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.

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Seminar cancelled due to spring break.

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