The CUNY Probability Seminar is typically held on Tuesdays from 4:15 to 5:15 pm at the CUNY Graduate Center Math Department in room 5417. The exact dates, times and locations are mentioned below. If you want to talk at the seminar, or want to be added to/removed from the seminar mailing list, then please contact the Seminar Coordinator.

*Toby Johnson* (https://www.math.csi.cuny.edu/~tobiasljohnson/)

## Upcoming Seminars

**Tuesday, December 4, 2018, 4:15-5:15**

**Room 5417**

**Speaker: **Chiranjib Mukherjee, Institut für Mathematische Statistik, Universität Münster

**Title:** The smoothed KPZ equation in dimension three and higher: Edwards-Wilkinson regime of its fluctuations and its localization properties

**Abstract:** We study the Kardar-Parisi-Zhang equation in dimension with space-time white noise which is smoothed in space. There is a natural disorder parameter attached to this equation which measures the intensity of the noise. We show that when the disorder is small, the approximating solution converges to a well-defined limit (with the limit depending on both the disorder and the mollification procedure), while the re-scaled fluctuations converge to a Gaussian limit as predicted by the Edwards-Wilkionson regime.

We also study the associated stochastic heat equation with multiplicative noise, which carries a natural Gaussian mutiplicative noise (GMC) on the Wiener space. When the disorder is large, we also show that the total mass of the GMC converges to zero, while the endpoint distribution of a Brownian path under the (renormlaized) GMC measure is purely atomic.

Based on joint works with, A. Shamov & O. Zeitouni, F. Comets & C. Cosco as well Y. Broeker.

## Previous Seminars

**Tuesday, September 4, 2018, 4:15-5:15**

**Room 5417**

**Speaker: **Philip Matchett Wood, U. Wisconsin

**Title: **Limiting eigenvalue distribution for the non-backtracking matrix of an Erdős-Rényi random graph

**Abstract:** A non-backtracking random walk on a graph is a directed walk with the constraint that the last edge crossed may not be immediately crossed again in the opposite direction. This talk will give a precise description of the eigenvalues of the adjacency matrix for the non-backtracking walk when the underlying graph is an Erdős-Rényi random graph on *n* vertices, where edges present independently with probability *p*. We allow *p* to be constant or decreasing with *n*, so long as tends to infinity. The key ideas in the proof are partial derandomization, applying the Tao-Vu Replacement Principle in a novel context, and showing that partial derandomization may be interpreted as a perturbation, allowing one to apply the Bauer-Fike Theorem. Joint work with Ke Wang at HKUST (Hong Kong University of Science and Technology).

**Tuesday, September 25, 2018, 4:30-5:30** *(please note that this talk starts at 4:30, not 4:15!)*

**Room 5417**

**Speaker: **Qiang Zeng, Queens College

**Title:** Replica Symmetry Breaking for mean field spin glass models

**Abstract:** Mean field spin glass models were introduced as an approximation of the physical short range models in the 1970s. The typical mean field models include the Sherrington-Kirkpatrick (SK) model, the (Ising) mix p-spin model and the spherical mixed p-spin model. Starting in 1979, the physicist Giorgio Parisi wrote a series of groundbreaking papers introducing the idea of replica symmetry breaking (RSB), which allowed him to predict a solution for the SK model by breaking the symmetry of replicas infinitely many times at low temperature. In this talk, we will show that Parisi’s prediction holds at zero temperature for the more general mixed p-spin model. On the other hand, we will show that there exist two-step RSB spherical mixed p-spin glass models at zero temperature, which are the first natural examples beyond the replica symmetric, one-step RSB and Full-step RSB phases.

This talk is based on joint works with Antonio Auffinger (Northwestern University) and Wei-Kuo Chen (University of Minnesota).

**Tuesday, October 2, 2018, 4:15-5:15**

**Room 5417**

**Speaker: **Erik Slivken, University of Paris VII

**Title:** Pattern-avoiding permutations and Dyson Brownian motion

**Abstract:** Let denote the set of permutations of length *n*. For a permutation we say τ contains a pattern if there is a subsequence such that has the the same relative order of σ. If τ contains no pattern σ, we say that τ avoids σ. We denote the set of σ-avoiding permutations of length *n* by . Recently, there have been a number of results that help describe the geometric properties of a uniformly random element in . Many of these geometric properties are related to well-studied random objects that appear in other settings. For example, if , then a permutation chosen uniformly in converges, in some appropriate sense, to Brownian excursion. Furthermore for σ = 123, 312, or 231, we can describe properties like the number and location of fixed points in terms of Brownian excursion. Larger patterns are much more difficult to understand. Currently even the simplest question, enumeration, is unknown for the pattern σ = 4231. However, for the monotone decreasing pattern , a permutation chosen uniformly from can be coupled with a random walk in a cone that, in some appropriate sense, converges to a traceless Dyson Brownian motion.

**Tuesday, October 9, 2018, 4:15-5:15**

**Room 5417**

**Speaker: **Matthew Junge, Duke University

**Title:** Diffusion-Limited Annihilating Systems

**Abstract:** We study a two-type annihilating system in which particles are placed with equal density on the integer lattice. Particles perform simple random walk and annihilate when they contact a particle of different type. The occupation probability of the origin was known to exhibit anomalous behavior in low-dimension when particles have equal speeds. Describing the setting with asymmetric speeds has been open for over 20 years. We prove a lower bound that matches physicists’ conjectures and discuss partial progress towards an upper bound. Joint with Michael Damron, Hanbaek Lyu, Tobias Johnson, and David Sivakoff.

**Tuesday, October 16, 2018, 4:15-5:15**

**Room 5417**

**Speaker: **Xin Sun, Columbia University

**Title:** Natural measures on random fractals

**Abstract:** Random fractals arise naturally as the scaling limit of large discrete models at criticality. These fractals usually exhibit strong self similarity and spacial independence. In this talk, we will explain how these additional properties should give the existence of a natural occupation measure on the fractal set, defined to be the limit of the properly rescaled Lebesgue measure restricted to small neighborhoods of the fractal. Moreover, the occupation measure is also the scaling limit of the normalized counting measure over the corresponding discrete set. In two dimension, when putting an independent Liouville quantum gravity background over such a planar fractal, the quantum version of the occupation measure still exists, where the scaling dimension is related to the Euclidean one via the famous KPZ relation due to Knizhnik-Polyakov-Zamolodchikov and Duplantier-Sheffield. The quantum occupation measure is supposed to be the scaling limit of the normalized counting measure of the corresponding discrete set on certain random planar maps. The picture described above is expected to be true in great generality yet it is only established for a few models to various extents. In this talk, we report a fairly complete picture for planar percolation on the regular and random triangular lattice.

**Tuesday, October 23, 2018, 4:15-5:15**

**Room 5417**

**Speaker: **Ofer Zeitouni, Weizmann Institute/NYU

**Title:** Noise (in)stability of the spectrum of random matrices

**Abstract:** Non-Hermitian matrices have a spectrum that is notoriously unstable to small perturbations. This fact is well captured by the notion of pseudo-spectrum, which deals with “worse case” perturbations. I will discuss recent advances on the study of the spectrum of such matrices subject to vanishingly small noise. Joint work with Anirban Basak and Elliot Paquette.

**Tuesday, October 30, 2018, 4:15-5:15**

**Room 5417**

**Speaker: **Ruojun Huang, NYU

**Title:** On some instances of random walk in changing environment

**Abstract:** We will talk about two recent results on random walk in changing environment. Relying on heat kernel estimates, we show that SRW evolving independently (a) on growing-in-time internal diffusion limited aggregation random cluster is recurrent, when dimension larger than two; (b) among time-increasing uniformly elliptic conductances on graphs including Z^d, a sharp, but not fully explicit, criterion to determine transience versus recurrence. The latter is joint work with Amir Dembo and Tianyi Zheng. We also discuss universality conjectures put forward by G. Amir, I. Benjamini, O. Gurel-Gurevich, and G. Kozma in arXiv:1504.04870.

**Tuesday, November 6, 2018, 4:15-5:15**

**Room 5417**

**Speaker: **Lisa Hartung, Courant Institute

**Title:** From 1 to 6 in branching Brownian motion

**Abstract:** Brownian motion is a classical process in probability theory belonging to the class of ‘Log-correlated random fields’. It is well known due to Bramson that the order of the maximum has a different logarithmic correction as the corresponding independent setting.

In this talk we look at a version of branching Brownian motion where we slightly vary the diffusion parameter in a way that, when looking at the order of the maximum, we can smoothly interpolate between the logarithmic correction for independent random variables () and the logarithmic correction of BBM () and the logarithmic correction of 2-speed BBM with increasing variances (). We also establish in all cases the asymptotic law of the maximum and characterise the extremal process, which turns out to coincide essentially with that of standard BBM. We will see that the key to the above results is a precise understanding of the entropic repulsion experienced by an extremal particle. (joint work with A. Bovier)

**Tuesday, November 13, 2018, 4:15-5:15**

**Room 5417**

**Speaker: **Thomas Mountford, École Polytechnique Fédérale de Lausanne

**Title:** Critical values for renewal contact processes

**Abstract:** A renewal contact process is a (non-Markov) process similar to the classical contact process but where the rate one Poisson processes governing “recovery” are replaced by renewal processes (transmissions are still modelled by rate lambda Poisson processes). We show that the critical values are zero if the renewal distribution has very heavy tails but is strictly positive if a moment higher than one exists (under some strict regularity condition). This is joint work with D. Marchetti and R. Fontes of USP and M.E. Vares of UFRJ.

**Tuesday, November 27, 2018, 4:05-5:05** *(please note that this talk starts at 4:05, not 4:15!)*

**Room 5417**

**Speaker: **Guillaume Remy, Columbia University

**Title:** Exact formulas for Gaussian multiplicative chaos and Liouville theory

**Abstract:** We will present recent progress that has been made to prove exact formulas on the Gaussian multiplicative chaos (GMC) measures which are constructed by exponentiating a log-correlated field. We will give the law of the total mass of the GMC measure on the unit circle (the Fyodorov-Bouchaud formula) and on the unit interval (in collaboration with T. Zhu). The techniques of proof come from the link between GMC and Liouville conformal field theory studied by David-Kupiainen-Rhodes-Vargas. We will also discuss connections with the Duplantier-Miller-Sheffield approach to Liouville quantum gravity and identify the law of the total mass of the quantum disk (in collaboration with X. Sun).