# Probability Seminar

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# 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.

## 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, in person: 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)

Path properties of the KPZ Equation and related polymers

The KPZ equation is a fundamental stochastic PDE that can be viewed as the log-partition function of continuum directed random polymer (CDRP). In this talk, we will first focus on the fractal properties of the tall peaks of the KPZ equation. This is based on separate joint works with Li-Cheng Tsai and Promit Ghosal. In the second part of the talk, we will study the KPZ equation through the lens of polymers. In particular, we will discuss localization aspects of CDRP that will shed light on certain properties of the KPZ equation such as ergodicity and limiting Bessel behaviors around the maximum. This is based on joint work with Weitao Zhu.

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

Potential-weighted connective constants and uniqueness of Gibbs measures

Classical gases (or Gibbs point processes) are models of gases or fluids, with particles interacting in the continuum via a density against background Poisson processes. The major questions about these models are about phase transitions, and while these models have been studied in mathematics for over 70 years, some of the most fundamental questions remain open. After giving some background on these models, I will describe a new method for proving absence of phase transition (uniqueness of infinite volume Gibbs measures) in the low-density regime of processes interacting via a repulsive pair potential. The method involves defining a new quantity, the "potential-weighted connective constant", and is motivated by techniques from computer science. Joint work with Marcus Michelen.

## April 7, 2022, in person and on ZOOM: Eliza O'Reilly (Caltech)

Stochastic Geometry for Machine Learning

The Mondrian process is a stochastic process that produces a recursive partition of space with random axis-aligned cuts. It has been used in machine learning to build random forests and Laplace kernel approximations. The construction allows for efficient online algorithms, but the restriction to axis-aligned cuts does not capture dependencies between features. By viewing the Mondrian as a special case of the stable under iterated (STIT) process in stochastic geometry, we resolve open questions about the generalization of cut directions. We utilize the theory of stationary random tessellations to show that STIT random forests achieve minimax rates for Lipschitz and $C^2$ functions and STIT random features approximate a large class of stationary kernels. This work opens many new questions at the intersection of stochastic geometry and machine learning. Based on joint work with Ngoc Mai Tran.

## April 14, 2022, ZOOM: Eric Foxall (UBC-Okanagan)

Bifurcation theory of density-dependent Markov chains, well-mixed stochastic population models, and diffusively perturbed dynamical systems

Abstract: A common choice for modeling a finite, interacting population of individuals is to specify a variable system size parameter $N$, and to otherwise assume that interactions are well-mixed. When the model is cast in continuous time, has finitely many types, and is otherwise fairly simple, the vector $X$ of population sizes is a density-dependent Markov chain (DDMC), and the population density vector $\frac{X}{N}$ can be roughly seen as a diffusive perturbation of a deterministic system, with noise parameter $\varepsilon = 1/\sqrt{N}$. The law of large numbers and central limit theorem of these models has been known since the work of Thomas Kurtz in the 1970s. Here, we study bifurcations of parametrized models of this type, as a function of $\varepsilon$. We introduce the notion of limit scales to understand the shape of fluctuations in and around the bifurcation point, and give a three-step framework for finding them. The result is an enhanced bifurcation diagram that encodes this information with the help of a few well-chosen functions. We develop the theory in detail for one-dimensional models and illustrate it for simple bifurcation types including transcritical, saddle-node, pitchfork, and the Hopf bifurcation.

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

Metric growth dynamics in Liouville quantum gravity

Abstract: Liouville quantum gravity (LQG) is a canonical model of random geometry. Associated with the planar Gaussian free field, this geometry with conformal symmetries was introduced in the physics literature by Polyakov in the 80’s and is conjectured to describe the scaling limit of random planar maps. In this talk, I will introduce LQG as a metric measure space and discuss recent results on the associated metric growth dynamics. The primary focus will be on the dynamics of the trace of the free field on the boundary of growing LQG balls. Based on a joint work with Julien Dubédat.

## April 28, 2022, in person: Amol Aggarwal (Columbia/IAS)

ASEP Speed Process

We consider the asymmetric simple exclusion process, started under step initial data, with a single second class particle at the origin. We show that the trajectory of the second class particle almost surely follows a line, whose slope is a uniform random variable in [-1, 1]. The proof is based on a combination of probabilistic couplings and effective hydrodynamic bounds arising from the ASEP's solvability under specific choices of initial data. This is joint work with Ivan Corwin and Promit Ghosal.

## May 5, 2022, in person: David Harper (Georgia Tech)

Dynamical critical $2d$ first-passage percolation

In first-passage percolation (FPP), we let $\tau_v$ be i.i.d. nonnegative weights on the vertices of a graph and study the weight of the minimal path between distant vertices. If $F$ is the distribution function of $\tau_v$, there are different regimes: if $F(0)$ is small, this weight typically grows like a linear function of the distance, and when $F(0)$ is large, the weight is typically of order one. In between these is the critical regime in which the weight can diverge, but does so sublinearly. This talk will consider a dynamical version of critical FPP on the triangular lattice where vertices resample their weights according to independent rate-one Poisson processes. We will discuss results which show that if sum of $F^{-1}(1/2+1/2^k)$ diverges, then a.s. there are exceptional times at which the weight grows atypically, but if sum of $k^{7/8} F^{-1}(1/2+1/2^k)$ converges, then a.s. there are no such times. Furthermore, in the former case, we compute the Hausdorff and Minkowski dimensions of the exceptional set and show that they can be but need not be equal. These results show a wider range of dynamical behavior than one sees in subcritical (usual) FPP. This is a joint work with M. Damron, J. Hanson, W.-K. Lam.