Related papers: Selected problems in probability theory
We introduce and investigate the escape problem for random walkers that may eventually die, decay, bleach, or lose activity during their diffusion towards an escape or reactive region on the boundary of a confining domain. In the case of a…
This paper is concerned with the limit theory of the extreme order statistics derived from random walks. We establish the joint convergence of the order statistics near the minimum of a random walk in terms of the Feller chains. Detailed…
We consider two-dimensional percolation in the scaling limit close to criticality and use integrable field theory to obtain universal predictions for the probability that at least one cluster crosses between opposite sides of a rectangle of…
There is a close connection between intersections of Brownian motion paths and percolation on trees. Recently, ideas from probability on trees were an important component of the multifractal analysis of Brownian occupation measure, in joint…
We describe a list of open problems in random matrix theory and the theory of integrable systems that was presented at the conference Asymptotics in Integrable Systems, Random Matrices and Random Processes and Universality, Centre de…
Prediction is a complex notion, and different predictors (such as people, computer programs, and probabilistic theories) can pursue very different goals. In this paper I will review some popular kinds of prediction and argue that the theory…
A celebrated result in probability theory is that a simple symmetric random walk on the $d$-dimensional lattice $\mathbb{Z}^d$ is recurrent for $d=1,2$ and transient for $d\geq 3$. In this note, we derive a closed-form expression, in terms…
This text provides an introduction to distributed local algorithms -- an area at the intersection of theoretical computer science and discrete mathematics. We collect recent results in the area and demonstrate how they lead to a clean…
We present an overview of possible reasons for the appearance of heavy-tailed distributions in applications to the natural sciences. These distributions include the laws of Pareto, Lotka, and some new ones. The reasons are illustrated using…
We discuss the concept of discrete scale invariance and how it leads to complex critical exponents (or dimensions), i.e. to the log-periodic corrections to scaling. After their initial suggestion as formal solutions of renormalization group…
Walks in a directed graph can be given a partially ordered structure that extends to possibly unconnected objects, called hikes. Studying the incidence algebra on this poset reveals unsuspected relations between walks and self-avoiding…
We consider the general branching random walk under minimal assumptions, which in particular guarantee that the empirical particle distribution admits an almost sure central limit theorem. For such a process, we study the large time decay…
Percolation is one of the simplest and nicest models in probability theory/statistical mechanics which exhibits critical phenomena. Dynamical percolation is a model where a simple time dynamics is added to the (ordinary) percolation model.…
Sampling theory concerns the problem of reconstruction of functions from the knowledge of their values at some discrete set of points. In this paper we derive an orthogonal sampling theory and associated Lagrange interpolation formulae from…
Solitons, which describe the propagation of concentrated beams of light through nonlinear media, can exhibit a variety of behaviors as a result of the intrinsic dissipation, diffraction, and the nonlinear effects. One of these phenomena,…
Spreading of either information or matter can often be treated as a network problem. It can be of great importance to be able to estimate the likelihood that spreading through a network reaches essentially the entire network while still not…
Run-and-tumble particles confined between two walls seem like a simple enough problem to possess analytical tractability. Yet up to date, no satisfactory analysis is available for dimensions higher than one. This work contributes to the…
We present a perturbation theory by extending a prescription due to Feynman for computing the probability density function for the random flight motion. The method can be applied to a wide variety of otherwise difficult circumstances. The…
In probabilistic logic entailments, even moderate size problems can yield linear constraint systems with so many variables that exact methods are impractical. This difficulty can be remedied in many cases of interest by introducing a three…
There have been extensive studies of a random walk among a field of immobile traps (or obstacles), where one is interested in the probability of survival as well as the law of the random walk conditioned on its survival up to time $t$. In…