Submitted - 2107.12971.pdf

", "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.", "date": "2021-09-27", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20210924-202253573", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-202253573", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "European Research Council (ERC)", "grant_number": "804166" }, { "agency": "Natural Sciences and Engineering Research Council of Canada (NSERC)" } ] }, "doi": "10.48550/arXiv.2107.12971", "primary_object": { "basename": "2107.12971.pdf", "url": "/records/h9k7j-fzm15/files/2107.12971.pdf" }, "resource_type": "monograph", "pub_year": "2021", "author_list": "Hutchcroft, Tom; Michta, Emmanuel; et el." }, { "id": "https://authors.library.caltech.edu/records/vw7fw-2em04", "eprint_id": 111036, "eprint_status": "archive", "datestamp": "2023-08-19 23:59:46", "lastmod": "2023-10-23 20:00:40", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Hutchcroft-Tom", "name": { "family": "Hutchcroft", "given": "Tom" }, "orcid": "0000-0003-0061-593X" }, { "id": "Sousi-Perla", "name": { "family": "Sousi", "given": "Perla" } } ] }, "title": "Logarithmic corrections to scaling in the four-dimensional uniform spanning tree", "ispublished": "unpub", "full_text_status": "public", "note": "Perla Sousi's research was supported by the Engineering and Physical Sciences Research Council: EP/R022615/1.\n\nSubmitted - 2010.15830.pdf

", "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.", "date": "2021-09-27", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20210924-202133236", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-202133236", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Engineering and Physical Sciences Research Council (EPSRC)", "grant_number": "EP/R022615/1" } ] }, "doi": "10.48550/arXiv.2010.15830", "primary_object": { "basename": "2010.15830.pdf", "url": "/records/vw7fw-2em04/files/2010.15830.pdf" }, "resource_type": "monograph", "pub_year": "2021", "author_list": "Hutchcroft, Tom and Sousi, Perla" }, { "id": "https://authors.library.caltech.edu/records/k4eb1-c9n17", "eprint_id": 111030, "eprint_status": "archive", "datestamp": "2023-08-19 18:12:44", "lastmod": "2023-10-23 20:00:27", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Hutchcroft-Tom", "name": { "family": "Hutchcroft", "given": "Tom" }, "orcid": "0000-0003-0061-593X" } ] }, "title": "Non-intersection of transient branching random walks", "ispublished": "unpub", "full_text_status": "public", "note": "We thank Itai Benjamini, Jonathan Hermon, Asaf Nachmias, and Elisabetta Candellero for useful discussions. In particular, we thank Asaf for discussions that led to a substantially simpler proof of Theorem 3.3. We also thank the anonymous referee for their careful reading and helpful suggestions.\n\nAccepted Version - 1910.01018.pdf

", "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.", "date": "2021-09-27", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20210924-202112742", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-202112742", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "doi": "10.48550/arXiv.1910.01018", "primary_object": { "basename": "1910.01018.pdf", "url": "/records/k4eb1-c9n17/files/1910.01018.pdf" }, "resource_type": "monograph", "pub_year": "2021", "author_list": "Hutchcroft, Tom" }, { "id": "https://authors.library.caltech.edu/records/r0bnm-b8563", "eprint_id": 111039, "eprint_status": "archive", "datestamp": "2023-08-20 02:42:50", "lastmod": "2023-10-23 20:00:48", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Hutchcroft-Tom", "name": { "family": "Hutchcroft", "given": "Tom" }, "orcid": "0000-0003-0061-593X" }, { "id": "Tointon-Matthew", "name": { "family": "Tointon", "given": "Matthew" }, "orcid": "0000-0001-8086-9280" } ] }, "title": "Non-triviality of the phase transition for percolation on finite transitive graphs", "ispublished": "unpub", "full_text_status": "public", "note": "The first author was supported in part by ERC starting grant 804166 (SPRS) and thanks Gabor Pete for helpful discussions. The second author was partially supported by the Stokes Research Fellowship at Pembroke College, Cambridge. He is also grateful to Itai Benjamini and Ariel Yadin for introducing him to the problems considered in this paper, and to Romain Tessera for helpful conversations.\n\nSubmitted - 2104.05607.pdf

", "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.", "date": "2021-09-27", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20210924-202143857", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-202143857", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "European Research Council (ERC)", "grant_number": "804166" }, { "agency": "Pembroke College, Cambridge" } ] }, "doi": "10.48550/arXiv.2104.05607", "primary_object": { "basename": "2104.05607.pdf", "url": "/records/r0bnm-b8563/files/2104.05607.pdf" }, "resource_type": "monograph", "pub_year": "2021", "author_list": "Hutchcroft, Tom and Tointon, Matthew" }, { "id": "https://authors.library.caltech.edu/records/b7n9w-z9n48", "eprint_id": 111032, "eprint_status": "archive", "datestamp": "2023-08-19 20:03:59", "lastmod": "2023-10-23 20:00:32", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Angel-Omer", "name": { "family": "Angel", "given": "Omer" }, "orcid": "0000-0002-6451-8242" }, { "id": "Hutchcroft-Tom", "name": { "family": "Hutchcroft", "given": "Tom" }, "orcid": "0000-0003-0061-593X" }, { "id": "J\u00e1rai-Antal-A", "name": { "family": "J\u00e1rai", "given": "Antal A." }, "orcid": "0000-0003-3522-498X" } ] }, "title": "On the tail of the branching random walk local time", "ispublished": "unpub", "full_text_status": "public", "note": "The authors are grateful to the organizers of the Oberwolfach Workshop Strongly Correlated Interacting Processes, where this work was initiated. We thank Ed Perkins and Jean-Fran\u00e7ois Le Gall for helpful discussions on the literature. OA is supported in part by an NSERC discovery grant.\n\nSubmitted - 2002.12188.pdf

", "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.", "date": "2021-09-27", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20210924-202119558", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-202119558", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Natural Sciences and Engineering Research Council of Canada (NSERC)" } ] }, "doi": "10.48550/arXiv.2002.12188", "primary_object": { "basename": "2002.12188.pdf", "url": "/records/b7n9w-z9n48/files/2002.12188.pdf" }, "resource_type": "monograph", "pub_year": "2021", "author_list": "Angel, Omer; Hutchcroft, Tom; et el." }, { "id": "https://authors.library.caltech.edu/records/rsb7g-z9w39", "eprint_id": 111034, "eprint_status": "archive", "datestamp": "2023-08-19 23:02:37", "lastmod": "2023-10-23 20:00:37", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Hutchcroft-Tom", "name": { "family": "Hutchcroft", "given": "Tom" }, "orcid": "0000-0003-0061-593X" } ] }, "title": "Power-law bounds for critical long-range percolation below the upper-critical dimension", "ispublished": "unpub", "full_text_status": "public", "note": "Dedicated to Harry Kesten, November 19, 1931 - March 29, 2019. \n\nWe thank Jonathan Hermon for his careful reading of an earlier version of this manuscript, and thank Gordon Slade for helpful discussions on the physics literature. We also thank the anonymous referee for their helpful comments and corrections.\n\nAccepted Version - 2008.11197.pdf

", "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", "date": "2021-09-27", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20210924-202126385", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-202126385", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "doi": "10.48550/arXiv.2008.11197", "primary_object": { "basename": "2008.11197.pdf", "url": "/records/rsb7g-z9w39/files/2008.11197.pdf" }, "resource_type": "monograph", "pub_year": "2021", "author_list": "Hutchcroft, Tom" }, { "id": "https://authors.library.caltech.edu/records/8bzn7-k1747", "eprint_id": 111031, "eprint_status": "archive", "datestamp": "2023-08-19 19:55:55", "lastmod": "2023-10-23 20:00:30", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Hutchcroft-Tom", "name": { "family": "Hutchcroft", "given": "Tom" }, "orcid": "0000-0003-0061-593X" } ] }, "title": "Slightly supercritical percolation on nonamenable graphs I: The distribution of finite clusters", "ispublished": "unpub", "full_text_status": "public", "note": "We thank Jonathan Hermon and Asaf Nachmias for many helpful discussions, and thank Remco van der Hofstad for helpful comments on an earlier version of this manuscript. We also thank Antoine Godin for sharing his simplified proof of Proposition 3.1 with us.\n\nSubmitted - 2002.02916.pdf

", "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.", "date": "2021-09-27", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20210924-202116146", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-202116146", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "doi": "10.48550/arXiv.2002.02916", "primary_object": { "basename": "2002.02916.pdf", "url": "/records/8bzn7-k1747/files/2002.02916.pdf" }, "resource_type": "monograph", "pub_year": "2021", "author_list": "Hutchcroft, Tom" }, { "id": "https://authors.library.caltech.edu/records/qstde-m2w06", "eprint_id": 111038, "eprint_status": "archive", "datestamp": "2023-08-20 02:27:06", "lastmod": "2023-10-23 20:00:45", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Hutchcroft-Tom", "name": { "family": "Hutchcroft", "given": "Tom" }, "orcid": "0000-0003-0061-593X" } ] }, "title": "The critical two-point function for long-range percolation on the hierarchical lattice", "ispublished": "unpub", "full_text_status": "public", "note": "This research was supported by ERC starting grant 804166 (SPRS). We thank Gordon Slade for helpful comments on a previous version of the manuscript.\n\nSubmitted - 2103.17013.pdf

", "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.", "date": "2021-09-27", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20210924-202140311", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-202140311", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "European Research Council (ERC)", "grant_number": "804166" } ] }, "doi": "10.48550/arXiv.2103.17013", "primary_object": { "basename": "2103.17013.pdf", "url": "/records/qstde-m2w06/files/2103.17013.pdf" }, "resource_type": "monograph", "pub_year": "2021", "author_list": "Hutchcroft, Tom" }, { "id": "https://authors.library.caltech.edu/records/9ak9w-nca68", "eprint_id": 111040, "eprint_status": "archive", "datestamp": "2023-08-20 03:40:07", "lastmod": "2023-10-23 20:00:51", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Hutchcroft-Tom", "name": { "family": "Hutchcroft", "given": "Tom" }, "orcid": "0000-0003-0061-593X" } ] }, "title": "On the derivation of mean-field percolation critical exponents from the triangle condition", "ispublished": "unpub", "full_text_status": "public", "note": "The author was supported in part by ERC starting grant 804166 (SPRS). We thank Vivek Dewan, Emmanuel Michta, Stephen Muirhead, and Gordon Slade for helpful comments on a previous version of the manuscript.\n\nSubmitted - 2106.06400.pdf

", "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.", "date": "2021-06-11", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20210924-202147400", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-202147400", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "European Research Council (ERC)", "grant_number": "804166" } ] }, "doi": "10.48550/arXiv.2106.06400v2", "primary_object": { "basename": "2106.06400.pdf", "url": "/records/9ak9w-nca68/files/2106.06400.pdf" }, "resource_type": "monograph", "pub_year": "2021", "author_list": "Hutchcroft, Tom" }, { "id": "https://authors.library.caltech.edu/records/ksvt4-kdg91", "eprint_id": 111037, "eprint_status": "archive", "datestamp": "2023-08-20 00:21:33", "lastmod": "2023-10-23 20:00:42", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Hutchcroft-Tom", "name": { "family": "Hutchcroft", "given": "Tom" }, "orcid": "0000-0003-0061-593X" } ] }, "title": "Transience and recurrence of sets for branching random walk via non-standard stochastic orders", "ispublished": "unpub", "full_text_status": "public", "note": "We thank Toby Johnson and Matt Junge for helpful discussions.\n\nSubmitted - 2011.06402.pdf

", "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.", "date": "2020-11-12", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20210924-202136797", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-202136797", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "doi": "10.48550/arXiv.2011.06402", "primary_object": { "basename": "2011.06402.pdf", "url": "/records/ksvt4-kdg91/files/2011.06402.pdf" }, "resource_type": "monograph", "pub_year": "2020", "author_list": "Hutchcroft, Tom" }, { "id": "https://authors.library.caltech.edu/records/2xfrr-8xy91", "eprint_id": 111033, "eprint_status": "archive", "datestamp": "2023-08-19 22:34:30", "lastmod": "2023-10-23 20:00:35", "type": "monograph", "metadata_visibility": "show", "creators": { "items": [ { "id": "Hutchcroft-Tom", "name": { "family": "Hutchcroft", "given": "Tom" }, "orcid": "0000-0003-0061-593X" } ] }, "title": "Continuity of the Ising phase transition on nonamenable groups", "ispublished": "unpub", "full_text_status": "public", "note": "We thank Jonathan Hermon for making us aware of Freedman's work on maximal inequalities for martingales [33], which inspired Lemmas 3.4 and 3.5. We also thank Hugo Duminil-Copin, Geoffrey Grimmett, and Russ Lyons for helpful comments on an earlier version of the manuscript.\n\nSubmitted - 2007.15625.pdf

", "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.", "date": "2020-07-30", "date_type": "published", "publisher": "arXiv", "id_number": "CaltechAUTHORS:20210924-202122977", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210924-202122977", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "doi": "10.48550/arXiv.2007.15625", "primary_object": { "basename": "2007.15625.pdf", "url": "/records/2xfrr-8xy91/files/2007.15625.pdf" }, "resource_type": "monograph", "pub_year": "2020", "author_list": "Hutchcroft, Tom" } ]