Article records
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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenMon, 27 Nov 2023 17:56:52 +0000Collisions of random walks in reversible random graphs
https://resolver.caltech.edu/CaltechAUTHORS:20210923-215831853
Authors: Hutchcroft, Tom; Peres, Yuval
Year: 2015
DOI: 10.1214/ecp.v20-4330
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 Z².https://authors.library.caltech.edu/records/rwyak-pyf27Critical percolation on any quasi-transitive graph of exponential growth has no infinite clusters
https://resolver.caltech.edu/CaltechAUTHORS:20210924-190635043
Authors: Hutchcroft, Tom
Year: 2016
DOI: 10.1016/j.crma.2016.07.013
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.https://authors.library.caltech.edu/records/qy3jd-0m146Unimodular hyperbolic triangulations: circle packing and random walk
https://resolver.caltech.edu/CaltechAUTHORS:20210923-184021534
Authors: Angel, Omer; Hutchcroft, Tom; Nachmias, Asaf; Ray, Gourab
Year: 2016
DOI: 10.1007/s00222-016-0653-9
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–Schramm 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.https://authors.library.caltech.edu/records/a1rdy-13148Wired cycle-breaking dynamics for uniform spanning forests
https://resolver.caltech.edu/CaltechAUTHORS:20210924-190634976
Authors: Hutchcroft, Tom
Year: 2016
DOI: 10.1214/15-AOP1063
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–Watson tree, answering a question of Benjamini, Lyons, Peres and Schramm [Ann. Probab. 29 (2001) 1–65].
Our proof introduces and exploits a family of Markov chains under which the oriented WUSF is stationary, which we call the wired cycle-breaking dynamics.https://authors.library.caltech.edu/records/3eaf4-f6079Indistinguishability of trees in uniform spanning forests
https://resolver.caltech.edu/CaltechAUTHORS:20210923-225150172
Authors: Hutchcroft, Tom; Nachmias, Asaf
Year: 2017
DOI: 10.1007/s00440-016-0707-3
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):1–65, 2001). We also answer positively two additional questions of Benjamini et al. (Ann Probab 29(1):1–65, 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.https://authors.library.caltech.edu/records/9vs1v-3e586Boundaries of planar graphs: a unified approach
https://resolver.caltech.edu/CaltechAUTHORS:20210924-183504452
Authors: Hutchcroft, Tom; Peres, Yuval
Year: 2017
DOI: 10.1214/17-ejp116
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.
We 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.https://authors.library.caltech.edu/records/ds0pp-v5489The Hammersley-Welsh bound for self-avoiding walk revisited
https://resolver.caltech.edu/CaltechAUTHORS:20210924-184806499
Authors: Hutchcroft, Tom
Year: 2018
DOI: 10.1214/17-ECP94
The Hammersley-Welsh bound (Quart. J. Math., 1962) states that the number c_n of length n self-avoiding walks on Z^d satisfies
c_n ≤ exp[O(n^(1/2))]μ^n_c,
where μ_c = μ_c(d) is the connective constant of Z^d. While stronger estimates have subsequently been proven for d ≥ 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
c_n ≤ exp[o^(n^(1/2))] μ^n_c.
The 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.https://authors.library.caltech.edu/records/9c6re-9k844Interlacements and the wired uniform spanning forest
https://resolver.caltech.edu/CaltechAUTHORS:20210924-190634891
Authors: Hutchcroft, Tom
Year: 2018
DOI: 10.1214/17-aop1203
We extend the Aldous–Broder 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) 1702–1725], while the third extends a recent result of the author.
Finally, 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.https://authors.library.caltech.edu/records/ak4z3-yey29Hyperbolic and Parabolic Unimodular Random Maps
https://resolver.caltech.edu/CaltechAUTHORS:20210923-184021127
Authors: Angel, Omer; Hutchcroft, Tom; Nachmias, Asaf; Ray, Gourab
Year: 2018
DOI: 10.1007/s00039-018-0446-y
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–Schramm limit of finite maps.https://authors.library.caltech.edu/records/21ast-60592Coalescing random walk on unimodular graphs
https://resolver.caltech.edu/CaltechAUTHORS:20210923-184021815
Authors: Foxall, Eric; Hutchcroft, Tom; Junge, Matthew
Year: 2018
DOI: 10.1214/18-ecp136
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.https://authors.library.caltech.edu/records/xf9ak-cr102Finitely dependent cycle coloring
https://resolver.caltech.edu/CaltechAUTHORS:20210922-193309713
Authors: Holroyd, Alexander E.; Hutchcroft, Tom; Levy, Avi
Year: 2018
DOI: 10.1214/18-ECP118
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.https://authors.library.caltech.edu/records/40548-8c694Harmonic Dirichlet functions on planar graphs
https://resolver.caltech.edu/CaltechAUTHORS:20210922-193309575
Authors: Hutchcroft, Tom
Year: 2019
DOI: 10.1007/s00454-019-00057-2
Benjamini and Schramm (Invent Math 126(3):565–587, 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.https://authors.library.caltech.edu/records/228dv-bnt78Statistical physics on a product of trees
https://resolver.caltech.edu/CaltechAUTHORS:20210922-193309508
Authors: Hutchcroft, Tom
Year: 2019
DOI: 10.1214/18-aihp906
Let G be the product of finitely many trees T₁ × T₂ × ⋯ × 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.
We 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.https://authors.library.caltech.edu/records/x43w2-e4p85Percolation on hyperbolic graphs
https://resolver.caltech.edu/CaltechAUTHORS:20210922-193309440
Authors: Hutchcroft, Tom
Year: 2019
DOI: 10.1007/s00039-019-00498-0
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 ∇_p-c < ∞ 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.https://authors.library.caltech.edu/records/exzam-6kr78Self-avoiding walk on nonunimodular transitive graphs
https://resolver.caltech.edu/CaltechAUTHORS:20210922-193309236
Authors: Hutchcroft, Tom
Year: 2019
DOI: 10.1214/18-AOP1322
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 × Z^d of a k-regular tree (k ≥ 3) with Z^d, for which these results were previously only known for large k.https://authors.library.caltech.edu/records/m6mec-31e56Uniform spanning forests of planar graphs
https://resolver.caltech.edu/CaltechAUTHORS:20210922-193309305
Authors: Hutchcroft, Tom; Nachmias, Asaf
Year: 2019
DOI: 10.1017/fms.2019.14
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.https://authors.library.caltech.edu/records/4264h-6jw70The component graph of the uniform spanning forest: transitions in dimensions 9,10,11, ...
https://resolver.caltech.edu/CaltechAUTHORS:20210922-193309372
Authors: Hutchcroft, Tom; Peres, Yuval
Year: 2019
DOI: 10.1007/s00440-018-0884-3
We prove that the uniform spanning forests of Z^d and Z^ℓ have qualitatively different connectivity properties whenever ℓ > d ≥ 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.https://authors.library.caltech.edu/records/mcd2a-8fa05Mallows permutations and finite dependence
https://resolver.caltech.edu/CaltechAUTHORS:20210922-193308087
Authors: Holroyd, Alexander E.; Hutchcroft, Tom; Levy, Avi
Year: 2020
DOI: 10.1214/19-AOP1363
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.https://authors.library.caltech.edu/records/reagx-64650Universality of high-dimensional spanning forests and sandpiles
https://resolver.caltech.edu/CaltechAUTHORS:20210923-213625943
Authors: Hutchcroft, Tom
Year: 2020
DOI: 10.1007/s00440-019-00923-3
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 ≥ 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 ≥ 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.https://authors.library.caltech.edu/records/7dw4y-4c316Locality of the critical probability for transitive graphs of exponential growth
https://resolver.caltech.edu/CaltechAUTHORS:20210922-193308018
Authors: Hutchcroft, Tom
Year: 2020
DOI: 10.1214/19-AOP1395
Around 2008, Schramm conjectured that the critical probabilities for Bernoulli bond percolation satisfy the following continuity property: If (G_n)_(n ≥ 1) is a sequence of transitive graphs converging locally to a transitive graph G and limsup_(n → ∞)p_c(G_n) < 1, then p_c(G_n) → p_c(G) as n → ∞. 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.
In the unimodular case, this result is obtained as a corollary to the following theorem of independent interest: For every g > 1 and M < ∞, there exist positive constants C = C(g,M) and δ = δ(g,M) such that if G is a transitive unimodular graph with degree at most M and growth gr(G):= inf_(r ≥ 1)|B(o,r)|^(1/r) ≥ g, then P_(p_c)(|K_o| ≥ n) ≤ C_n^(−δ) for every n ≥ 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.https://authors.library.caltech.edu/records/d2qn7-4t233Indistinguishability of collections of trees in the uniform spanning forest
https://resolver.caltech.edu/CaltechAUTHORS:20210922-193308155
Authors: Hutchcroft, Tom
Year: 2020
DOI: 10.1214/19-AIHP988
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.https://authors.library.caltech.edu/records/cqd0v-w8754Kazhdan groups have cost 1
https://resolver.caltech.edu/CaltechAUTHORS:20210922-193307950
Authors: Hutchcroft, Tom; Pete, Gábor
Year: 2020
DOI: 10.1007/s00222-020-00967-6
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.https://authors.library.caltech.edu/records/ba2tq-frh31Nonuniqueness and mean-field criticality for percolation on nonunimodular transitive graphs
https://resolver.caltech.edu/CaltechAUTHORS:20210922-193307757
Authors: Hutchcroft, Tom
Year: 2020
DOI: 10.1090/jams/953
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 Häggström, 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.
All our results apply, for example, to the product T_k x Z^d of a k-regular tree with Z^d for k ≥ 3 and d ≥ 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.https://authors.library.caltech.edu/records/w8j7r-3fk10Non-intersection of transient branching random walks
https://resolver.caltech.edu/CaltechAUTHORS:20210922-193307881
Authors: Hutchcroft, Tom
Year: 2020
DOI: 10.1007/s00440-020-00964-z
Let G be a Cayley graph of a nonamenable group with spectral radius ρ < 1. It is known that branching random walk on G with offspring distribution μ is transient, i.e., visits the origin at most finitely often almost surely, if and only if the expected number of offspring μ[bar] satisfies μ[bar] ≤ ρ − 1. Benjamini and Müller (2010) conjectured that throughout the transient supercritical phase 1< μ[bar] ≤ ρ − 1, and in particular at the recurrence threshold μ[bar] = ρ − 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.
A 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.https://authors.library.caltech.edu/records/m8n9v-j1t44Anomalous diffusion of random walk on random planar maps
https://resolver.caltech.edu/CaltechAUTHORS:20210922-193307816
Authors: Gwynne, Ewain; Hutchcroft, Tom
Year: 2020
DOI: 10.1007/s00440-020-00986-7
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 γ-Liouville quantum gravity (LQG) universality class for γ∈(0,2)---including spanning tree-weighted maps, bipolar-oriented maps, and mated-CRT maps---typically travels graph distance n^(1/d_γ + o_n(1)) in n units of time, where dγ is the growth exponent for the volume of a metric ball on the map, which was shown to exist and depend only on γ by Ding and Gwynne (2018). Since d_γ > 2, this shows that the simple random walk on each of these maps is subdiffusive.
Our 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.https://authors.library.caltech.edu/records/0an9f-d4t30The L² boundedness condition in nonamenable percolation
https://resolver.caltech.edu/CaltechAUTHORS:20210922-193307690
Authors: Hutchcroft, Tom
Year: 2020
DOI: 10.1214/20-ejp525
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) = ℙ_(p_c) (u↔v) is bounded as an operator T_(p_c) : L²(V)→L²(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.
In this paper we further investigate the consequences of the L² 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→2); ii) If p_c < p_(2→2), then the critical exponent governing the extrinsic diameter of a critical cluster is 1; iii) Below p_(2→2), 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→2), then ‖T_(p_c) ‖_(q→q) ≍ (q−1)−1 and p_(q→q) − p_c ≍ q − 1 as q↓1. v) If p_c < p_(2→2), 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 − p_c)³ as p ↓ p_c. All of these results are new even in the Gromov hyperbolic case.
Finally, 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.https://authors.library.caltech.edu/records/s777e-ew214New critical exponent inequalities for percolation and the random cluster model
https://resolver.caltech.edu/CaltechAUTHORS:20210924-201102398
Authors: Hutchcroft, Tom
Year: 2020
DOI: 10.2140/pmp.2020.1.147
We apply a variation on the methods of Duminil-Copin, Raoufi, and Tassion (Ann. of Math. (2) 189:1 (2019), 75–99) to establish a new differential inequality applying to both Bernoulli percolation and the Fortuin–Kasteleyn random cluster model. This differential inequality has a similar form to that derived for Bernoulli percolation by Menshikov (Dokl. Akad. Nauk 288:6 (1986), 1308–1311) 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:
1. The critical exponent inequalities γ ≤ δ − 1 and Δ ≤ γ + 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 ∈ [1,2).
2. 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.https://authors.library.caltech.edu/records/ave15-h5f50Geometric and spectral properties of causal maps
https://resolver.caltech.edu/CaltechAUTHORS:20210922-193307624
Authors: Curien, Nicolas; Hutchcroft, Tom; Nachmias, Asaf
Year: 2020
DOI: 10.4171/jems/1001
We study the random planar map obtained from a critical, finite variance, Galton–Watson 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 Ambjørn 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−o(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 α-stable law for α ∈ (1,2), for which our understanding is much less complete.https://authors.library.caltech.edu/records/ty32r-s8w85Large, lengthy graphs look locally like lines
https://resolver.caltech.edu/CaltechAUTHORS:20210922-193307537
Authors: Benjamini, Itai; Hutchcroft, Tom
Year: 2021
DOI: 10.1112/blms.12436
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–Hausdorff metric.https://authors.library.caltech.edu/records/1nt9d-yhc77Supercritical percolation on nonamenable graphs: isoperimetry, analyticity, and exponential decay of the cluster size distribution
https://resolver.caltech.edu/CaltechAUTHORS:20210922-193306644
Authors: Hermon, Jonathan; Hutchcroft, Tom
Year: 2021
DOI: 10.1007/s00222-020-01011-3
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
P_p(n ≤ |K| < ∞) ≤ e^(−c_p)n)
for every n ≥ 1, where K is the cluster of the origin. We deduce the following two corollaries:
1. 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).
2. 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.https://authors.library.caltech.edu/records/1j2jy-xp413No Percolation at Criticality on Certain Groups of Intermediate Growth
https://resolver.caltech.edu/CaltechAUTHORS:20210924-202109319
Authors: Hermon, Jonathan; Hutchcroft, Tom
Year: 2021
DOI: 10.1093/imrn/rnz265
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) ≤ exp[−Ω(n^γ)] for some γ > ½. The result is new in the case that the graph is of intermediate volume growth.https://authors.library.caltech.edu/records/b2avf-5rb22Collisions of random walks in dynamic random environments
https://resolver.caltech.edu/CaltechAUTHORS:20210924-202129801
Authors: Halberstam, Noah; Hutchcroft, Tom
Year: 2022
DOI: 10.1214/21-EJP738
We study dynamic random conductance models on ℤ² 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.https://authors.library.caltech.edu/records/39xfc-ekz38What are the limits of universality?
https://resolver.caltech.edu/CaltechAUTHORS:20210924-203748960
Authors: Halberstam, Noah; Hutchcroft, Tom
Year: 2022
DOI: 10.1098/rspa.2021.0857
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 Z⁴ 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 τ. This gives strong evidence against a conjecture of Balankin et al. (2018 Phys. Lett. A382, 12–19 (doi:10.1016/j.physleta.2017.10.035)).https://authors.library.caltech.edu/records/dv0h8-wnc25Slightly supercritical percolation on non-amenable graphs I: The distribution of finite clusters
https://resolver.caltech.edu/CaltechAUTHORS:20220907-386218000
Authors: Hutchcroft, Tom
Year: 2022
DOI: 10.1112/plms.12474
We study the distribution of finite clusters in slightly supercritical (p↓pc) 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≤|K|<∞)≍n−1/2exp[−Θ(|p−pc|2n)] and Pp(r≤Rad(K)<∞)≍r−1exp[−Θ(|p−pc|r)] for every p∈(pc−δ,pc+δ) and n,r≥1, where all implicit constants depend only on G. We deduce in particular that the critical exponents γ′ and Δ′ describing the rate of growth of the moments of a finite cluster as p↓pc 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 ρ<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 ℤd 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.https://authors.library.caltech.edu/records/0pmve-wtk48On the Derivation of Mean-Field Percolation Critical Exponents from the Triangle Condition
https://resolver.caltech.edu/CaltechAUTHORS:20220823-628154700.756
Authors: Hutchcroft, Tom
Year: 2022
DOI: 10.1007/s10955-022-02967-7
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 ∇pc is large. In contrast to earlier methods, our approach continues to yield bounds of reasonable order when the triangle diagram ∇p is unbounded but diverges slowly as p↑pc, 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↑pc 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.https://authors.library.caltech.edu/records/zwvhy-s1238Sharp hierarchical upper bounds on the critical two-point function for long-range percolation on ℤᵈ
https://resolver.caltech.edu/CaltechAUTHORS:20221129-370786800.2
Authors: Hutchcroft, Tom
Year: 2022
DOI: 10.1063/5.0088450
Consider long-range Bernoulli percolation on ℤᵈ in which we connect each pair of distinct points x and y by an edge with probability 1 − exp(−β‖x − y‖^(−d−α)), where α > 0 is fixed and β ⩾ 0 is a parameter. We prove that if 0 < α < d, then the critical two-point function satisfies (1/|Λ_r|)∑_(xϵΛ_(r))P_(β_(c))(0 ↔ x) ≤ r^(−d+a) for every r ⩾ 1, where Λ_r = [−r,r]ᵈ ∩ ℤᵈ. In other words, the critical two-point function on ℤᵈ 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 α strictly below the crossover value α_(c)(d), where the values of several critical exponents for long-range percolation on ℤᵈ and the hierarchical lattice are believed to be equal.https://authors.library.caltech.edu/records/c1dgm-xs860Transience and anchored isoperimetric dimension of supercritical percolation clusters
https://resolver.caltech.edu/CaltechAUTHORS:20230725-745306000.2
Authors: Hutchcroft, Tom
Year: 2023
DOI: 10.1214/23-ejp905
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.https://authors.library.caltech.edu/records/95j8r-pwq77The bunkbed conjecture holds in the p ↑ 1 limit
https://resolver.caltech.edu/CaltechAUTHORS:20230103-818063100.57
Authors: Hutchcroft, Tom; Kent, Alexander; Nizić-Nikolac, Petar
Year: 2023
DOI: 10.1017/s096354832200027x
Let G = (V, E) be a countable graph. The Bunkbed graph of G is the product graph G x K₂, 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 ϵ V. Kasteleyn's Bunkbed conjecture states that for each u, v ϵ V and p ϵ [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 ↑ 1 limit in the sense that for each finite graph G there exists ε(G) > 0 such that the bunkbed conjecture holds for p ⩾ 1 - ε(G).https://authors.library.caltech.edu/records/a7hr1-srd50High-dimensional near-critical percolation and the torus plateau
https://resolver.caltech.edu/CaltechAUTHORS:20230411-695015900.4
Authors: Hutchcroft, Tom; Michta, Emmanuel; Slade, Gordon
Year: 2023
DOI: 10.1214/22-aop1608
We consider percolation on Zᵈ and on the d-dimensional discrete torus, in dimensions d ≥ 11 for the nearest-neighbour model and in dimensions d > 6 for spread-out models. For Zᵈ 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
P_(p_c − ε)(0 ↔ x) ≤ C/(∥x∥ᵈ⁻²)e^(−cε^(1/2)∥x∥),
(c/r²)e^(−Cε^((1/2)r)) ≤ P_(pc−ε)(0 ↔ ∂[−r,r]ᵈ) ≤ C/(r²)e^(−cε(1/2)r).
Using this, we prove that throughout the critical window the torus two-point function has a "plateau," meaning that it decays for small x as ∥x∥⁻⁽ᵈ⁻²⁾ but for large x is essentially constant and of order V^(−2/3) 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 Zᵈ are separated by a multiple of V^(−1/3). 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 Zᵈ two-point function. In particular, we use results derived from the lace expansion on Zᵈ, but in contrast to previous work on high-dimensional torus percolation, we do not need or use a separate torus lace expansion.https://authors.library.caltech.edu/records/em9gs-e7c18Logarithmic Corrections to Scaling in the Four-dimensional Uniform Spanning Tree
https://resolver.caltech.edu/CaltechAUTHORS:20230420-614686900.16
Authors: Hutchcroft, Tom; Sousi, Perla
Year: 2023
DOI: 10.1007/s00220-023-04686-w
We compute the precise logarithmic corrections to mean-field scaling for various quantities describing the uniform spanning tree of the four-dimensional hypercubic lattice Z⁴. 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⁻¹, that the probability that the past containsat least n vertices is of order (log n)^(1/6)n^(−1/2), and that the probability that the past reaches the boundary of the box [−n, n]⁴ is of order (log n)^(2/3+o(1))n⁻². 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.https://authors.library.caltech.edu/records/y9141-cjk45