MIT Probability Seminar

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Spring 2021

*** Seminar is online for Spring 2021 semester ***

Monday 4.15 - 5.15 pm

Room 2-147

Talks will be held on Zoom, Monday 4:15-5:15 pm. A link to a Zoom classroom will appear here!!

Online Schedule

• February 15

President's day (no seminar)

• February 22

*** 11:15 am - 12:15 pm (Special time)***

Hariharan Narayanan (TIFR, Mumbai)

Random discrete concave functions on an equilateral lattice with periodic boundary conditions

Abstract: Motivated by connections to random matrices, Littlewood-Richardson coefficients and square-triangle tilings, we study random discrete concave functions on an equilateral lattice, where the Hessian satisfies periodic boundary conditions and has a given average s. Defining surface tension sigma to be the negative of a certain limiting differential entropy per site, we show that sigma is a well defined convex function of s. When s is such that sigma is strictly convex, we show that the corresponding rescaled random surfaces concentrate in the sup norm as the length scale of the periodicity n tends to infinity. We also show that concentration occurs when the gradient of sigma belongs to a certain cone, and in this case obtain quantitative bounds for the concentration.

A preprint can be found at https://arxiv.org/abs/2005.13376 .

• March 1

Tom Alberts (Utah)

Loewner Dynamics for the Multiple SLE($0$) Process

Abstract: Recently Peltola and Wang introduced the multiple SLE($0$) process as the deterministic limit of the random multiple SLE($\kappa$) curves as $\kappa$ goes to zero. They prove this result by means of a small $\kappa$" large deviations principle, but the limiting curves also turn out to have important geometric characterizations that are independent of their relation to SLE($\kappa$). In particular, they show that the SLE($0$) curves can be generated by a deterministic Loewner evolution driven by multiple points, and the vector field describing the evolution of these points must satisfy a particular system of algebraic equations. We show how to generate solutions to these algebraic equations in two ways: first in terms of the poles and critical points of an associated real rational function, and second via the well-known Calogero-Moser integrable system with particular initial velocities. Although our results are purely deterministic they are again motivated by taking limits of probabilistic constructions, which I will explain.

• March 8

Student holiday (no seminar)

• March 15

Benjamin McKenna (NYU)

Random determinants and landscape complexity beyond invariance

Abstract: The Kac-Rice formula allows one to study the complexity of high-dimensional Gaussian random functions (meaning asymptotic counts of critical points) via the determinants of large random matrices. We present a new result on determinant asymptotics for non-invariant random matrices, and use it to compute (annealed) complexity for several types of landscapes. These include (i) the elastic manifold, where we identify the "Larkin mass" separating order and disorder, verifying results of Fyodorov-Le Doussal, and (ii) soft spins in an anisotropic well, where we find a new phase transition with universal quadratic and cubic near-critical behavior. This extends the pioneering complexity results of Fyodorov and Auffinger-Ben Arous-Cerny. Joint work with Gerard Ben Arous and Paul Bourgade.

• March 22

Shirshendu Ganguly (UC Berkeley)

Stability and chaos in dynamical last passage percolation

Abstract:Many complex disordered systems in statistical mechanics are characterized by intricate energy landscapes. The ground state, the configuration with the lowest energy, lies at the base of the deepest valley. In important examples, such as Gaussian polymers and spin glass models, the landscape has many valleys and the abundance of near-ground states (at the base of the valleys) indicates the phenomenon of chaos, under which the ground state alters profoundly when the disorder of the model is slightly perturbed.

In this talk, we will discuss a recent work with Alan Hammond computing the critical exponent that governs the onset of chaos in a dynamic manifestation of a canonical planar last passage percolation model in the Kardar-Parisi-Zhang universality class. We expect this exponent to be universal across a wide range of interface and stochastic growth models. The arguments rely on Chatterjee's harmonic analytic theory of equivalence of super-concentration and chaos in Gaussian spaces and a refined understanding of the corresponding static landscape geometry.

• March 29

*** 10:00 am - 11:00 am (Special time)***

Antonio Auffinger (Northwestern)

TAP equations and ground states of generalized spin glass models.

Abstract: In this talk, I will survey models of spin glasses where the spins take values either in a ball in $\mathbb{R}^d$ or in a large subset of the integers. I will discuss two important quantities: the TAP equations, a system of self-consistent equations relating the spin magnetization at high temperature, and the ground-state energy, the minimum of the Hamiltonian. During the talk, I will stress the differences and the new difficulties that appear when one compares these models to classical models such as the Sherrington-Kirkpatrick or the spherical p-spin model. Based on joint works with Cathy Chen (Northwestern) and Yuxin Zhou (Northwestern).

• April 5

Masha Gordina (UConn)

Uniform volume doubling and functional inequalities on Lie groups.

Abstract:On a compact Lie group with a left-invariant Riemannian metric, many important functional inequalities for the Laplacian (such as Poincaré inequality, parabolic Harnack inequality, etc.) can be proved using only the volume doubling property. That is, constants in these inequalities can be controlled by the doubling constant of the metric; this can be strictly more powerful than classical techniques involving Ricci curvature lower bounds. It can happen that there is a uniform bound on the doubling constants of all left-invariant metrics on a given Lie group; such a group is called uniformly doubling. In such a case, the implicit constants in the functional inequalities will also be uniformly bounded over all left-invariant metrics. We show that this happens for the special unitary group $SU(2)$, via explicit uniform volume estimates and describe the consequences (heat kernel estimates, Weyl counting function etc)

This is joint work with Nate Eldredge and Laurent Saloff-Coste.

• April 12

Duncan Dauvergne (Princeton)

The directed landscape.

Abstract: The directed landscape is a random `directed metric' on the spacetime plane that arises as the scaling limit of integrable models of last passage percolation. It is expected to be the universal scaling limit for all models in the KPZ universality class for random growth. In this talk, I will describe its construction in terms of the Airy line ensemble, give an extension of this construction for optimal length disjoint paths, and discuss probabilistic consequences of these constructions. Based on joint work with J. Ortmann, B. Virag, and L. Zhang.

• April 19

Patriots' day (no seminar)

• April 26

Christopher Shriver (UCLA)

Free energy, Gibbs measures, and Glauber dynamics on trees.

Abstract: I will introduce some ideas from sofic entropy theory and use them to define a notion of free energy density in the context of finite-alphabet, nearest-neighbor interactions (like the Ising model) indexed by infinite regular trees. This free energy is used to prove that shift-invariant measures are Gibbs if and only if they are Glauber-invariant. We also establish a metastability phenomenon for the corresponding dynamics on finite locally-tree-like regular graphs. These results can be combined to characterize maximal-entropy joinings of Gibbs measures.

• May 3

Youngtak Sohn (Stanford)

Replica symmetry breaking for random regular NAE-SAT.

Abstract:In a wide class of random constraint satisfaction problems, ideas from statistical physics predict that there is a rich set of phase transitions governed by one-step replica symmetry breaking(1RSB). In particular, it is conjectured that there is the condensation regime below the satisfiability threshold, where the solution space condenses into the large clusters. We establish this phenomenon for the random regular NAE-SAT model by showing that most of the solutions lie in a bounded number of clusters and the overlap of two independent solutions concentrates on two points. Central to the proof is to calculate the moments of the number of clusters whose size is in an O(1) window.

This is joint work with Danny Nam and Allan Sly.

• May 10

Morris Ang (MIT)

Integrability of the conformal loop ensemble

Abstract: For $\frac{8}{3} < \kappa < 8$, the conformal loop ensemble $\mathrm{CLE}_{\kappa}$ is a canonical random ensemble of loops which is conformally invariant in law, and whose loops locally look like Schramm-Loewner evolution with parameter ?. It describes the scaling limits of the Ising model, percolation, and other models. When $\kappa \leq 4$ the loops are simple curves. In this regime we compute the three-point function of $\mathrm{CLE}_{\kappa}$ on the sphere, and show it agrees with the imaginary DOZZ formula of Zamolodchikov (2005). We also verify a conjecture of Kenyon and Wilson on the electrical thickness of $\mathrm{CLE}_{\kappa}$ on the sphere. Our arguments depend on couplings of $\mathrm{CLE}$ with Liouville quantum gravity and the integrability of Liouville conformal field theory.

Based on joint work with Xin Sun, which builds on our recent work with Holden and Remy.

• May 17

Minjae Park (MIT)

Wilson loop expectations as sums over surfaces in 2D

Abstract: Although lattice Yang-Mills theory on $\mathbb{Z}^d$ is easy to rigorously define, the construction of a satisfactory continuum theory on $\mathbb{R}^d$ is a major open problem when $d \geq 3$. Such a theory should assign a Wilson loop expectation to each collection of loops in $\mathbb{R}^d$. One of the proposed approaches involves representing this quantity as a sum over surfaces having the loops as their boundary. There are some formal/heuristic ways to make sense of this notion, but they typically yield an ill-defined difference of infinities. The goal of this talk is to make sense of Yang-Mills integrals as surface sums in the special case that $d=2$, where the existence of a well-defined continuum theory is already well known. We also obtain an alternative proof of the Makeenko-Migdal equation, and Levy's formula based on the Schur-Weyl duality.

Joint work with Joshua Pfeffer, Scott Sheffield, and Pu Yu.