Drewitz, Alexander; Prévost, Alexis; Rodriguez, Pierre-François. Critical exponents for a percolation model on transient graphs. (English) Zbl 07662556 Invent. Math. 232, No. 1, 229-299 (2023).

The authors study the bond percolation model obtained by considering the clusters of a weighted graph G (transient for the random walk on G) induced by the excursion sets of the Gaussian free field ϕ on the cable system G~ associated to G. They give two theorems describing the near-critical regime of the phase transition for the corresponding percolation model and derive various associated critical exponents, all of them functions of two parameters, ν and α, describing resp. the decay of correlations and the volume growth of G. The proofs make use of continuity and strong Markov properties and of potential theory

Extendable self-avoiding walks

The connective constant mu of a graph is the exponential growth rate of the number of n-step self-avoiding walks starting at a given vertex. A self-avoiding walk is said to be forward (respectively, backward) extendable if it may be extended forwards (respectively, backwards) to a singly infinite self-avoiding walk. It is called doubly extendable if it may be extended in both directions simultaneously to a doubly infinite self-avoiding walk. We prove that the connective constants for forward, backward, and doubly extendable self-avoiding walks, denoted respectively by mu^F, mu^B, mu^FB, exist and satisfy mu = mu^F = mu^B = mu^FB for every infinite, locally finite, strongly connected, quasi-transitive directed graph. The proofs rely on a 1967 result of Furstenberg on dimension, and involve two different arguments depending on whether or not the graph is unimodular.Comment: Accepted versio

Percolation and local isoperimetric inequalities

In this paper we establish some relations between percolation on a given graph G and its geometry. Our main result shows that, if G has polynomial growth and satisfies what we call the local isoperimetric inequality of dimension d > 1, then p_c(G) < 1. This gives a partial answer to a question of Benjamini and Schramm. As a consequence of this result we derive, under the additional condition of bounded degree, that these graphs also undergo a non-trivial phase transition for the Ising-Model, the Widom-Rowlinson model and the beach model. Our techniques are also applied to dependent percolation processes with long range correlations. We provide results on the uniqueness of the infinite percolation cluster and quantitative estimates on the size of finite components. Finally we leave some remarks and questions that arise naturally from this work.Comment: 21 pages, 2 figure

Interlacement percolation on transient weighted graphs

In this article, we first extend the construction of random interlacements, introduced by A.S. Sznitman in [arXiv:0704.2560], to the more general setting of transient weighted graphs. We prove the Harris-FKG inequality for this model and analyze some of its properties on specific classes of graphs. For the case of non-amenable graphs, we prove that the critical value u_* for the percolation of the vacant set is finite. We also prove that, once G satisfies the isoperimetric inequality IS_6 (see (1.5)), u_* is positive for the product GxZ (where we endow Z with unit weights). When the graph under consideration is a tree, we are able to characterize the vacant cluster containing some fixed point in terms of a Bernoulli independent percolation process. For the specific case of regular trees, we obtain an explicit formula for the critical value u_*.Comment: 25 pages, 2 figures, accepted for publication in the Elect. Journal of Pro

Geometry of non-transitive graphs

In this note, we study non-transitive graphs and prove a number of results when they satisfy a coarse version of transitivity. Also, for each finitely generated group $G$, we produce continuum many pairwise non-quasi-isometric regular graphs that have the same growth rate, number of ends, and asymptotic dimension as $G$.Comment: 13 pages, 7 figure

A survey on graphs with polynomial growth

AbstractIn this paper we give an overview on connected locally finite transitive graphs with polynomial growth. We present results concerning the following topics: &#x02022;Automorphism groups of graphs with polynomial growth.&#x02022;Groups and graphs with linear growth.&#x02022;S-transitivity.&#x02022;Covering graphs.&#x02022;Automorphism groups as topological groups

Percolation and isoperimetry on roughly transitive graphs

In this paper we study percolation on a roughly transitive graph G with polynomial growth and isoperimetric dimension larger than one. For these graphs we are able to prove that p_c < 1, or in other words, that there exists a percolation phase. The main results of the article work for both dependent and independent percolation processes, since they are based on a quite robust renormalization technique. When G is transitive, the fact that p_c < 1 was already known before. But even in that case our proof yields some new results and it is entirely probabilistic, not involving the use of Gromov's theorem on groups of polynomial growth. We finish the paper giving some examples of dependent percolation for which our results apply.Comment: 32 pages, 2 figure

Geodesics in Transitive Graphs

AbstractLetPbe a double ray in an infinite graphX, and letdanddPdenote the distance functions inXand inPrespectively. One callsPageodesicifd(x, y)=dP(x, y), for all verticesxandyinP. We give situations when every edge of a graph belongs to a geodesic or a half-geodesic. Furthermore, we show the existence of geodesics in infinite locally-finite transitive graphs with polynomial growth which are left invariant (set-wise) under “translating” automorphisms. As the main result, we show that an infinite, locally-finite, transitive, 1-ended graph with polynomial growth is planar if and only if the complement of every geodesic has exactly two infinite components

Extendable self-avoiding walks

The connective constant mu of a graph is the exponential growth rate of the number of n-step self-avoiding walks starting at a given vertex. A self-avoiding walk is said to be forward (respectively, backward) extendable if it may be extended forwards (respectively, backwards) to a singly infinite self-avoiding walk. It is called doubly extendable if it may be extended in both directions simultaneously to a doubly infinite self-avoiding walk. We prove that the connective constants for forward, backward, and doubly extendable self-avoiding walks, denoted respectively by mu^F, mu^B, mu^FB, exist and satisfy mu = mu^F = mu^B = mu^FB for every infinite, locally finite, strongly connected, quasi-transitive directed graph. The proofs rely on a 1967 result of Furstenberg on dimension, and involve two different arguments depending on whether or not the graph is unimodular