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The seminar is jointly sponsored by Temple and Penn. The organizers are Brian Rider and Atilla Yilmaz (Temple), and Jiaoyang Huang, Jiaqi Liu, Robin Pemantle and Xin Sun (Penn).
Talks are Tuesdays 3:30 - 4:30 pm and are held either in Wachman Hall (Temple) or David Rittenhouse Lab (Penn) as indicated below.
For a chronological listing of the talks, click the year above.
Mokshay Madiman, University of Delaware
The ordered elements in several one-dimensional Coulomb gas ensembles arising in probability and mathematical physics are shown to have log-concave distributions. Examples include the beta ensembles with convex potentials (in the continuous setting) and the orthogonal polynomial ensembles (in the discrete setting). In particular, we prove the log-concavity of the Tracy-Widom β distributions, Airy distribution, and Airy-2 process. Log-concavity of last passage times in percolation is proven using their connection to Meixner ensembles. We then obtain the log-concavity of top rows of Young diagrams under Poissonized Plancherel measure, which is the Poissonized version of a conjecture of Chen. This is ongoing joint work with Jnaneshwar Baslingker and Manjunath Krishnapur.
Zhengjiang Lin, Courant Institute, NYU
We will briefly discuss some asymptotic topological statistics of Gaussian random zero sets, which include a random distribution on knots as a special case. We will also discuss some results on zero sets of random Laplacian eigenfunctions, which are related to Courant’s nodal domain theorem and Milnor-Thom’s theorem on Betti numbers of real algebraic varieties.
Yuxin Zhou, University of Chicago
In this talk, I will discuss the spherical mixed p-spin glass model at zero temperature. I will present some recent results that classify the possible structure of the functional ordered parameter. For spherical p+s spin glasses, we classify all possibilities for the Parisi measure as a function of the model. Moreover, for the spherical spin models with n components, the Parisi measure at zero temperature is at most n-RSB or n-FRSB. Some of these results are jointly with Antonio Auffinger.
Brian Rider, Temple University
There are three basic flavors of local limit theorems in random matrix theory, connected to the spectral bulk and the so-called soft and hard edges. There also abound a collection of more exotic limits which arise in models that posses degenerate (or “non-regular”) points in their equilibrium measure. What is more, there is typically a natural double scaling about these non-regular points, producing limit laws that transition between the more familiar basic flavors. I will describe a general beta matrix model for which the appropriate double scaling limit is the Stochastic Airy Operator, conditioned on having no eigenvalues below a fixed level. I know of no other random matrix double scaling fully characterized outside of beta = 2. This is work in progress with J. Ramirez (University of Costa Rica).
Fraydoun Rezakhanlou, University of California, Berkeley
According to Conley-Zehnder's theorem, any periodic Hamiltonian ODE in $\mathbb{R}^{2n}$ has at least $2n+1$ geometrically distinct periodic orbits. For a stochastically stationary Hamiltonian ODE, the set of periodic orbits yields a translation invariant random process. In this talk, I will discuss an ergodic theorem for the density of periodic orbits, and formulate some open questions which are the stochastic variants of Conley-Zehnder's theorem.
Sayan Das, University of Chicago
We consider one-dimensional simple random walks whose all one-step transition probabilities are iid [0,1]-valued mean 1/2 random variables. In this talk, we will explain how under a certain moderate deviation scaling the quenched density of the walk converges weakly to the Stochastic Heat Equation with multiplicative noise. Our result captures universality in the sense that it holds for all non-trivial laws for random environments. Time permitting, we will discuss briefly how our proof techniques depart from the existing techniques in the literature. Based on a joint work with Hindy Drillick and Shalin Parekh.
Oanh Nguyen, Brown University
The contact process serves as a model for the spread of epidemics on networks, with three popular variations: the Susceptible-Infected-Recovered-Susceptible (SIRS), SIR, and SIS. Our focus lies in understanding the temporal evolution of these processes, especially regarding survival time and its associated phase transitions. I will provide a brief overview of related literature, recent progress, and open problems.
Paul Jung, Fordham University
We discuss a class of partially exchangeable random arrays which generalizes the notion of hierarchical exchangeability introduced in Austin and Panchenko (2014). We say that our partially exchangeable arrays are DAG-exchangeable since their partially exchangeable structure is governed by a collection of Directed Acyclic Graphs (DAG). More specifically, such a random array is indexed by N^|V| for some DAG, G = (V,E), and its exchangeability structure is governed by the edge set E. We prove a representation theorem which generalizes the Aldous-Hoover and Austin-Panchenko representation theorems.
Zhongyang Li, University of Connecticut
We investigate limit shapes and height fluctuations in statistical mechanical models, such as dimers and lecture hall tableaux, through the asymptotics of symmetric polynomials. Confirming a conjecture by Corteel, Keating, and Nicoletti, we show that the rescaled height functions' slopes in the scaling limit of lecture hall tableaux adhere to a complex Burgers equation.
Yujin Kim, Courant Institute, NYU
The extremal process of branching Brownian motion (BBM) —i.e., the collection of particles furthest from the origin— has gained lots of attention in dimension $d = 1$ due to its significance to the universality class of log-correlated fields, as well as to certain PDEs. In recent years, a description of the extrema of BBM in $d > 1$ has been obtained. In this talk, we address the following geometrical question that can only be asked in $d > 1$. Generate a BBM at a large time, and draw the outer envelope of the cloud of particles: what is its shape? Macroscopically, the shape is known to be a sphere; however, we focus on the outer envelope around an extremal point— the "front" of the BBM. We describe the scaling limit for the front, with scaling exponent 3/2, as an explicit, rotationally-symmetric random surface. Based on joint works with Julien Berestycki, Bastien Mallein, Eyal Lubetzky, and Ofer Zeitouni.
Dana Randall, Georgia Tech
Programmable matter explores how collections of computationally limited agents acting locally and asynchronously can achieve some useful coordinated behavior. We take a stochastic approach using techniques from randomized algorithms and statistical physics to develop distributed algorithms for emergent collective behaviors that give guarantees and are robust to failures.
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