Probability Seminar

Spring 2022

Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom

We usually end for questions at 3:20 PM.

ZOOM LINK. Valid only for online seminars.

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February 3, 2022, ZOOM: Zhipeng Liu (University of Kansas)

One-point distribution of the geodesic in directed last passage percolation

In the recent twenty years, there has been a huge development in understanding the universal law behind a family of 2d random growth models, the so-called Kardar-Parisi-Zhang (KPZ) universality class. Especially, limiting distributions of the height functions are identified for a number of models in this class. Different from the height functions, the geodesics of these models are much less understood. There were studies on the qualitative properties of the geodesics in the KPZ universality class very recently, but the precise limiting distributions of the geodesic locations remained unknown.

In this talk, we will discuss our recent results on the one-point distribution of the geodesic of a representative model in the KPZ universality class, the directed last passage percolation with iid exponential weights. We will give the explicit formula of the one-point distribution of the geodesic location joint with the last passage times, and its limit when the parameters go to infinity under the KPZ scaling. The limiting distribution is believed to be universal for all the models in the KPZ universality class. We will further give some applications of our formulas.

February 10, 2022, ZOOM: Jacob Calvert (U.C. Berkeley)

Harmonic activation and transport

Models of Laplacian growth, such as diffusion-limited aggregation (DLA), describe interfaces which move in proportion to harmonic measure. I will introduce a model, called harmonic activation and transport (HAT), in which a finite subset of Z^2 is rearranged according to harmonic measure. HAT exhibits a phenomenon called collapse, whereby the diameter of the set is reduced to its logarithm over a number of steps comparable to this logarithm. I will describe how collapse can be used to prove the existence of the stationary distribution of HAT, which is supported on a class of sets viewed up to translation. Lastly, I will discuss the problem of quantifying the least positive harmonic measure as a function of set cardinality, which arises in the study of HAT, and a partial resolution of which rules out predictions about DLA from the physics literature. Based on joint work with Shirshendu Ganguly and Alan Hammond.

February 17, 2022, in person: Pax Kivimae (Northwestern University)

The Ground-State Energy and Concentration of Complexity in Spherical Bipartite Models

Bipartite spin glass models have been gaining popularity in the study of glassy systems with distinct interacting species. Recently, the annealed complexity of the pure spherical bipartite model was obtained by B. McKenna. In this talk, I will explain how to show that the low-lying complexity actually concentrates around this value, and how from this one can obtain a formula for the ground-state energy.

February 24, 2022, ZOOM: Lucas Benigni (University of Chicago)

Optimal delocalization for generalized Wigner matrices

We consider eigenvector statistics of large symmetric random matrices. When the matrix entries are sampled from independent Gaussian random variables, eigenvectors are uniformly distributed on the sphere and numerous properties can be computed exactly. In particular, we can bound their extremal coordinates with high probability. There has been an extensive amount of work on generalizing such a result, known as delocalization, to more general entry distributions. After giving a brief overview of the previous results going in this direction, we present an optimal delocalization result for matrices with sub-exponential entries for all eigenvectors. The proof is based on the dynamical method introduced by Erdos-Yau, an analysis of high moments of eigenvectors as well as new level repulsion estimates which will be presented during the talk. This is based on a joint work with P. Lopatto.

March 3, 2022, in person: David Keating (UW-Madison)

$k$-tilings of the Aztec diamond

We study $k$-tilings ($k$-tuples of domino tilings) of the Aztec diamond of rank $m$. We assign a weight to each $k$-tiling, depending on the number of vertical dominos and also on the number of "interactions" between the different tilings. We will compute the generating polynomials of the $k$-tilings by relating them to an integrable colored vertex model. We will then prove some combinatorial results about $k$-tilings in certain limits of the interaction strength.

March 10, 2022, format TBD: Qiang Wu (University of Illinois Urbana-Champaign)

Mean field spin glass models under weak external field

We study the fluctuation and limiting distribution of free energy in mean-field spin glass models with Ising spins under weak external fields. We prove that at high temperature, there are three regimes concerning the strength of external field $h \approx \rho N^{-\alpha}$ with $\rho,\alpha\in (0,\infty)$. In the super-critical regime $\alpha < 1/4$, the variance of the log-partition function is $\approx N^{1-4\alpha}$. In the critical regime $\alpha = 1/4$, the fluctuation is of constant order but depends on $\rho$. Whereas, in the sub-critical regime $\alpha>1/4$, the variance is $\Theta(1)$ and does not depend on $\rho$. We explicitly express the asymptotic mean and variance in all three regimes and prove Gaussian central limit theorems. Our proofs mainly follow two approaches. One utilizes quadratic coupling and Guerra's interpolation scheme for Gaussian disorder, extending to many other spin glass models. However, this approach can prove the CLT only at very high temperatures. The other one is a cluster-based approach for general symmetric disorders, first used in the seminal work of Aizenman, Lebowitz, and Ruelle (Comm. Math. Phys. 112 (1987), no. 1, 3-20) for the zero external field case. It was believed that this approach does not work if the external field is present. We show that if the external field is present but not too strong, it still works with a new cluster structure. In particular, we prove the CLT up to the critical temperature in the Sherrington-Kirkpatrick (SK) model when $\alpha \ge 1/4$. We further address the generality of this cluster-based approach. Specifically, we give similar results for the multi-species SK model and diluted SK model. Based on joint work with Partha S. Dey.

March 24, 2022, in person: Sayan Das (Columbia University)

TBA

March 31, 2022, in person: Will Perkins (University of Illinois Chicago)

TBA

April 7, 2022, ZOOM: Eliza O'Reilly (Caltech)

TBA

April 14, 2022, in person: Eric Foxall (UBC-Okanagan)

TBA

April 21, 2022, ZOOM: Hugo Falconet (NYU)

TBA

Past Seminars