相关论文: Quantitative indistinguishability and sparse and d…

We introduce a general framework to show the indistinguishability of infinite clusters (ergodicity of the cluster subrelation) in group-invariant percolation processes with a weaker version of the finite energy property: the possibility of…

We show that when percolation produces infinitely many infinite clusters on a Cayley graph, one cannot distinguish the clusters from each other by any invariantly defined property. This implies that uniqueness of the infinite cluster is…

We study the distribution of finite clusters in slightly supercritical ($p \downarrow p_c$) Bernoulli bond percolation on transitive nonamenable graphs, proving in particular that if $G$ is a transitive nonamenable graph satisfying the…

We introduce infinitesimal weak containment for measure-preserving actions of a countable group $\Gamma$: an action $(X,\mu)$ is infinitesimally contained in $(Y,\nu)$ if the statistics of the action of $\Gamma$ on small measure subsets of…

Answering a question of Benjamini, we present an isometry-invariant random partition of the Euclidean space $\mathbb{R}^d$, $d\geq 3$, into infinite connected indistinguishable pieces, such that the adjacency graph defined on the pieces is…

We show that the uniqueness thresholds for Poisson-Voronoi percolation in symmetric spaces of connected higher rank semisimple Lie groups with property (T) converge to zero in the low-intensity limit. This phenomenon is fundamentally…

We study invariant percolation processes on the d-regular tree that are obtained as a factor of an iid process. We show that the density of any factor of iid site percolation process with finite clusters is asymptotically at most (log d)/d…

We prove that for every class of graphs $\mathcal{C}$ which is nowhere dense, as defined by Nesetril and Ossona de Mendez, and for every first order formula $\phi(\bar x,\bar y)$, whenever one draws a graph $G\in \mathcal{C}$ and a subset…

Using functional and harmonic analysis methods, we study Kazhdan sets in topological groups which do not necessarily have Property (T). We provide a new criterion for a generating subset $Q$ of a group $G$ to be a Kazhdan set; it relies on…

We study Poisson--Voronoi percolation and its discrete analogue Bernoulli--Voronoi percolation in spaces with a non-amenable product structure. We develop a new method of proving smallness of the uniqueness threshold $p_u(\lambda)$ at small…

We consider homogeneous factor models on uniformly sparse graph sequences converging locally to a (unimodular) random tree $T$, and study the existence of the free energy density $\phi$, the limit of the log-partition function divided by…

This paper explores the link between the ergodicity of the clus-ter equivalence relation restricted to its infinite locus and the indis-tinguishability of infinite clusters. It is an important element of the dictionary connecting orbit…

We study percolation on nonamenable groups at the uniqueness threshold $p_u$, the critical value that separates the phase in which there are infinitely many infinite clusters from the phase in which there is a unique infinite cluster. The…

Given a graph $G$, we form a random subgraph $G_p$ by including each edge of $G$ independently with probability $p$. We provide an asymptotic expansion of the expected number of independent sets in random subgraphs of regular bipartite…

In this paper we consider Bernoulli percolation on an infinite connected bounded degrees graph $G$. Assuming the uniqueness of the infinite open cluster and a quasi-multiplicativity of crossing probabilities, we prove the existence of…

For a large class of amenable transient weighted graphs $G$, we prove that the sign clusters of the Gaussian free field on $G$ fall into a regime of strong supercriticality, in which two infinite sign clusters dominate (one for each sign),…

In this paper, we establish the existence and equivalence of four types of incipient infinite clusters (IICs) for the critical Gaussian free field (GFF) level-set and the critical loop soup on the metric graph $\widetilde{\mathbb{Z}}^d$ for…

We give an example of an $\mathsf{FIID}$ vertex-labeling of $\mathbb{T}_3$ whose marginals are uniform on $[0,1]$, and if we delete the edges between those vertices whose labels are different, then some of the remaining clusters are…

We show that a totally dissipative system has all nonsingular systems as factors, but that this is no longer true when the factor maps are required to be finitary. In particular, if a nonsingular Bernoulli shift satisfies the Doeblin…

Complex numbers are central to the formulation of quantum mechanics, yet their role as a genuine resource is only beginning to be understood. In this work, we demonstrate that quantum states with intrinsically complex amplitudes provide a…