相关论文: A multivariate extension of the Erd\"os-Taylor the…

We consider the nonlinear Schr\"odinger equation set on a flat torus, in the regime which is conjectured to lead to the kinetic wave equation; in particular, the data are random, and spread up to high frequency in a weakly nonlinear regime.…

This paper studies particle propagation in a one-dimensional inhomogeneous medium where the laws of motion are generated by chaotic and deterministic local maps. Assuming that the particle's initial location is random and uniformly…

We prove that the simple random walk on the uniform infinite planar triangulation (UIPT) typically travels graph distance at most $n^{1/4 + o_n(1)}$ in $n$ units of time. Together with the complementary lower bound proven by Gwynne and…

Consider a critical branching random walk on $\mathbb{Z}^d$, $d\geq 1$, started with a single particle at the origin, and let $L(x)$ be the total number of particles that ever visit a vertex $x$. We study the tail of $L(x)$ under suitable…

Given a sequence $\mathscr{A}=\{a_0<a_1<a_2\ldots\}\subseteq \mathbb{N}$, let $r_{\mathscr{A},h}(n)$ denote the number of ways $n$ can be written as the sum of $h$ elements of $\mathscr{A}$. Fixing $h\geq 2$, we show that if $f$ is a…

The log-partition function $ \log W_N(\beta)$ of the two-dimensional directed polymer in random environment is known to converge in distribution to a normal distribution when considering temperature in the subcritical regime…

We study products of random isometries acting on Euclidean space. Building on previous work of the second author, we prove a local limit theorem for balls of shrinking radius with exponential speed under the assumption that a Markov…

Random walks on expanders play a crucial role in Markov Chain Monte Carlo algorithms, derandomization, graph theory, and distributed computing. A desirable property is that they are rapidly mixing, which is equivalent to having a spectral…

The probability of a random walker to return to its starting point in dimensions one and two is unity, a theorem first proven by G. Polya. The recurrence probability -- the probability to be found at the origin at a time t, is a power law…

We study a random walk in a N dimensional hypercube and exhibit results about stopping times when N diverges. The first theorem discusses the time in which two coupling processes spend to meet. A corollary provides a majorant for the…

The classical Erd\H{o}s-Littlewood-Offord theorem says that for nonzero vectors $a_1,\dots,a_n\in \mathbb{R}^d$, any $x\in \mathbb{R}^d$, and uniformly random $(\xi_1,\dots,\xi_n)\in\{-1,1\}^n$, we have…

Place an obstacle with probability $1-p$ independently at each vertex of $\mathbb Z^d$, and run a simple random walk until hitting one of the obstacles. For $d\geq 2$ and $p$ strictly above the critical threshold for site percolation, we…

We extend a result of Lyons (2016) from fractional tiling of finite graphs to a version for infinite random graphs. The most general result is as follows. Let $\bf P$ be a unimodular probability measure on rooted networks $(G, o)$ with…

Analytical results for the distribution of first hitting times of random walks on Erd\H{o}s-R\'enyi networks are presented. Starting from a random initial node, a random walker hops between adjacent nodes until it hits a node which it has…

We give several quantum dynamical analogs of the classical Kronecker-Weyl theorem, which says that the trajectory of free motion on the torus along almost every direction tends to equidistribute. As a quantum analog, we study the quantum…

We study graph-theoretic properties of the trace of a random walk on a random graph. We show that for any $\varepsilon>0$ there exists $C>1$ such that the trace of the simple random walk of length $(1+\varepsilon)n\ln{n}$ on the random…

We study the statistics of the number of records R_{n,N} for N identical and independent symmetric discrete-time random walks of n steps in one dimension, all starting at the origin at step 0. At each time step, each walker jumps by a…

Let $X_1, X_2, \dots$ be independent, identically distributed random variables taking values from a compact metrizable group $G$. We prove that the random walk $S_k=X_1 X_2 \cdots X_k$, $k=1,2,\dots$ equidistributes in any given Borel…

We show that the classical Pollard rho algorithm for discrete logarithms produces a collision in expected time O(sqrt(n)(log n)^3). This is the first nontrivial rigorous estimate for the collision probability for the unaltered Pollard rho…

Let $X_1,X_2,...$ be a sequence of independent and identically distributed random variables, and put $S_n=X_1+...+X_n$. Under some conditions on the positive sequence $\tau_n$ and the positive increasing sequence $a_n$, we give necessary…