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

Feb 3: Louis-Pierre Arguin (Baruch)

Large Deviations of the Riemann Zeta Function and Random Walks. In this talk, I will present a proof that the measures of level sets of the Riemann zeta function have Gaussian tail, up to a constant C, for values in suitable regimes, conditionally on the Riemann Hypothesis. As a corollary, we recover the best-known bounds on the moments on the critical line. The proof relies on the recursive scheme of prior work with Bourgade & Radziwill that is inspired by a random walk heuristic. It also combines ideas of Soundararajan and Harper. We will discuss possible improvements to the constant C as well as the connections with the Keating-Snaith Conjecture from Random Matrix Theory for the optimal constant. This is joint work with Emma Bailey & Asher Roberts and Nathan Creighton.

Feb 10: Christophe Garban (Université Lyon/NYU)

One-arm exponents of the high-dimensional Ising model. In a joint work with Diederik van Engelenburg, Romain Panis and Franco Severo, we study the probability that the origin is connected to the boundary of the box of size $n$ (the one-arm probability) in several percolation models related to the Ising model. We prove that different universality classes emerge at criticality and that the FK-Ising model has upper-critical dimension equal to 6, in contrast to the Ising model, where it is known to be (less or) equal to 4. I will start the talk with a short introduction on the Ising model on Z^d.

Mar 3: Roland Bauerschmidt (NYU)

The massless sine-Gordon model – from solitons to integrability and spontaneous symmetry breaking. The massless sine-Gordon model is a central example in 2d probabilistic quantum field theory. It is dual to the 2d Coulomb gas and conjecturally features in various different contexts such as in the continuum limits of height functions of near-critical dimer and 6-vertex models. This talk starts with a brief overview of the probabilistic perspective on quantum field theory and then explains various aspects in the instance of the massless sine-Gordon model. This includes in particular a recent result with Christian Webb (Helsinki) and Scott Mason (NYU) in which we prove the conjecture that the fractional correlation functions of the sine-Gordon model at the free fermion point are given by renormalized determinants of twisted Dirac operators — the tau functions of Sato-Miwa-Jimbo.

Mar 10: Pranav Chinmay (CUNY)

The chemical distance in high dimensional critical percolation. The chemical distance is the observable that encapsulates the metric structure of percolation clusters. At criticality, heuristics suggest that the chemical distance between two connected points scales quadratically in the extrinsic distance, in line with the analogy to branching random walk. Our work presents an exact statement of this result, where the rescaled two-point chemical distance converges in distribution to a random variable whose density is expressible as a Brownian motion hitting time. The method relies on the robust incipient infinite cluster constructed in our previous work to enforce a decoupling argument that separates neighborhoods of distant pivotal edges. This decoupling tool yields further applications towards studying the mass structure of percolation clusters, i.e. k-point functions, which is necessary in the steps towards a full scaling limit result for the IIC. These projects are joint work with Shirshendu Chatterjee, Jack Hanson, and Philippe Sosoe. The preprint can be found at https://arxiv.org/abs/2509.06236.

Mar 24: Corrine Yap (Georgia Tech)

Fixed-Magnetization Ising Model on Random Graph. Spin models on random graphs are a source of many interesting questions in probability, statistical physics, algorithms, and combinatorics. On the random d-regular graph G(n,d), the classical Ising model is well-understood, largely due to the locally tree-like structure of the graph. We study the fixed-magnetization Ising model (where the number of positive spins is fixed) on G(n,d) in the non-reconstruction regime. By analyzing the free energy density and carrying out the second moment method in a planted model, we characterize the local structure of the Gibbs measure, showing it weakly converges to a specific measure on the infinite d-regular tree. From this, we show that the Ising model Glauber dynamics from a uniformly random start mixes in subexponential time. Finally, we provide the first positive-temperature resolution of the Zdeborová–Boettcher conjecture, which in the $\beta \to \infty$ limit states that the max-cut and min-bisection sizes on G(n,d) asymptotically sum to nd/2. Joint work with Reza Gheissari and Will Perkins.

April 21: Victor de le Pena (Columbia)

Sharp Decoupling of the Variance of arbitrarily  Dependent Variables. We present sharp decoupling inequalities for the variance and second moment of sums of arbitrarily dependent random variables. The key result bounds Var(∑di)≤2Var(∑ei), where (ei) is a conditionally independent tangent sequence, with the constant 2 being optimal. Unlike prior work (Hitczenko 1994), no non-negativity assumption is required — only square-integrability. We demonstrate applications to randomly stopped sums with non-zero drift, improving bounds of de la Peña and Govindarajulu (1992). The framework extends naturally to U-statistics and other dependent structures arising in sequential analysis.

April 28: Elena Kosygina (Baruch)

Limits of self-interacting random walks via joint Ray-Knight theorems. For several classes of self-interacting random walks (SIRWs) on the integers B. Tóth proved the generalized Ray-Knight theorems and used them to study the limiting distributions of these walks. A natural question is whether these generalized Ray-Knight theorems uniquely identify the limiting process. Recently, in a joint work with T. Mountford and J. Peterson, we showed that this need not be the case. Our current work (with L. Marêché, T. Mountford, and J. Peterson) shows that, under mild conditions, the generalized joint Ray-Knight theorems for SIRWs can be “inverted” to prove convergence of the rescaled walks and hence construct a unique limiting process. We apply our general results to “true” self-avoiding walks and polynomially self-repelling walks. The second application gives new class of non-markovian processes — polynomially self-repelling motions.

May 5: Michael Damron (Georgia Tech)

Percolation and first-passage percolation on logarithmic subgraphs of Z^2.  In two-dimensional Bernoulli percolation, we declare each edge of the square grid Z^2 to be open with probability p or closed with probability 1-p, independently from edge to edge. There is a critical value p_c = 1/2, such that for p < p_c, all components of open edges are finite, and for p > p_c, there is a unique infinite component of open edges. In ’83, Grimmett introduced the following variant. Let f be a nonnegative real function on [0,\infty), and consider the subgraph G_f of Z^2 induced by the edges between the positive first coordinate axis and the graph of f. Grimmett found that if f(u) \sim a \log u as u \to \infty, then the critical value p_c(f) for percolation on G_f equals a specific function of a only. In ’86, Chayes-Chayes considered the function f(u) = a \log(1+u) + b \log(1+\log(1+u)) and showed that if b > 2a, then the percolation G_f has an infinite open component at the critical point (that is, a discontinuous phase transition). In joint work with Wai-Kit Lam, we prove that the phase transition is discontinuous if and only if b > a, and we compute sharp asymptotics for all p, a, and b of the expected passage time in G_f from the origin to the vertical line x=n in the related first-passage percolation model, improving results of Ahlberg. We also find asymptotics for the variance and prove a central limit theorem.

May 12: Erik Slivken (UNC Wilmington)

Limit shape of online separable permutations. Given a permutation of size n, we can create a new permutation of size n+1 by appending a value k between 1 and n+1 to the end of the permutation, increasing previous values that are at least k, so the new sequence remains a permutation. This process is a natural way to generate a uniformly random permutation where at each step we chose k uniformly from {1,…,n+1}. If we restrict our choices of k at each step so that the new permutation avoids a particular pattern, we get interesting distributions on pattern-avoiding permutations that differ somewhat drastically from the uniform distribution on the same pattern-avoiding class. We will focus on the class of separable online permutations and describe a few different ways in which they are qualitatively different from other models of separable permutations. We show a permuton limit which is deterministic (versus random in the other known models). We also show that fluctuations about the deterministic limit converge to Brownian motion.

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