Related papers: On the upper tail large deviation rate function fo…

Many real-world networks exhibit the so-called small-world phenomenon: their typical distances are much smaller than their sizes. One mathematical model for this phenomenon is a long-range percolation graph on a $d$-dimensional box $\{0, 1,…

We consider the standard first passage percolation model on $\mathbb Z^d$ with bounded and bounded away from zero weights. We show that the rescaled passage time $\widetilde{\mathbf T}_{n,X}$ restricted to a compact set $X$ satisfies a…

We consider the extinction time of the contact process on increasing sequences of finite graphs obtained from a variety of random graph models. Under the assumption that the infection rate is above the critical value for the process on the…

We identify the asymptotic distribution of the chemical distance in high-dimensional critical Bernoulli percolation. Namely, we show that the distance between the origin and a distant vertex conditioned to lie in the cluster of the origin…

In this paper, we study lower tail probabilities of the height function $\mathfrak{h}(M,N)$ of the stochastic six-vertex model. We introduce a novel combinatorial approach to demonstrate that the tail probabilities…

We show quenched large deviations for the simple random walk on a certain class of percolations with long-range correlations. This class contains the supercritical Bernoulli percolations, the model considered by Drewitz, R'ath and…

We consider an i.i.d. supercritical bond percolation on Z^d , every edge is open with a probability p > p\_c (d), where p\_c (d) denotes the critical parameter for this percolation. We know that there exists almost surely a unique infinite…

We provide sufficient conditions for a regular graph $G$ of growing degree $d$, guaranteeing a phase transition in its random subgraph $G_p$ similar to that of $G(n,p)$ when $p\cdot d\approx 1$. These conditions capture several well-studied…

We perform an analytical analysis of the long-range degree correlation of the giant component in an uncorrelated random network by employing generating functions. By introducing a characteristic length, we find that a pair of nodes in the…

We consider a jump-diffusion process on a bounded domain with reflection at the boundary, and establish long-term results for a general additive process of its path. This includes the long-term behaviour of its occupation time in the…

Following the recent investigations of Baik and Suidan in \cite{baik2005gcl} and Bodineau and Martin in \cite{bodineau2005upl}, we prove large deviation properties for a last-passage percolation model in $\mathbb{Z}^{2}_{+}$ whose paths are…

The focus of this article is on the different behavior of large deviations of random subadditive functionals above the mean versus large deviations below the mean in two random media models. We consider the point-to-point first passage…

There is a renewed surge in percolation-induced transport properties of diverse nano-particle composites (cf. RSC Nanoscience & Nanotechnology Series, Paul O'Brien Editor-in-Chief). We note in particular a broad interest in nano-composites…

Let $d\ge 3$ be a fixed integer, $p\in (0,1)$, and let $n\geq 1$ be a positive integer such that $dn$ is even. Let $\mathbb{G}(n, d, p)$ be a (random) graph on $n$ vertices obtained by drawing uniformly at random a $d$-regular (simple)…

We consider spread-out models of self-avoiding walk, bond percolation, lattice trees and bond lattice animals on the d-dimensional hyper cubic lattice having long finite-range connections, above their upper critical dimensions d=4…

Let $X_k$ denote the number of $k$-term arithmetic progressions in a random subset of $\mathbb{Z}/N\mathbb{Z}$ or $\{1, \dots, N\}$ where every element is included independently with probability $p$. We determine the asymptotics of $\log…

We consider supercritical bond percolation in $\mathbb{Z}^d$ for $d \geq 3$. The origin lies in a finite open cluster with positive probability, and, when it does, the diameter of this cluster has an exponentially decaying tail. For each…

A systematic convolutional encoder of rate $(n-1)/n$ and maximum degree $D$ generates a code of free distance at most ${\cal D} = D+2$ and, at best, a column distance profile (CDP) of $[2,3,\ldots,{\cal D}]$. A code is \emph{Maximum…

Consider the upper tail probability that the homomorphism count of a fixed graph $H$ within a large sparse random graph $G_n$ exceeds its expected value by a fixed factor $1+\delta$. Going beyond the Erd\H{o}s-R\'enyi model, we establish…

We study the tail behavior for the maximum of discrete Gaussian free field on a 2D box with Dirichlet boundary condition after centering by its expectation. We show that it exhibits an exponential decay for the right tail and a double…