The CUNY Probability Seminar will be held by videoconference for the entire semester. Its usual time will be Tuesdays from 4:30 to 5:30 pm EST. The exact dates, times, and seminar links are mentioned below. If you are interested in speaking at the seminar or would like to be added or to be removed from the seminar mailing list, then please contact the Seminar Coordinator.

Shirshendu Chatterjee and Jack Hanson

## Seminar Schedule:

**Tuesday, 4:30-5:30 pm EDT****Videoconference Link: Zoom**

(https://ccny.zoom.us/j/85229381369)

**Time: February 09, 5 – 6 pm EST**

**Speaker: Michel Pain, Courant Institute of Mathematical Sciences, NYU**

**Seminar Link:** **Zoom**

**Title: **Optimal local law and central limit theorem for beta-ensembles

**Abstract: **In this talk, I will present a joint work with Paul Bourgade and Krishnan Mody. We consider beta-ensembles with general potentials (or equivalently a log-gas in dimension 1), which are a generalization of Gaussian beta-ensembles and of classical invariant ensembles of random matrices. We prove a multivariate central limit theorem for the logarithm of the characteristic polynomial, showing that it behaves as a log-correlated field. A key ingredient is an optimally sharp local law for the the Stieljes transform of the empirical measure which can be of independent interest. Both the proofs of the CLT and the local law are based essentially on loop equations techniques.

**Time: February 16, 3:30 – 4:30 pm EST**

**Speaker: Guillaume Remy, Columbia University**

**Seminar Link:** **Zoom**

**Title: **A probabilistic construction of conformal blocks for Liouville CFT.

**Abstract: **Liouville theory is a fundamental example of a conformal field theory (CFT) first introduced by A. Polyakov in the context of string theory. In recent years it has been rigorously studied using probabilistic techniques. In this talk we will present a probabilistic construction of the conformal blocks of Liouville CFT on the torus. These are the fundamental objects that allow to understand the integrable structure of CFT using the conformal bootstrap equation. We will also mention the connection with the AGT correspondence. Based on joint work with Promit Ghosal, Xin Sun, and Yi Sun.

**Time: February 23, 4:30 – 5:30 pm EST**

**Speaker: Nicholas Cook, Duke University**

**Seminar Link:** **Zoom**

**Title: **Universality for the minimum modulus of random trigonometric polynomials

**Abstract: **We consider the restriction to the unit circle of random degree-n polynomials with iid normalized coefficients (Kac polynomials). Recent work of Yakir and Zeitouni shows that for Gaussian coefficients, the minimum modulus (suitably rescaled) follows a limiting exponential distribution. We show this is a universal phenomenon, extending their result to arbitrary sub-Gaussian coefficients, such as Rademacher signs. Our approach relates the joint distribution of small values at several angles to that of a random walk in high-dimensional phase space, for which we obtain strong central limit theorems. For discrete coefficients this requires dealing with arithmetic structure among the angles. Based on joint work with Hoi Nguyen.

**Time: March 02, 4:30 – 5:30 pm EST**

**Speaker: Erik Bates, **.University of Wisconsin, Madison

**Seminar Link:** **Zoom**

**Title: **Empirical measures, geodesic lengths, and a variational formula in first-passage percolation

**Abstract: **We consider the standard first-passage percolation model on Z^d, in which each edge is assigned an i.i.d. nonnegative weight, and the passage time between any two points is the smallest total weight of a nearest-neighbor path between them. Our primary interest is in the empirical measures of edge-weights observed along geodesics from 0 to [n\xi], where \xi is a fixed unit vector. For various dense families of edge-weight distributions, we prove that these measures converge weakly to a deterministic limit as n tends to infinity. The key tool is a new variational formula for the time constant. In this talk, I will derive this formula and discuss its implications for the convergence of both empirical measures and lengths of geodesics.

**Time: March 09, 3:30 – 4:30 pm EST**

**Speaker: Wai-Kit Lam, University of Minnesota**

**Seminar Link:** **Zoom**

**Title: **Near-critical avalanches in 2D frozen percolation and forest fires

**Abstract: **We study two closely related processes on the triangular lattice: frozen percolation, where connected components of occupied vertices freeze as soon as they contain at least $N$ vertices, and forest fire processes, where connected components burn at rate $\zeta > 0$. When the density of occupied sites approaches the critical threshold for Bernoulli percolation, both processes display a striking phenomenon: the appearance of what we call near-critical “avalanches”. We analyze the whole avalanches, all the way up to the natural characteristic scale $m_{\infty}$ of each model. For frozen percolation, we show in particular that around any given vertex, the number of frozen clusters is asymptotically equivalent to $(\log(96/5))^{-1} \log\log N$ as $N \to \infty$. A similar mechanism underlies forest fires, enabling us to obtain an analogous result, but with substantially more work: the number of burnt clusters is equivalent to $(\log(96/41))^{-1} \log\log (\zeta^{-1})$ as $\zeta \to 0$. This constitutes an important step toward understanding the self-organized critical behavior of such processes. Based on joint work with Pierre Nolin.

**Time: March 16, 3:30 – 4:30 pm EST**

**Speaker: Theo Assiotis**, **University of Edinburgh**

**Seminar Link:** **Zoom**

**Title: **On the joint moments of characteristic polynomials of random unitary matrices.

**Abstract: **I will talk about the joint moments of characteristic polynomials of random unitary matrices and their derivatives. In joint work with Jon Keating and Jon Warren we establish the asymptotics of these quantities for general real values of the exponents as the size N of the matrix goes to infinity. This proves a conjecture of Hughes from 2001. In subsequent joint work with Benjamin Bedert, Mustafa Alper Gunes and Arun Soor we focus on the leading order coefficient in the asymptotics, we connect this to Painleve equations for general values of the exponents and obtain explicit expressions corresponding to the so-called classical solutions of these equations.

**Time: March 23, 4:30 – 5:30 pm EST**

**Speaker: Christophe Garban**, **Université Lyon 1**

**Seminar Link:** **Zoom**

**Title: Vortex fluctuations in continuous spin systems and lattice gauge theory**

**Abstract: **Topological phase transitions were discovered by Berezinskii-Kosterlitz-Thouless (BKT) in the 70’s. They describe intriguing phase transitions for classical statistical physics models such as – the 2d XY model (spins on Z^2 with values in the unit circle) – the 2d Coulomb gas – the integer-valued Gaussian Free Field (or Z-ferromagnet) – Abelian lattice gauge theory on Z^4In this talk, I will explain a new technique to obtain quantitative lower bounds on the fluctuations induced by the topological defects (vortices) on such systems at low temperature. We will see in particular that the fluctuations generated by the vortices are at least of the same order of magnitude as the ones produced by the so-called “spin-wave”. Our approach is non-perturbative but it gives matching lower bounds with the fluctuations predicted from RG analysis. I will start the talk by giving an overview of the above models. The talk is based on joint works with Avelio Sepúlveda.

**Time: April 06, 4:30 – 5:30 pm EST**

**Speaker: Laurent Tournier**, Université Sorbonne Paris-Nord (Paris 13) & NYU Shanghai

**Seminar Link:** **Zoom**

**Title: The phase transition in three-speed ballistic annihilation**

**Abstract: ** In the ballistic annihilation model, particles are emitted from a Poisson point process on the line, move at constant speeds, chosen randomly, i.i.d. at initial time, and mutually annihilate when they collide. This model was introduced in the 1990’s in physics as an alternative to more classical diffusion limited reaction models, however its asymptotic behavior remains very poorly understood as soon as the speeds may take at least three values. We will focus on this minimal case when speeds may be -1, 0 or 1, with symmetric probabilities, and show in particular that a phase transition takes place when 0-speed particles have probability 1/4, while discussing remarkable combinatorial properties of this model. This talk will be based on joint works with J.Haslegrave and V.Sidoravicius.

**Time: April 13, 4:30 – 5:30 pm EST**

**Speaker:** Ioan Manolescu, University of Fribourg

**Seminar Link:** **Zoom**

**Title: **Rotational invariance in planar FK-percolation

**Abstract: ** We prove the asymptotic rotational invariance of the critical FK-percolation model on the square lattice with any cluster-weight between 1 and 4. These models are expected to exhibit conformally invariant scaling limits that depend on the cluster weight, thus covering a continuum of universality classes. The rotation invariance of the scaling limit is a strong indication of the wider conformal invariance, and may indeed serve as a stepping stone to the latter.

Our result is obtained via a universality theorem for FK-percolation on certain isoradial lattices. This in turn is proved via the star-triangle (or Yang-Baxter) transformation, which may be used to gradually change the square lattice into any of these isoradial lattices, while preserving certain features of the model. It was previously proved that throughout this transformation, the large scale geometry of the model is distorted by at most a limited amount. In the present work we argue that the distortion becomes insignificant as the scale increases. This hinges on the interplay between the inhomogeneity of isoradial models and their embeddings, which compensate each other at large scales.

As a byproduct, we obtain the asymptotic rotational invariance also for models related to FK-percolation, such as the Potts and six-vertex ones. Moreover, the approach described here is fairly generic and may be adapted to other systems which possess a Yang-Baxter transformation.

Based on joint work with Hugo Duminil-Copin, Karol Kajetan Kozlowski, Dmitry Krachun and Mendes Oulamara.

**Time: April 20, 4:30 – 5:30 pm EST**

**Speaker:** Kenneth Alexander, USC

**Seminar Link:** **Zoom**

**Title: **Geodesics, bigeodesics, and coalescence in first passage percolation

**Abstract: **In first passage percolation, independent identically distributed bond passage times are attached to the bonds of the lattice Z^{d}; these may alternatively be viewed as random lengths. This creates a random distance on the lattice: the geodesic from x to y is the lattice path which minimizes the sum of bond passage times, and this minimum is the passage time T(x,y), which may also be viewed as a distance. We are interested in infinite geodesics for this distance: a θ-ray is a path to ∞ with asymptotic direction θ for which every finite segment is a geodesic, and a bigeodesic is an analogous doubly infinite geodesic path. It is known that under mild hypotheses, for each starting point x and direction θ, there is a θ-ray from x. In d = 2 it is a.s. unique, and furthermore, for all x, y the θ-rays from x and y eventually coalesce, and there are no bigeodesics with θ as either asymptotic direction. We show that in general dimension, under somewhat stronger hypotheses, a weak form of coalescence called bundling occurs: we take all θ-rays starting next to a hyperplane H_{0}, translate the hyperplane forward by distance R to give HR, and consider the density of sites in HR where one of the θ-rays first crosses H_{R}. We show this density approaches 0 as R → ∞, with near-optimal bounds on the rate. Essentially as a consequence, we show that there are no bigeodesics in any direction.

**Time: April 27, 4:30 – 5:30 pm EDT**

**Speaker:** Eric Foxall, The University of British Columbia

**Seminar Link:** **Zoom**

**Title: **Dominance, fixation and identifiability in finite neutral genealogy models.

**Abstract: ** Given a population history (population size, offspring numbers and deaths over time), the corresponding neutral model is constructed by assigning deaths and parentage uniformly at random in each generation, giving a directed graph. Using an intriguing representation of this graph, known as the “lookdown” because of the way it arranges vertices in each generation, we explore two properties of both theoretical and applied interest: (i) dominance: whether or not one vertex in each generation has a significantly larger number of offspring than any other (one lineage mostly takes over) and (ii) fixation: whether the graph has a unique infinite path (one lineage takes over completely). These properties are closely related to the question of identifiability: is it possible to recover some or all of the lookdown representation simply by observing the (unlabelled) family trees?

ArXiv link: https://arxiv.org/abs/2104.00193

**Time: May 04, 3:30 – 4:30 pm EDT**

**Speaker:** Alice Contat, Universit´e Paris-Saclay

**Seminar Link:** **Zoom**

**Title: **Parking on Cayley trees & Frozen Erdös-Rényi

**Abstract: **The talk is based on joint work with Nicolas Curien. Consider a uniform rooted Cayley tree Tn with n vertices and let m cars arrive sequentially, independently, and uniformly on its vertices. Each car tries to park on its arrival node, and if the spot is already occupied, it drives towards the root of the tree and parks as soon as possible. Using combinatorial enumeration, Lackner & Panholzer established a phase transition for this process when m is approximately n/2 . We couple this model with a variation of the classical Erdös–Rényi random graph process. This enables us to describe completely the phase transition for the size of the components of parked cars using a modification of the standard multiplicative coalescent which we named the frozen multiplicative coalescent.

**Time: May 11, 4:30 – 5:30 pm EDT**

**Speaker:** Malin Palö Forsström, KTH Royal Institute of Technology

**Seminar Link:** **Zoom**

**Title: **Correlations in finite Abelian lattice gauge theories, with and without a Higgs field

**Abstract: **Lattice gauge theories are discrete approximations of Yang-Mills gauge theory and have since their introduction been used to predict properties of elementary particles. However, many of these predictions are not rigorous and only supported by simulations and heuristic arguments. Moreover, it is not clear how to obtain a limit as the lattice spacing tends to zero even in the simplest cases. A natural first step for both these problems is to attempt to understand the decay of correlations of local functions, such as plaquette spins and so-called Wilson loops. In this talk, we will introduce Abelian lattice gauge theories from a probabilistic perspective and discuss some of their properties and natural observables. Also, several useful ideas which can be used to study these models will be described. In particular, we will present recent results on the expected value of Wilson loops in $\mathbb{Z}_4$ in the presence of a Higgs field. This talk is based on joint work with Jonatan Lenells and Fredrik Viklund.