We meet Tuesdays 3:00–4:00pm at the CUNY Graduate Center. If you are interested in joining our mailing list, please contact Matthew Junge.

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Seminar Schedule

February 11: Evita Nestoridi (Stony Brook)

Shuffling via transpositions. In their seminal work, Diaconis and Shahshahani proved that shuffling a deck of n cards sufficiently well via random transpositions takes 1/2 n log n steps. Their argument was algebraic and relied on the combinatorics of the symmetric group. In this talk, I will focus on two other shuffles, generalizing random transpositions and I will discuss the underlying combinatorics for understanding their mixing behavior and indeed proving cutoff. The talk will be based on joint works with A. Yan and S. Arfaee.

February 25: Josh Meisel (CUNY)

Cutoff for activated random walk. Self-organized criticality (SOC) is an influential theory that explains how critical behavior, like that seen in lab-tuned phase transitions, occurs throughout nature despite no apparent external tuning. The originally proposed model for SOC, the abelian sandpile, was gradually disqualified primarily due to slow mixing. Activated random walk (ARW) is a stochastic variant of the abelian sandpile conjectured to remedy this and other issues. We prove that ARW on an interval mixes as desired in a model of SOC; it exhibits cutoff at a time proportional to the critical density for ARW on Z. Based on joint work with Chris Hoffman, Toby Johnson and Matt Junge.

March 4: Roger Van Peski (Columbia)

Integer random matrices, fluctuations of random groups, and an interacting particle system. I will discuss recent work with Hoi Nguyen (https://arxiv.org/abs/2409.03099 ) on products of discrete random matrices with iid integer entries from any non-degenerate distribution. Roughly speaking, the analogues of singular values for these products converge to the single-time marginal distribution of an interacting particle system, the reflecting Poisson sea. I will also discuss the technique we developed to show this, which is a general-purpose ‘rescaled moment method’ for fluctuations of random abelian groups.

March 11: James MacLaurin (NJIT)

Kinetic Theory for High-Dimensional Models of Interacting Neurons. I study high-dimensional interacting particle models inspired by Neuroscience. Some of the specific features include (i) a disordered and sparse connectivity structure, (ii) delays in the interactions, and (iii) very strong interactions that can be both excitatory and inhibitory. I determine the large N kinetic limit for these systems, and I conduct a bifurcation analysis to determine the existence of patterns, waves and limit cycle oscillations in the kinetic limit. This work involves contributions from Daniele Avitabile and Pedro Vilanova.

March 18: Bhargav Narayana (Rutgers)

Littlewood–Offord sans Fourier. Given unit vectors v_1,…,v_n in Euclidean space, what can we say about the probability that a randomly-signed sum of these vectors lands in a small (say unit) ball around the origin? It turns out that both the natural problems here, namely, finding optimal upper bounds and finding optimal lower bounds for this “small-ball” probability, are quite interesting. The strongest results for both problems have typically come from harmonic analysis; in this talk, I’ll share some recently-developed (elementary) techniques that do just as well (and occasionally better).

March 25: Pranav Chinmay (CUNY)

Robust construction of the incipient infinite cluster in high-dimensional percolation. The incipient infinite cluster was first proposed by physicists in the 1970s as a canonical example of a two-dimensional medium on which random walk is subdiffusive. It is the measure obtained in critical percolation by conditioning on the existence of an infinite cluster, which is a probability zero event. Kesten presented the first rigorous two-dimensional construction of this object as a weak limit of the one-arm event. In high dimensions, van der Hofstad and Jarai constructed the IIC as a weak limit of the two-point connection using the lace expansion. Our work presents a new high-dimensional construction which is “robust”, establishing that the weak limit is independent of the choice of conditioning. The main tools used are Kesten’s original two-dimensional construction combined with Kozma and Nachmias’ regularity method. Our robustness allows for several applications, such as the explicit computation of the limiting distribution of the chemical distance, which forms the content of our upcoming project. This is joint work with Shirshendu Chatterjee, Jack Hanson, and Philippe Sosoe.

April 1: Alisa Knizel (Barnard)

Random Lozenge Waterfall: Dimensional Collapse of Gibbs Measures. I will discuss a two-parameter family of random lozenge tilings related to the q-Racah orthogonal polynomials. The model is a generalization of the uniform and q-weighted random tilings and due to the presence of additional parameters it displays a new interesting behavior. In particular, I will explain a new phenomenon that can be observed in the so-called waterfall region, where two-dimensional pure states collapse into one dimension.

April 8: Evan Sorensen (Columbia)

Viscous shock fluctuations in KPZ. I will discuss a recent preprint with Alex Dunlap, where we study “V-shaped” solutions to the KPZ equation. These are solutions having asymptotic slopes \theta > 0 and -\theta at plus and minus infinity, respectively. We show that there are no V-shaped invariant measures for the KPZ equation, which, combined with recent work of Janjigian, Rassoul-Agha, and Seppalainen, completes the classification of the extremal invariant measures for the KPZ equation. To accomplish this, we study the fluctuations of viscous shocks in the KPZ equation under some special choices of initial data. While V-shaped invariant measures in a fixed frame of reference do not exist, we give an explicit description of a family of V-shaped invariant measures from the perspective of a shock.

April 22: David Sivakoff (Ohio State)

Polluted Modified Bootstrap Percolation. The standard bootstrap percolation model is a cellular automaton model for nucleation and growth in the 2-dimensional square lattice. Each site begins as either occupied (1) or empty (0). Empty sites become occupied if at least two of their nearest-neighbors are occupied, and occupied sites remain occupied forever. In the modified bootstrap percolation model, an empty site only becomes occupied if it has at least one occupied neighbor to the north or south, and at least one occupied neighbor to the east or west. Gravner and McDonald (1997) introduced the polluted bootstrap percolation model, wherein sites are initially closed with probability q, open and occupied with probability p, and open and empty otherwise; and closed sites never change their states. They proved that for the standard model, there exist constants C,c>0 such that pollution stops occupation from reaching the origin if q>Cp^2, but does not if q<cp^2 when p tends to 0. For the modified model, they proved that such a transition happens for q between order (p / log p)^2 and p^2. We give more precise bounds, showing that for the polluted modified model the transition happens around p^2 / (log 1/p)^{1+o(1)}. I will describe the methods involved in proving these results, and possibly some ongoing work on related models. Based on joint work with Janko Gravner, Alexander Holroyd and Sangchul Lee.

April 29: Vittoria Silvestri (Sapienza Università di Roma)

Branching Internal DLA. Internal DLA is a random aggregation process in which the growth of discrete clusters is governed by the harmonic measure seen from an internal point. That is, a simple random walk is released from inside the cluster, and its exit location is added to it. The asymptotic shape of IDLA on Zd starting from a single seed has long been known to be a Euclidean ball, with very small fluctuations. In this talk I will discuss a natural variant of IDLA, namely Branching IDLA, in which the particles that drive the process perform critical branching random walks rather than simple random walks. We will show that BIDLA has a strikingly different phenomenology, namely we prove a phase transition from macroscopic fluctuations in low dimension to the existence of a shape theorem in higher dimension. Based on a joint work with Amine Asselah (Paris-Est Créteil) and Lorenzo Taggi (Rome La Sapienza). 

May 6: Christine Chang (CUNY)

Hybrid Statistics of the Maxima of a Random Model of the Zeta Function over Short Intervals. We discuss how the extreme value distribution of a random model of the Riemann zeta function evolves over intervals of varying length.  Specifically, we will present a matching upper and lower bound for the right tail probability of the maximum over short intervals. As the interval length varies, the distributional behavior transitions between that of log-correlated and IID random variables. We will also discuss a new normalization for the moments over short intervals. This work extends recent developments by Arguin, Dubach, and Hartung and is inspired by a conjecture by Fyodorov-Hiary-Keating on the local maximum over short intervals.

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CUNY Probabilists: Louis-Pierre Arguin, Christian Benes, Shirshendu Chatterjee, Olympia Hadjiliadis, Jack Hanson, Tobias Johnson, Matthew Junge, Elena Kosygina, Yanghui Liu, Jay Rosen, Indranil SenGupta