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Given $n>0$, let $S\subset [0,1]^2$ be a set of $n$ points, chosen uniformly at random. Let $R\cup B$ be a random partition, or coloring, of $S$ in which each point of $S$ is included in $R$ uniformly at random with probability $1/2$.…
We address the problem of sampling colorings of a graph $G$ by Markov chain simulation. For most of the article we restrict attention to proper $q$-colorings of a path on $n$ vertices (in statistical physics terms, the one-dimensional…
A well-studied randomized election algorithm proceeds as follows: In each round the remaining candidates each toss a coin and leave the competition if they obtain heads. Of interest is the number of rounds required and the number of…
In probability theory, the independence is a very fundamental concept, but with a little mystery. People can always easily manipulate it logistically but not geometrically, especially when it comes to the independence relationships among…
Consider a continuous time Markov chain with rates Q in the state space \Lambda\cup\{0\} with 0 as an absorbing state. In the associated Fleming-Viot process N particles evolve independently in \Lambda with rates Q until one of them…
Disordered pinning models deal with the (de)localization tran- sition of a polymer in interaction with a heterogeneous interface. In this paper, we focus on two models where the inhomogeneities at the interface are not independent but given…
We study the sample complexity of learning revenue-optimal multi-item auctions. We obtain the first set of positive results that go beyond the standard but unrealistic setting of item-independence. In particular, we consider settings where…
This simple note lays out a few observations which are well known in many ways but may not have been said in quite this way before. The basic idea is that when comparing two different Markov chains it is useful to couple them is such a way…
Reinforcement Learning Algorithms are predominantly developed for stationary environments, and the limited literature that considers nonstationary environments often involves specific assumptions about changes that can occur in transition…
Diffusion models have achieved huge empirical success in data generation tasks. Recently, some efforts have been made to adapt the framework of diffusion models to discrete state space, providing a more natural approach for modeling…
This paper shows the existence of independent random matching of a large (continuum) population in both static and dynamic systems, which has been popular in the economics and genetics literatures. We construct a joint agent-probability…
The Prisoner's Dilemma, a 2-person game in which the players can either cooperate or defect, is a common paradigm for studying the evolution of cooperation, when individuals exhibit variable degrees of cooperation. It is known that in the…
The well-known Hammersley-Clifford theorem states (under certain conditions) that any Markov random field is a Gibbs state for a nearest neighbor interaction. In this paper we study Markov random fields for which the proof of the…
We prove a non-asymptotic central limit theorem for vector-valued martingale differences using Stein's method, and use Poisson's equation to extend the result to functions of Markov Chains. We then show that these results can be applied to…
Markov Chain Monte Carlo (MCMC) algorithms are often used for approximate inference inside learning, but their slow mixing can be difficult to diagnose and the approximations can seriously degrade learning. To alleviate these issues, we…
How many parameters are required for a model to execute a given task? It has been argued that large language models, pre-trained via self-supervised learning, exhibit emergent capabilities such as multi-step reasoning as their number of…
We consider a two-type oriented competition model on the first quadrant of the two-dimensional integer lattice. Each vertex of the space may contain only one particle of either Red type or Blue type. A vertex flips to the color of a…
The Prisoner's Dilemma (PD) game is used in several fields due to the emergence of cooperation among selfish players. Here, we have considered a one-dimensional lattice, where each cell represents a player, that can cooperate or defect.…
A multi-armed bandit with finitely many arms is studied when each arm is a homogeneous Markov process on an underlying finite state space. The transition law of one of the arms, referred to as the odd arm, is different from the common…
We examine a discrete-time Markovian particle system on the quarter-plane introduced by M. Defosseux. The vertical boundary acts as a reflecting wall. The particle system lies in the Anisotropic Kardar-Parisi-Zhang with a wall universality…