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In this paper, we consider the hypothesis testing of correlation between two m-uniform hypergraphs on n unlabelled nodes. Under the null hypothesis, the hypergraphs are independent, while under the alternative hypothesis, the hyperdges have the same marginal distributions as in the null hypothesis but are correlated after some unknown node permutation. We focus on two scenarios: the hypergraphs are generated from the Gaussian-Wigner model and the dense Erdös-Rényi model. We derive the sharp information-theoretic testing threshold. Above the threshold, there exists a powerful test to distinguish the alternative hypothesis from the null hypothesis. Below the threshold, the alternative hypothesis and the null hypothesis are not distinguishable. The threshold involves m and decreases as m gets larger. This indicates testing correlation of hypergraphs (m = 3) becomes easier than testing correlation of graphs (m = 2). [ABSTRACT FROM AUTHOR]

In this article we show, in a concise manner, a result of uniform in time propagation of chaos for non exchangeable systems of particles interacting according to a random graph. Provided the interaction is Lipschitz continuous, the restoring force satisfies a general one-sided Lipschitz condition (thus allowing for non-convex confining potential) and the graph is dense enough, we use a coupling method suggested by Eberle (2016) known as reflection coupling to obtain uniform in time mean-field limit with bounds that depend explicitly on the graph structure. [ABSTRACT FROM AUTHOR]

In this article we study a one dimensional model for a polymer in a poor solvent: the random walk on Z penalized by its range. More precisely, we consider a Gibbs transformation of the law of the simple symmetric random walk by a weight exp(-hn|Rn|), with |Rn| the number of visited sites and hn a size-dependent positive parameter. We use gambler's ruin estimates to obtain exact asymptotics for the partition function, that enables us to obtain a precise description of trajectories, in particular scaling limits for the center and the amplitude of the range. A phase transition for the fluctuations around an optimal amplitude is identified at hn n1/4, inherent to the underlying lattice structure. [ABSTRACT FROM AUTHOR]

We consider the two-opinion voter model on a regular random graph with n vertices and degree d = 3. It is known that consensus is reached on time scale n and that on this time scale the volume of the set of vertices with one opinion evolves as a Fisher-Wright diffusion. We are interested in the evolution of the number of discordant edges (i.e., edges linking vertices with different opinions), which can be thought as the perimeter of the set of vertices with one opinion, and is the key observable capturing how consensus is reached. We show that if initially the two opinions are drawn independently from a Bernoulli distribution with parameter u (0, 1), then on time scale 1 the fraction of discordant edges decreases and stabilises to a value that depends on d and u, and is related to the meeting time of two random walks on an infinite tree of degree d starting from two neighbouring vertices. Moreover, we show that on time scale n the fraction of discordant edges moves away from the constant plateau and converges to zero in an exponential fashion. Our proofs exploit the classical dual system of coalescing random walks and use ideas from Cooper et al. (2010) built on the so-called First Visit Time Lemma. We further introduce a novel technique to derive concentration properties from weak-dependence of coalescing random walks on moderate time scales. [ABSTRACT FROM AUTHOR]

We consider the standard model of i.i.d. bond percolation on Zd of parameter p. When p > pc where pc denotes the critical parameter, there exists almost surely a unique infinite cluster C8. Using the recent techniques of Cerf and Dembin (2022), we prove that the variance of the graph distance in C8 between two points of C8 is sublinear. This result extends the works of Benjamini et al. (2003), Benaïm and Rossignol (2008) and Damron et al. (2015) for the study of the variance of passage times in first passage percolation without moment conditions on the edge-weight distribution. [ABSTRACT FROM AUTHOR]

We study first-passage percolation on random simple triangulations and their dual maps with independent identically distributed link weights. Our main result shows that the first-passage percolation distance concentrates in an op(n1/4) window around a constant multiple of the graph distance. Our approach follows the proof strategy by Curien and Le Gall (Ann. Sci. Éc. Norm. Supér., 2019), but we have to overcome several obstacles specific to simple triangulations. [ABSTRACT FROM AUTHOR]