"pages": "2801-2829"
},
{
"id": "authors:exzam-6kr78",
"collection": "authors",
"collection_id": "exzam-6kr78",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210922-193309440",
"type": "article",
"title": "Percolation on hyperbolic graphs",
"author": [
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
}
],
"abstract": "We prove that Bernoulli bond percolation on any nonamenable, Gromov hyperbolic, quasi-transitive graph has a phase in which there are infinitely many infinite clusters, verifying a well-known conjecture of Benjamini and Schramm (1996) under the additional assumption of hyperbolicity. In other words, we show that p_c < p_u for any such graph. Our proof also yields that the triangle condition \u2207_p-c < \u221e holds at criticality on any such graph, which is known to imply that several critical exponents exist and take their mean-field values. This gives the first family of examples of one-ended groups all of whose Cayley graphs are proven to have mean-field critical exponents for percolation.",
"doi": "10.1007/s00039-019-00498-0",
"issn": "1016-443X",
"publisher": "Springer",
"publication": "Geometric and Functional Analysis",
"publication_date": "2019-06",
"series_number": "3",
"volume": "29",
"issue": "3",
"pages": "766-810"
},
{
"id": "authors:x43w2-e4p85",
"collection": "authors",
"collection_id": "x43w2-e4p85",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210922-193309508",
"type": "article",
"title": "Statistical physics on a product of trees",
"author": [
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
}
],
"abstract": "Let G be the product of finitely many trees T\u2081 \u00d7 T\u2082 \u00d7 \u22ef \u00d7 T_N, each of which is regular with degree at least three. We consider Bernoulli bond percolation and the Ising model on this graph, giving a short proof that the model undergoes a second order phase transition with mean-field critical exponents in each case. The result concerning percolation recovers a result of Kozma (2013), while the result concerning the Ising model is new. \n\nWe also present a new proof, using similar techniques, of a lemma of Schramm concerning the decay of the critical two-point function along a random walk, as well as some generalizations of this lemma.",
"doi": "10.1214/18-aihp906",
"issn": "0246-0203",
"publisher": "Institut Henri Poincar\u00e9",
"publication": "Annales de l'Institut Henri Poincar\u00e9, Probabilit\u00e9s et Statistiques",
"publication_date": "2019-05",
"series_number": "2",
"volume": "55",
"issue": "2",
"pages": "1001-1010"
},
{
"id": "authors:228dv-bnt78",
"collection": "authors",
"collection_id": "228dv-bnt78",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210922-193309575",
"type": "article",
"title": "Harmonic Dirichlet functions on planar graphs",
"author": [
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
}
],
"abstract": "Benjamini and Schramm (Invent Math 126(3):565\u2013587, 1996) used circle packing to prove that every transient, bounded degree planar graph admits non-constant harmonic functions of finite Dirichlet energy. We refine their result, showing in particular that for every transient, bounded degree, simple planar triangulation T and every circle packing of T in a domain D, there is a canonical, explicit bounded linear isomorphism between the space of harmonic Dirichlet functions on T and the space of harmonic Dirichlet functions on D.",
"doi": "10.1007/s00454-019-00057-2",
"issn": "0179-5376",
"publisher": "Springer",
"publication": "Discrete & Computational Geometry",
"publication_date": "2019-04",
"series_number": "3",
"volume": "61",
"issue": "3",
"pages": "479-506"
},
{
"id": "authors:40548-8c694",
"collection": "authors",
"collection_id": "40548-8c694",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210922-193309713",
"type": "article",
"title": "Finitely dependent cycle coloring",
"author": [
{
"family_name": "Holroyd",
"given_name": "Alexander E.",
"clpid": "Holroyd-Alexander-E"
},
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
},
{
"family_name": "Levy",
"given_name": "Avi",
"clpid": "Levy-Avi"
}
],
"abstract": "We construct stationary finitely dependent colorings of the cycle which are analogous to the colorings of the integers recently constructed by Holroyd and Liggett. These colorings can be described by a simple necklace insertion procedure, and also in terms of an Eden growth model on a tree. Using these descriptions we obtain simpler and more direct proofs of the characterizations of the 1- and 2-color marginals.",
"doi": "10.1214/18-ECP118",
"issn": "1083-589X",
"publisher": "Institute of Mathematical Statistics and Bernoulli Society",
"publication": "Electronic Communications in Probability",
"publication_date": "2018-09-15",
"volume": "23",
"pages": "Art. No. 64"
},
{
"id": "authors:xf9ak-cr102",
"collection": "authors",
"collection_id": "xf9ak-cr102",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210923-184021815",
"type": "article",
"title": "Coalescing random walk on unimodular graphs",
"author": [
{
"family_name": "Foxall",
"given_name": "Eric",
"clpid": "Foxall-Eric"
},
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
},
{
"family_name": "Junge",
"given_name": "Matthew",
"clpid": "Junge-Matthew"
}
],
"abstract": "Coalescing random walk on a unimodular random rooted graph for which the root has finite expected degree visits each site infinitely often almost surely. A corollary is that an opinion in the voter model on such graphs has infinite expected lifetime. Additionally, we deduce an adaptation of our main theorem that holds uniformly for coalescing random walk on finite random unimodular graphs with degree distribution stochastically dominated by a probability measure with finite mean.",
"doi": "10.1214/18-ecp136",
"issn": "1083-589X",
"publisher": "Institute of Mathematical Statistics",
"publication": "Electronic Communications in Probability",
"publication_date": "2018-09-13",
"volume": "23",
"pages": "Art. No. 62"
},
{
"id": "authors:21ast-60592",
"collection": "authors",
"collection_id": "21ast-60592",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210923-184021127",
"type": "article",
"title": "Hyperbolic and Parabolic Unimodular Random Maps",
"author": [
{
"family_name": "Angel",
"given_name": "Omer",
"orcid": "0000-0002-6451-8242",
"clpid": "Angel-Omer"
},
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
},
{
"family_name": "Nachmias",
"given_name": "Asaf",
"orcid": "0000-0002-4852-5645",
"clpid": "Nachmias-Asaf"
},
{
"family_name": "Ray",
"given_name": "Gourab",
"clpid": "Ray-Gourab"
}
],
"abstract": "We show that for infinite planar unimodular random rooted maps. many global geometric and probabilistic properties are equivalent, and are determined by a natural, local notion of average curvature. This dichotomy includes properties relating to amenability, conformal geometry, random walks, uniform and minimal spanning forests, and Bernoulli bond percolation. We also prove that every simply connected unimodular random rooted map is sofic, that is, a Benjamini\u2013Schramm limit of finite maps.",
"doi": "10.1007/s00039-018-0446-y",
"issn": "1016-443X",
"publisher": "Springer",
"publication": "Geometric and Functional Analysis",
"publication_date": "2018-07",
"series_number": "4",
"volume": "28",
"issue": "4",
"pages": "879-942"
},
{
"id": "authors:ak4z3-yey29",
"collection": "authors",
"collection_id": "ak4z3-yey29",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-190634891",
"type": "article",
"title": "Interlacements and the wired uniform spanning forest",
"author": [
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
}
],
"abstract": "We extend the Aldous\u2013Broder algorithm to generate the wired uniform spanning forests (WUSFs) of infinite, transient graphs. We do this by replacing the simple random walk in the classical algorithm with Sznitman's random interlacement process. We then apply this algorithm to study the WUSF, showing that every component of the WUSF is one-ended almost surely in any graph satisfying a certain weak anchored isoperimetric condition, that the number of 'excessive ends' in the WUSF is nonrandom in any graph, and also that every component of the WUSF is one-ended almost surely in any transient unimodular random rooted graph. The first two of these results answer positively two questions of Lyons, Morris and Schramm [Electron. J. Probab. 13 (2008) 1702\u20131725], while the third extends a recent result of the author. \n\nFinally, we construct a counterexample showing that almost sure one-endedness of WUSF components is not preserved by rough isometries of the underlying graph, answering negatively a further question of Lyons, Morris and Schramm.",
"doi": "10.1214/17-aop1203",
"issn": "0091-1798",
"publisher": "Institute of Mathematical Statistics",
"publication": "Annals of Probability",
"publication_date": "2018-03",
"series_number": "2",
"volume": "46",
"issue": "2",
"pages": "1170-1200"
},
{
"id": "authors:9c6re-9k844",
"collection": "authors",
"collection_id": "9c6re-9k844",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-184806499",
"type": "article",
"title": "The Hammersley-Welsh bound for self-avoiding walk revisited",
"author": [
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
}
],
"abstract": "The Hammersley-Welsh bound (Quart. J. Math., 1962) states that the number c_n of length n self-avoiding walks on Z^d satisfies \n\nc_n \u2264 exp[O(n^(1/2))]\u03bc^n_c, \n\nwhere \u03bc_c = \u03bc_c(d) is the connective constant of Z^d. While stronger estimates have subsequently been proven for d \u2265 3, for d = 2 this has remained the best rigorous, unconditional bound available. In this note, we give a new, simplified proof of this bound, which does not rely on the combinatorial analysis of unfolding. We also prove a small, non-quantitative improvement to the bound, namely \n\nc_n \u2264 exp[o^(n^(1/2))] \u03bc^n_c. \n\nThe improved bound is obtained as a corollary to the sub-ballisticity theorem of Duminil-Copin and Hammond (Commun. Math. Phys., 2013). We also show that any quantitative form of that theorem would yield a corresponding quantitative improvement to the Hammersley-Welsh bound.",
"doi": "10.1214/17-ECP94",
"issn": "1083-589X",
"publisher": "Institute of Mathematical Statistics",
"publication": "Electronic Communications in Probability",
"publication_date": "2018-02",
"volume": "23",
"pages": "Art. No. 5"
},
{
"id": "authors:5d27n-a1k66",
"collection": "authors",
"collection_id": "5d27n-a1k66",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210922-193309642",
"type": "book_section",
"title": "Counterexamples for percolation on unimodular random graphs",
"book_title": "Unimodularity in randomly generated graphs",
"author": [
{
"family_name": "Angel",
"given_name": "Omer",
"orcid": "0000-0002-6451-8242",
"clpid": "Angel-Omer"
},
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
}
],
"contributor": [
{
"family_name": "Sobieczky",
"given_name": "Florian",
"orcid": "0000-0001-5228-0153",
"clpid": "Sobieczky-Florian"
}
],
"abstract": "We construct an example of a bounded degree, nonamenable, unimodular random rooted graph with p_c = p_u for Bernoulli bond percolation, as well as an example of a bounded degree, unimodular random rooted graph with p_c < 1 but with an infinite cluster at criticality. These examples show that two well-known conjectures of Benjamini and Schramm are false when generalised from transitive graphs to unimodular random rooted graphs.",
"doi": "10.1090/conm/719/14465",
"isbn": "978-1-4704-3914-9",
"publisher": "American Mathematical Society",
"place_of_publication": "Providence, RI",
"publication_date": "2018",
"pages": "11-28"
},
{
"id": "authors:ds0pp-v5489",
"collection": "authors",
"collection_id": "ds0pp-v5489",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-183504452",
"type": "article",
"title": "Boundaries of planar graphs: a unified approach",
"author": [
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
},
{
"family_name": "Peres",
"given_name": "Yuval",
"orcid": "0000-0001-5456-6323",
"clpid": "Peres-Yuval"
}
],
"abstract": "We give a new proof that the Poisson boundary of a planar graph coincides with the boundary of its square tiling and with the boundary of its circle packing, originally proven by Georgakopoulos [9] and Angel, Barlow, Gurel-Gurevich and Nachmias [2] respectively. Our proof is robust, and also allows us to identify the Poisson boundaries of graphs that are rough-isometric to planar graphs. \n\nWe also prove that the boundary of the square tiling of a bounded degree plane triangulation coincides with its Martin boundary. This is done by comparing the square tiling of the triangulation with its circle packing.",
"doi": "10.1214/17-ejp116",
"issn": "1083-6489",
"publisher": "Institute of Mathematical Statistics",
"publication": "Electronic Journal of Probability",
"publication_date": "2017-10",
"volume": "22",
"pages": "Art. No. 100"
},
{
"id": "authors:9vs1v-3e586",
"collection": "authors",
"collection_id": "9vs1v-3e586",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210923-225150172",
"type": "article",
"title": "Indistinguishability of trees in uniform spanning forests",
"author": [
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
},
{
"family_name": "Nachmias",
"given_name": "Asaf",
"orcid": "0000-0002-4852-5645",
"clpid": "Nachmias-Asaf"
}
],
"abstract": "We prove that in both the free and the wired uniform spanning forest (FUSF and WUSF) of any unimodular random rooted network (in particular, of any Cayley graph), it is impossible to distinguish the connected components of the forest from each other by invariantly defined graph properties almost surely. This confirms a conjecture of Benjamini et al. (Ann Probab 29(1):1\u201365, 2001). We also answer positively two additional questions of Benjamini et al. (Ann Probab 29(1):1\u201365, 2001) under the assumption of unimodularity. We prove that on any unimodular random rooted network, the FUSF is either connected or has infinitely many connected components almost surely, and, if the FUSF and WUSF are distinct, then every component of the FUSF is transient and infinitely-ended almost surely. All of these results are new even for Cayley graphs.",
"doi": "10.1007/s00440-016-0707-3",
"issn": "0178-8051",
"publisher": "Springer",
"publication": "Probability Theory and Related Fields",
"publication_date": "2017-06",
"series_number": "1-2",
"volume": "168",
"issue": "1-2",
"pages": "113-152"
},
{
"id": "authors:3eaf4-f6079",
"collection": "authors",
"collection_id": "3eaf4-f6079",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-190634976",
"type": "article",
"title": "Wired cycle-breaking dynamics for uniform spanning forests",
"author": [
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
}
],
"abstract": "We prove that every component of the wired uniform spanning forest (WUSF) is one-ended almost surely in every transient reversible random graph, removing the bounded degree hypothesis required by earlier results. We deduce that every component of the WUSF is one-ended almost surely in every supercritical Galton\u2013Watson tree, answering a question of Benjamini, Lyons, Peres and Schramm [Ann. Probab. 29 (2001) 1\u201365]. \n\nOur proof introduces and exploits a family of Markov chains under which the oriented WUSF is stationary, which we call the wired cycle-breaking dynamics.",
"doi": "10.1214/15-AOP1063",
"issn": "0091-1798",
"publisher": "Institute of Mathematical Statistics",
"publication": "Annals of Probability",
"publication_date": "2016-11",
"series_number": "6",
"volume": "44",
"issue": "6",
"pages": "3879-3892"
},
{
"id": "authors:a1rdy-13148",
"collection": "authors",
"collection_id": "a1rdy-13148",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210923-184021534",
"type": "article",
"title": "Unimodular hyperbolic triangulations: circle packing and random walk",
"author": [
{
"family_name": "Angel",
"given_name": "Omer",
"orcid": "0000-0002-6451-8242",
"clpid": "Angel-Omer"
},
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
},
{
"family_name": "Nachmias",
"given_name": "Asaf",
"orcid": "0000-0002-4852-5645",
"clpid": "Nachmias-Asaf"
},
{
"family_name": "Ray",
"given_name": "Gourab",
"clpid": "Ray-Gourab"
}
],
"abstract": "We show that the circle packing type of a unimodular random plane triangulation is parabolic if and only if the expected degree of the root is six, if and only if the triangulation is amenable in the sense of Aldous and Lyons [1]. As a part of this, we obtain an alternative proof of the Benjamini\u2013Schramm Recurrence Theorem [19]. Secondly, in the hyperbolic case, we prove that the random walk almost surely converges to a point in the unit circle, that the law of this limiting point has full support and no atoms, and that the unit circle is a realisation of the Poisson boundary. Finally, we show that the simple random walk has positive speed in the hyperbolic metric.",
"doi": "10.1007/s00222-016-0653-9",
"issn": "0020-9910",
"publisher": "Springer",
"publication": "Inventiones Mathematicae",
"publication_date": "2016-10",
"series_number": "1",
"volume": "206",
"issue": "1",
"pages": "229-268"
},
{
"id": "authors:qy3jd-0m146",
"collection": "authors",
"collection_id": "qy3jd-0m146",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-190635043",
"type": "article",
"title": "Critical percolation on any quasi-transitive graph of exponential growth has no infinite clusters",
"author": [
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
}
],
"abstract": "We prove that critical percolation on any quasi-transitive graph of exponential volume growth does not have a unique infinite cluster. This allows us to deduce from earlier results that critical percolation on any graph in this class does not have any infinite clusters. The result is new when the graph in question is either amenable or nonunimodular.",
"doi": "10.1016/j.crma.2016.07.013",
"issn": "1631-073X",
"publisher": "Elsevier",
"publication": "Comptes Rendus Mathematique",
"publication_date": "2016-09",
"series_number": "9",
"volume": "354",
"issue": "9",
"pages": "944-947"
},
{
"id": "authors:rwyak-pyf27",
"collection": "authors",
"collection_id": "rwyak-pyf27",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210923-215831853",
"type": "article",
"title": "Collisions of random walks in reversible random graphs",
"author": [
{
"family_name": "Hutchcroft",
"given_name": "Tom",
"orcid": "0000-0003-0061-593X",
"clpid": "Hutchcroft-Tom"
},
{
"family_name": "Peres",
"given_name": "Yuval",
"orcid": "0000-0001-5456-6323",
"clpid": "Peres-Yuval"
}
],
"abstract": "We prove that in any recurrent reversible random rooted graph, two independent simple random walks started at the same vertex collide infinitely often almost surely. This applies to the Uniform Infinite Planar Triangulation and Quadrangulation and to the Incipient Infinite Cluster in Z\u00b2.",
"doi": "10.1214/ecp.v20-4330",
"issn": "1083-589X",
"publisher": "Institute of Mathematical Statistics",
"publication": "Electronic Communications in Probability",
"publication_date": "2015-09",
"volume": "20",
"pages": "Art. No. 4330"
}
]