Day 1, May 29 Conference Program

8:00-8:45 AM

Registration

8:45-9:00 AM

Opening Remarks

Jeffrey Lehman, Vice Chancellor of NYU Shanghai

Tong Shijun, Chancellor of NYU Shanghai

Session Chair: Pierre Tarrès, NYU Shanghai / NYU

9:00-10:00 AM

Thermodynamic Limit of the First Lee-Yang Zero

By Charles Newman, New York University

In a recent paper (C.P.A.M., 2024), Jianping Jiang and I completed (for the standard Ising ferromagnet in d dimensions) the rigorous verification of the celebrated Yang-Lee (1952) and Lee-Yang (1952) proposal that thermodynamic singularities are exactly the limits in the real physical parameter space (say, the real line for the external magnetic field) of finite-volume singularities in the complex plane.  The missing ingredient was to prove that for *all* T > T_c, there is a zero-free disc of the partition function about the origin; this extends the 1971-73 high-temperature results of Ruelle. A key ingredient in the proof is the result by Federico Camia, Jianping Jiang and me (Commun. Math. Phys., 2023) that in zero field, the even (multi-site) Ursell functions u_{2k} satisfy: (-1)^k u_{2k} is increasing in each (ferromagnetic) coupling. The positivity of (-1)^k u_{2k} for all k was derived by Shlosman in 1986.

10:00-10:30 AM

Morning Tea Break

10:30-11:30 AM

Random matrices and logarithmically correlated fields: dimensions 2 and 3

By Paul Bourgade, New York University

The Liouville quantum gravity measure is a properly normalized exponential of 2d log-correlated fields, such as the Gaussian free field. I will explain how this naturally appears in random matrix theory either in space time from random matrix dynamics, or in space from the characteristic polynomial of random normal matrices. A 3d log-correlated field also naturally emerges in random matrix theory, from dynamics on non-Hermitian matrices.

11:30 AM-12:10 PM

Random aggregation on the complex plane and the Hastings-Levitov models

By Vittoria Silvestri, University of Rome La Sapienza

The ALE (Aggregate Loewner Evolution) models describe growing random clusters on the complex plane, built by iterated composition of random conformal maps. A striking feature of these models is that they can be used to define natural off-lattice analogues of several fundamental discrete models, such as Eden growth, Dielectric Breakdown and Diffusion Limited Aggregation, by tuning the correlation between the defining maps appropriately. In these talks I will introduce the ALE models, which include Hastings-Levitov models as particular cases, and discuss their large scale properties such as scaling limits and fluctuations. I will conclude by stating some conjectures and open questions. Based on joint work with James Norris (Cambridge, UK) and Amanda Turner (Leeds, UK).

12:10 PM-12:30 PM

12:30 PM-2:30 PM

Lunch Break

Session Chair: Hong-Bin Chen, Institut des Hautes Études Scientifiques (IHES)

2:30-3:30 PM

Four-dimensional Brownian motion in the continuum Derrida--Retaux system

By Zhan Shi, Academy of Mathematics and Systems Science, Chinese Academy of Sciences

I am going to make some elementary discussions on the presence of (the Euclidean norm of) four-dimensional Brownian motion in the continuum Derrida--Retaux system. Joint work with E. Aïdékon, B. Derrida and T. Duquesne.

3:30-4:00 PM

Afternoon Tea Break

4:00-4:40 PM

Quasi-stationary distributions for subcritical population models

By Leonardo Rolla, University of São Paulo

We consider population models with reproduction (branching process, contact process, branching random walks). This model has a phase transition in terms of the offspring distribution. In the subcritical case, every initial distribution is attracted to the empty configuration. In the lack of a non-trivial stationary distribution, one studies the quasi-stationary behavior of the system. In this talk we will discuss questions of existence, uniqueness and nonuniqueness of the quasi-stationary distribution, and their relation with spatial aspects of the dynamics. Joint work with Pablo Groisman and Célio A. Terra.

4:40-5:20 PM

Continuous symmetry breaking for rotator models and dimers

By Wei Wu, NYU Shanghai

We discuss two models of interest in statistical mechanics. The first of these models, the XY and the Villain models, are models which exhibit the celebrated Kosterlitz-Thouless phase transitions in two dimensions. The spin wave conjecture, originally proposed by Dyson and by Mermin and Wagner, predicts that at low temperature, spin correlations of these models are closely related to Gaussian free fields. I will discuss some recent progress on this conjecture in d>=3. The second is the dimer model and the monomer double dimer model. I will introduce a new (complex) spin representation for models belonging to this class, and derive a new proof of the Mermin-Wagner theorem which does not require the positivity of the Gibbs measure. Based on joint works with Paul Dario (CNRS) and with Lorenzo Taggi (Sapienza Univ. di Roma).