Berkeley Student Probability Seminar, Spring 2026

Deciding order of speakers and topics.

I will review basic definitions and results of markov chains, such as the markov property, irreducibility, recurrence and transience, and stationary distributions

I will review mixing time, total variation, and couplings, following LPW Chapters 4&5.

We'll study the lazy random walk on the hypercube, which Zoe introduced last time, and get a sharp understanding of the mixing behavior, showing that the chain mixes at times t = 1/2 n log n + s n as s goes from very negative to very positive. In particular, this is an example of the cutoff phenomenon. We will also use the ideas developed here to understand a generalization of this chain to a model with interaction called the Curie-Weiss model, which is a basic model of a ferromagnet. We will see that at the critical temperature, the mixing time of this chain is of order n^{3/2}, and there is no cutoff.

Talk Notes

Following Vilas’ talk about bounding mixing times using path coupling on random lazy walks on the hypercube, we will extend the techniques to analyze mixing times of the Ising model. In this talk, we will introduce the Ising model, prove a general lower bound on its mixing time, and show fast mixing at high temperatures. This talk follows section 4.3 of Roch’s Modern Discrete Probability.

Analyzing the mixing time of a Markov chain can be difficult — what techniques are there to bound the mixing time? We’ll discuss classical bounds for the mixing time of a reversible Markov chain in terms of the spectral gap (LPW Ch. 12).

But analyzing the spectral gap of a Markov chain can be difficult — what techniques are there to bound the spectral gap? For Markov chains arising from Glauber dynamics of ‘high-dimensional’ distributions, we can study the spectrum of an associated influence matrix to bound the spectral gap. We start to introduce the language needed for this approach, which includes verifying a property known as ‘spectral independence.

Following the previous talk on spectral approaches to bounding mixing times of Markov chains, we will continue to develop the theory of spectral independence, and in particular, the connection between spectral independence and local spectral expansion. Along the way, we will gain some intuition for the influence matrix that the framework of spectral independence studies.

We will then transition to an example where spectral independence implies rapid mixing: the hardcore model. After providing some background, it will be possible to indicate how one might actually establish spectral independence for the hardcore model in the regime where the corresponding Glauber dynamics is expected to mix rapidly.

The Gaussian Free Field (GFF) is a fundamental object in modern probability and mathematical physics, often viewed as the natural multi-dimensional generalization of the Brownian bridge. And the essay "Cover times, blanket times, and majorizing measures" gives a strong connection between cover times and blanket time of graphs and Gaussian processes. We will discuss GFF and some theorems shown in the essay.

Following the previous talk, we will continue exploring the relationship between the cover time of a graph and the maximum of the Gaussian Free Field on that graph. We will follow Zhai's paper https://arxiv.org/abs/1407.7617, which strengthens and simplifies the main result of Ding–Lee–Peres alluded to last week. This talk will be self-contained; background from the previous talk will be reviewed as needed.

Coupling two random walks such that they reach the same point (whether at the same time or not) is an interesting question and has many applications. In this talk, we will discuss the opposite question: can we couple two random walks such that they don't intersect with each other. We will first briefly show that this is impossible when d = 2 and trivial when d bigger than 4. Then we will see that such coupling exists when d = 4, which is an application of Hall's marriage theorem. The talk is based on recent work by Itai Benjamini and Gady Kozma Coupled but distant.

I will introduce a recent notion of the spectral gap of a nonreversible Markov chain, due to Sourav Chatterjee, in terms of the second smallest singular value of the generator of the chain. This definition, which is distinct from previous attempts to characterize spectral gaps of nonreversible chains, aligns nicely with the classical definition of the spectral gap of a reversible chain. I will survey some of the properties of Chatterjee's spectral gap, including its connection to both mixing times and convergence of empirical averages. I will also present several examples where this notion of spectral gap can be used to show that empirical averages converge faster than the mixing time of a chain.

This is a broad term that encompasses generalizations of standard Markov chains across different fields: in functional analysis, it refers to non-commuting Markov semigroups; in physics, to models of thermalization in open quantum systems; and in computer science, to algorithms that prepare quantum Gibbs states. An analogous but richer theory emerges, where concepts such as detailed balance and mixing time find more nuanced counterparts. I’ll survey these three points of view, show how they connect, and describe some recent breakthroughs. No knowledge of quantum mechanics will be assumed.

[Abstract]