Related papers: Giant Component and Vacant Set for Random Walk on …
Consider $N$ points randomly distributed along a line segment of unitary length. A walker explores this disordered medium moving according to a partially self-avoiding deterministic walk. The walker, with memory $\mu$, leaves from the…
Let $G$ be an infinite, locally finite graph. We investigate the relation between supercritical, transient branching random walk and the Martin boundary of its underlying random walk. We show results regarding the typical asymptotic…
Let $G$ be a discrete group, $\mu$ a measure on $G$ and $X$ a proper CAT(0) space. We show that if $G$ acts non-elementarily with a rank one element on $X$, then the pushforward $\{Z_n o \}_n$ to $X$ of the random walk generated by $\mu$…
Activated Random Walk is a system of interacting particles which presents a phase transition and a conjectured phenomenon of self-organized criticality. In this note, we prove that, in dimension 1, in the supercritical case, when a segment…
This paper focuses on the problem of modeling for small world effect on complex networks. Let's consider the supercritical Poisson continuous percolation on $d$-dimensional torus $T^d_n$ with volume $n^d$. By adding "long edges (short…
We study the random walk on dynamical percolation of $\mathbb{Z}^d$ (resp., the two-dimensional triangular lattice $\mathcal{T}$), where each edge (resp., each site) can be either open or closed, refreshing its status at rate $\mu\in…
We show the existence of rigid combinatorial objects which previously were not known to exist. Specifically, for a wide range of the underlying parameters, we show the existence of non-trivial orthogonal arrays, $t$-designs, and $t$-wise…
We consider the simple random walk on the infinite cluster of a general class of percolation models on $\mathbb{Z}^d$, $d\geq 3$, including Bernoulli percolation as well as models with strong, algebraically decaying correlations. For almost…
We consider random walks in which the walk originates in one set of nodes and then continues until it reaches one or more nodes in a target set. The time required for the walk to reach the target set is of interest in understanding the…
We consider the simple random walk on the (unique) infinite cluster of super-critical bond percolation in $\Z^d$ with $d\ge2$. We prove that, for almost every percolation configuration, the path distribution of the walk converges weakly to…
We consider (random) walks in a multidimensional orthant. Using the idea of universality in probability theory, one can associate a unique polyhedral domain to any given walk model. We use this connection to prove two sets of new results.…
We consider random walks on $\Z^d$ among nearest-neighbor random conductances which are i.i.d., positive, bounded uniformly from above but whose support extends all the way to zero. Our focus is on the detailed properties of the paths of…
Paths are important structural elements in complex networks because they are finite (unlike walks), related to effective node coverage (minimum spanning trees), and can be understood as being dual to star connectivity. This article…
This work proves that the fluctuations of the cover time of simple random walk in the discrete torus of dimension at least three with large side-length are governed by the Gumbel extreme value distribution. This result was conjectured for…
We consider random interlacements on Z^d, with d bigger or equal to 3, when their vacant set is in a strongly percolative regime. We derive an asymptotic upper bound on the probability that the random interlacements disconnect a box of…
Rotor walk is deterministic counterpart of random walk on graphs. We study that under a certain initial configuration in Z^d, n particles perform rotor walks from the origin consecutively. They would stop if they hit the origin or infinity.…
A temporal (directed) graph is a graph whose edges are available only at specific times during its lifetime, $\tau$. Paths are sequences of adjacent edges whose appearing times are either strictly increasing or non-strictly increasingly…
We study percolation of two-sided level sets for the discrete Gaussian free field (DGFF) in 2D. For a DGFF $\varphi$ defined in a box $B_N$ with side length $N$, for $C$ large enough, there exist low crossings in the set of vertices $z$…
Several cycle lexicographical orders are found to describe the relative likelihood of elements of the random walks on the symmetric group generated by the conjugacy classes of transpositions, 3-cycles, and n-cycles. Spectral analysis finds…
Let (Z_n)_{n\in\N_0} be a d-dimensional random walk in random scenery, i.e., Z_n=\sum_{k=0}^{n-1}Y_{S_k} with (S_k)_{k\in\N_0} a random walk in Z^d and (Y_z)_{z\in Z^d} an i.i.d. scenery, independent of the walk. We assume that the random…