Monograph records
https://feeds.library.caltech.edu/people/Hutchcroft-Tom/monograph.rss
A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenFri, 08 Dec 2023 12:17:26 +0000Continuity of the Ising phase transition on nonamenable groups
https://resolver.caltech.edu/CaltechAUTHORS:20210924-202122977
Authors: Hutchcroft, Tom
Year: 2020
DOI: 10.48550/arXiv.2007.15625
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 ⟨σo⟩+β,h is a locally Hölder-continuous function of the inverse temperature β and external field h throughout the non-negative quadrant (β,h)∈[0,∞)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.https://authors.library.caltech.edu/records/2xfrr-8xy91Transience and recurrence of sets for branching random walk via non-standard stochastic orders
https://resolver.caltech.edu/CaltechAUTHORS:20210924-202136797
Authors: Hutchcroft, Tom
Year: 2020
DOI: 10.48550/arXiv.2011.06402
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 μ and ν are supercritical offspring distributions with means μ¯<ν¯ then every space-time set that is recurrent with respect to the offspring distribution μ is also recurrent with respect to the offspring distribution ν and similarly that every space-time set that is transient with respect to the offspring distribution ν is also transient with respect to the offspring distribution μ. 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 μ is smaller than ν 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.https://authors.library.caltech.edu/records/ksvt4-kdg91On the derivation of mean-field percolation critical exponents from the triangle condition
https://resolver.caltech.edu/CaltechAUTHORS:20210924-202147400
Authors: Hutchcroft, Tom
Year: 2021
DOI: 10.48550/arXiv.2106.06400v2
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 ∇_(p_c) 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 ↑ p_c, as is expected to occur in percolation on ℤ^d 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/9ak9w-nca68The critical two-point function for long-range percolation on the hierarchical lattice
https://resolver.caltech.edu/CaltechAUTHORS:20210924-202140311
Authors: Hutchcroft, Tom
Year: 2021
DOI: 10.48550/arXiv.2103.17013
We prove up-to-constants bounds on the two-point function (i.e.,
point-to-point connection probabilities) for critical long-range percolation on
the d-dimensional hierarchical lattice. More precisely, we prove that if we
connect each pair of points x and y by an edge with probability
1-exp(-β||x-y||^(-d-α)), where 0 < α < d is fixed and β ≥ 0 is a parameter, then the critical two-point function satisfies P_(β_c)(x ↔ y)||x-y||^(-d+α) for
every pair of distinct points x and y. We deduce in particular that the
model has mean-field critical behaviour when α < d/3 and does not have
mean-field critical behaviour when α > d/3.https://authors.library.caltech.edu/records/qstde-m2w06Slightly supercritical percolation on nonamenable graphs I: The distribution of finite clusters
https://resolver.caltech.edu/CaltechAUTHORS:20210924-202116146
Authors: Hutchcroft, Tom
Year: 2021
DOI: 10.48550/arXiv.2002.02916
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/8bzn7-k1747Power-law bounds for critical long-range percolation below the upper-critical dimension
https://resolver.caltech.edu/CaltechAUTHORS:20210924-202126385
Authors: Hutchcroft, Tom
Year: 2021
DOI: 10.48550/arXiv.2008.11197
We study long-range Bernoulli percolation on ℤd in which each two vertices x and y are connected by an edge with probability 1−exp(−β‖x−y‖−d−α). It is a theorem of Noam Berger (CMP, 2002) that if 0<αhttps://authors.library.caltech.edu/records/rsb7g-z9w39On the tail of the branching random walk local time
https://resolver.caltech.edu/CaltechAUTHORS:20210924-202119558
Authors: Angel, Omer; Hutchcroft, Tom; Járai, Antal A.
Year: 2021
DOI: 10.48550/arXiv.2002.12188
Consider a critical branching random walk on ℤ^d, d≥1, 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.https://authors.library.caltech.edu/records/b7n9w-z9n48Non-triviality of the phase transition for percolation on finite transitive graphs
https://resolver.caltech.edu/CaltechAUTHORS:20210924-202143857
Authors: Hutchcroft, Tom; Tointon, Matthew
Year: 2021
DOI: 10.48550/arXiv.2104.05607
We prove that if (G_n)_(n ≥ 1) = ((V_n,E_n))_(n ≥ 1) is a sequence of
finite, vertex-transitive graphs with bounded degrees and |V_n|→∞ that
is at least (1+ϵ)-dimensional for some ϵ > 0 in the sense that
diam(G_n)=O(|V_n|^(1/(1+ϵ) as n → ∞ then this sequence of graphs has a non-trivial phase transition
for Bernoulli bond percolation. More precisely, we prove under these conditions
that for each 0<α<1 there exists p_c(α) < 1 such that for each
p ≥ p_c(α), Bernoulli-p bond percolation on G_n has a cluster of
size at least α|V_n| with probability tending to 1 as n → ∞.
In fact, we prove more generally that there exists a universal constant a
such that the same conclusion holds whenever diam(G_n) = 0(|V_n|/(log|V_n|α) as n → ∞.
This verifies a conjecture of Benjamini up to the value of the constant a,
which he suggested should be 1.
We also prove a generalization of this result to quasitransitive graph
sequences with a bounded number of vertex orbits and prove that one may indeed
take a = 1 when the graphs G_n are all Cayley graphs of Abelian groups. A key
step in our proof is to adapt the methods of Duminil-Copin, Goswami, Raoufi,
Severo, and Yadin from infinite graphs to finite graphs. This adaptation also
leads to an isoperimetric criterion for infinite graphs to have a nontrivial
uniqueness phase (i.e., to have p_u < 1) which is of independent interest. We
also prove that the set of possible values of the critical probability of an
infinite quasitransitive graph has a gap at 1 in the sense that for every
k,n < ∞ there exists ϵ > 0 such that every infinite graph G of
degree at most k whose vertex set has at most n orbits under Aut(G)
either has p_c = 1 or p_c ≤ 1 - ϵ.https://authors.library.caltech.edu/records/r0bnm-b8563Non-intersection of transient branching random walks
https://resolver.caltech.edu/CaltechAUTHORS:20210924-202112742
Authors: Hutchcroft, Tom
Year: 2021
DOI: 10.48550/arXiv.1910.01018
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¯ satisfies μ¯≤ρ−1. Benjamini and Müller (2010) conjectured that throughout the transient supercritical phase 1<μ¯≤ρ−1, and in particular at the recurrence threshold μ¯=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/k4eb1-c9n17Logarithmic corrections to scaling in the four-dimensional uniform spanning tree
https://resolver.caltech.edu/CaltechAUTHORS:20210924-202133236
Authors: Hutchcroft, Tom; Sousi, Perla
Year: 2021
DOI: 10.48550/arXiv.2010.15830
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 contains at 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/vw7fw-2em04High-dimensional near-critical percolation and the torus plateau
https://resolver.caltech.edu/CaltechAUTHORS:20210924-202253573
Authors: Hutchcroft, Tom; Michta, Emmanuel; Slade, Gordon
Year: 2021
DOI: 10.48550/arXiv.2107.12971
We consider percolation on Z^d 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 ℤ^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
ℙ_(p_c − ε) (0↔x) ≤ C/(‖x‖^(d−2)) e^(−cε^(1/2)‖x‖) and c/r^2 e^(−Cε^((1/2)r( ≤ ℙ_(p_c − ε)(0↔∂[−r,r]^d) ≤ C/r^2 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 ℤ^d 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 ℤ^d two-point function. In particular, we use results derived from the lace expansion on ℤ^d, 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/h9k7j-fzm15