Related papers: Moderate deviations and laws of the iterated logar…
Consider a random walk $S_n=\sum_{i=1}^n X_i$ with independent and identically distributed real-valued increments $X_i$ of zero mean and finite variance. Assume that $X_i$ is non-lattice and has a moment of order $2+\delta$. For any $x\geq…
For any graph having a suitable uniform Poincare inequality and volume growth regularity, we establish two-sided Gaussian transition density estimates and parabolic Harnack inequality, for constant speed continuous time random walks…
On certain self-similar substrates the time behavior of a random walk is modulated by logarithmic periodic oscillations on all time scales. We show that if disorder is introduced in a way that self-similarity holds only in average, the…
There is a condition (T'), such that it is the necessary condition that a random walk in random environment is ballistic. Under this condition, we show the law of the iterated logarithm for a random walk in random environment.
In this note, we demonstrate an instance of bounded-degree graphs of size $n$, for which the total variation mixing time for the random walk is decreased by a factor of $\log n/ \log\log n$ if we multiply the edge-conductances by bounded…
We consider the almost sure asymptotic behavior of the periodogram of stationary and ergodic sequences. Under mild conditions we establish that the limsup of the periodogram properly normalized identifies almost surely the spectral density…
We study both the positively and negatively step-reinforced random walks with parameter $p$. For a step distribution $\mu$ with finite second moment, the positively step-reinforced random walk with $p\in [1/2,1)$ and the negatively…
We present an infinite family of finite planar graphs $\{X_n\}$ with degree at most five and such that for some constant $c > 0$, $$ \lambda_1(X_n) \geq c(\frac{\log \diam(X_n)}{\diam(X_n)})^2\,, $$ where $\lambda_1$ denotes the smallest…
In this paper, we establish a quenched invariance principle for the random walk on a certain class of infinite, aperiodic, oriented random planar graphs called "T-graphs" [Kenyon-Sheffield04]. These graphs appear, together with the…
Suppose $\left \{ X_{i,k}; 1\le i \le p, 1\le k \le n \right \} $ is an array of i.i.d.~real random variables. Let $\left \{ p=p_{n}; n \ge1 \right \} $ be positive integers. Consider the maximum interpoint distance $M_{n}=\max_{1\le i<…
We consider a random walk on a supercritical Galton-Watson tree with leaves, where the transition probabilities of the walk are determined by biases that are randomly assigned to the edges of the tree. The biases are chosen independently on…
Let $\{X, X_{n}; n \geq 1\}$ be a sequence of i.i.d. non-degenerate real-valued random variables with $\mathbb{E}X^{2} < \infty$. Let $S_{n} = \sum_{i=1}^{n} X_{i}$, $n \geq 1$. Let $g(\cdot): ~[0, \infty) \rightarrow [0, \infty)$ be a…
We study the statistics of the number of real eigenvalues in the elliptic deformation of the real Ginibre ensemble. As the matrix dimension grows, the law of large numbers and the central limit theorem for the number of real eigenvalues are…
In this note, we compute the probability that a two-dimensional symmetric random walk visits more vertices than expected, for deviations on scales between the mean behavior and linear growth.
Consider a one dimensional simple random walk $X=(X_n)_{n\geq0}$. We form a new simple symmetric random walk $Y=(Y_n)_{n\geq0}$ by taking sums of products of the increments of $X$ and study the two-dimensional walk…
Consider a simple symmetric random walk on the integer lattice $\mathbb{Z}$. Let $E(n)$ denote a favorite edge of the random walk at time $n$. In this paper, we study the escape rate of $E(n)$, and show that almost surely…
We investigate reflected random walks in the quarter plane, with particular emphasis on the time spent along the reflection boundary axes. Assuming the drift of the random walk lies within the cone, the local time converges -- without the…
We give a local central limit theorem for simple random walks on Z^d, including Gaussian error estimates. The detailed proof combines standard large deviation techniques with Cramer-Edgeworth expansions for lattice distributions.
We prove a moderate deviation principle for the continuous time interpolation of discrete time recursive stochastic processes. The methods of proof are somewhat different from the corresponding large deviation result, and in particular the…
These notes were written for the mini-course "Extrema of log-correlated random variables: Principles and Examples" at the Introductory School held in January 2015 at the Centre International de Rencontres Math\'ematiques in Marseille. There…