arXiv subjects

Tom Hutchcroft

Publications and source records attributed to Tom Hutchcroft.

At least 73 records · Page 4Linked to original sources

arXiv2019

Percolation on hyperbolic graphs

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 $\nabla_{p_c}<\infty$ 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.

math.PR↗

arXiv2019

Geometric and spectral properties of causal maps

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 that 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 $α\in (1,2)$, for which our understanding is much less complete.

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arXiv2019

Harmonic Dirichlet Functions on Planar Graphs

Benjamini and Schramm (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 $D$.

math.PR↗

arXiv2019

Statistical physics on a product of trees

Let $G$ be the product of finitely many trees $T_1\times T_2 \times \cdots \times T_N$, 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.

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arXiv2018

Self-avoiding walk on nonunimodular transitive graphs

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 $n$th 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 these results apply in particular to the product $T_k \times \mathbb{Z}^d$ of a $k$-regular tree ($k\geq 3$) with $\mathbb{Z}^d$, for which these results were previously only known for large $k$.

math.PR↗

arXiv2018

The Component Graph of the Uniform Spanning Forest: Transitions in Dimensions $9,10,11,\ldots$

We prove that the uniform spanning forests of $\mathbb{Z}^d$ and $\mathbb{Z}^{\ell}$ have qualitatively different connectivity properties whenever $\ell >d \geq 4$. 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.

math.PR↗

arXiv2018

Indistinguishability of collections of trees in the uniform spanning forest

We prove the following indistinguishability theorem for $k$-tuples of trees in the uniform spanning forest of $\mathbb{Z}^d$: Suppose that $\mathscr{A}$ 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 $\mathscr{A}$ almost surely, or no $k$-tuple of distinct trees has property $\mathscr{A}$ 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.

math.PR↗

arXiv2018

Mallows permutations as stable matchings

We show that the Mallows measure on permutations of $1,\ldots,n$ arises as the law of the unique Gale-Shapley stable matching of the random bipartite graph conditioned to be perfect, where preferences arise from a total ordering of the vertices but are restricted to the (random) edges of the graph. We extend this correspondence to infinite intervals, for which the situation is more intricate. We prove that almost surely every stable matching of the random bipartite graph obtained by performing Bernoulli percolation on the complete bipartite graph $K_{\mathbb{Z},\mathbb{Z}}$ falls into one of two classes: a countable family $(σ_n)_{n\in\mathbb{Z}}$ of tame stable matchings, in which the length of the longest edge crossing $k$ is $O(\log |k|)$ as $k\to\pm \infty$, and an uncountable family of wild stable matchings, in which this length is $\exp Ω(k)$ as $k\to +\infty$. The tame stable matching $σ_n$ has the law of the Mallows permutation of $\mathbb{Z}$ (as constructed by Gnedin and Olshanski) composed with the shift $k\mapsto k+n$. The permutation $σ_{n+1}$ dominates $σ_{n}$ pointwise, and the two permutations are related by a shift along a random strictly increasing sequence.

math.PR↗

arXiv2018

Coalescing random walk on unimodular graphs

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.

math.PR↗

arXiv2018

Hyperbolic and Parabolic Unimodular Random Maps

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.

math.PR↗

arXiv2018

Uniform Spanning Forests of Planar Graphs

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.

math.PR↗

arXiv2017

The Hammersley-Welsh bound for self-avoiding walk revisited

The Hammersley-Welsh bound (1962) states that the number $c_n$ of length $n$ self-avoiding walks on $\mathbb{Z}^d$ satisfies \[ c_n \leq \exp \left[ O(n^{1/2}) \right] μ_c^n, \] where $μ_c=μ_c(d)$ is the connective constant of $\mathbb{Z}^d$. While stronger estimates have subsequently been proven for $d\geq 3$, 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 \[ c_n \leq \exp\left[ o(n^{1/2})\right] μ_c^n. \] The improved bound is obtained as a corollary to the sub-ballisticity theorem of Duminil-Copin and Hammond (2013). We also show that any quantitative form of that theorem would yield a corresponding quantitative improvement to the Hammersley-Welsh bound.

math.PR↗

arXiv2017

Counterexamples for percolation on unimodular random graphs

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.

math.PR↗

arXiv2017

Finitely dependent cycle coloring

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.

math.PR↗

arXiv2017

Mallows Permutations and Finite Dependence

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.

math.PR↗

arXiv2016

Interlacements and the Wired Uniform Spanning Forest

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 non-random 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, 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.

math.PR↗

arXiv2016

Unimodular Hyperbolic Triangulations: Circle Packing and Random Walk

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. As a part of this, we obtain an alternative proof of the Benjamini-Schramm Recurrence Theorem. 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.

math.PR↗

arXiv2016

Boundaries of Planar Graphs: A Unified Approach

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 and Angel, Barlow, Gurel-Gurevich and Nachmias 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.

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