Uniform dilations modulo a prime
26/02/2023
How large must k=k(epsilon,p) be, so that every set X of k integers in [1,p] must have a dilation nX modulo p, where all gaps are at most of length epsilon*p?
Yuval Peres
Mathematical open problems
This page is devoted to open problems in probability, analysis and computing that I have encountered over the years, which are still unsolved (to the best of my knowledge). If you know solutions to these problems, or have ideas on how to attack them, please write to me at yuval@yuvalperes.com
About Yuval Peres
Yuval Peres is a mathematician working in probability theory, analysis, combinatorics and computer science. He has co-authored books on Markov chains, Brownian notion, Fractals, Probability on graphs and game theory. His most cited research includes papers relating game theory to PDE, analyzing cover time and mixing time of random walks, broadcasting on trees, Bernoulli convolutions and related fractals, and uniform spanning trees.
He has published five papers in the Annals of Mathematics (see https://annals.math.princeton.edu). He has more than 200 coauthors, including two fields medalists, and recipients of the Abel prize and the Shaw prize.
Recent Open Problems
Recurrence and range of rotor walk
26/02/2023
Consider Glauber dynamics for the Ising model on a finite graph. Which pair of initial states maximizes the total variation distance at time 
?
Maximizing total variation distance for the Ising model
25/12/2022
Consider Glauber dynamics for the Ising model on a finite graph. Which pair of initial states maximizes the total variation distance at time ?
The power law for the mixing time of Swendsen-Wang dynamics for the Ising model
25/12/2022
Does Swendsen-Wang dynamics for the Ising model on an 
-vertex graph always mix in time 
?
A Markov chain arising from Gaussian elimination
03/10/2022
Consider a Markov chain on the set of n by n invertible matrices mod 2, where in each step, two distinct rows are chosen uniformly at random, and the first row is added to the second mod 2. How does the mixing time grow with n? And how does it change if only test functions that are computable in polynomial time are allowed?
Erdos problems on lacunary sequences
03/10/2022
Given a lacunary sequence of integers, Erdos asked for the chromatic number of the graph on the integers where two integers are linked by an edge is their difference is in the sequence. determining the optimal bound for this chromatic number is closely related to a question on Diophantine approximation considered by Khinchin, Erdos and Katznelson.