Schedule

• September 16

  Olivier Bernardi (Brandeis)

  Percolation on triangulations, and a bijective path to Liouville quantum gravity

  Abstract:I will discuss the percolation model on planar triangulations, and present a bijection that is key to relating this model to some fundamental probabilistic objects. I will attempt to achieve several goals:

  1. Present the site-percolation model on random planar triangulations.
  2. Provide an informal introduction to several probabilistic objects: the Gaussian free field, Schramm-Loewner evolutions, and the Brownian map.
  3. Present a bijective encoding of percolated triangulations by certain lattice paths, and explain its role in establishing exact relations between the above-mentioned objects.

  This is joint work with Nina Holden, and Xin Sun.

• September 23

  Two Speakers:

  3:00-4:00 in 2-139:

  Remco van der Hofstad (TU/e)

  Ising models on random graphs

  Abstract: The Ising model is one of the simplest statistical mechanics models that displays a phase transition. While invented by Ising and Lenz to model magnetism, for which the Ising model lives on regular lattices, it is now widely used for other real-world applications as a model for cooperative behavior and consensus between people. As such, it is natural to consider the Ising model on complex networks. Since complex networks are modelled using random graphs, this leads us to study the Ising model on random graphs. In this talk, we discuss some recent results on the stationary distribution of the Ising model on locally tree-like random graphs as well as on its approach to stationarity.

  Due to the randomness of the graphs on which the Ising model lives, there are different settings for the Ising model on it. The quenched setting describes the Ising model on the random graph as it is, while the averaged quenched setting takes the expectation w.r.t. the randomness of the graph. As such, it takes the expectation of the Boltzman distribution, which is a ratio of an exponential involving the Hamiltonian, and the partition function. In the annealed setting, the expectation is taken on both sides of the ratio. These different settings each describe different physical realities.

  We discuss the thermodynamic limit of the Ising model, which can be used to define the phase transition in the Ising model on locally tree-like random graphs, by describing when spontaneous magnetization exists and when not, extending work by Dembo and Montanari. We give an explicit expression for the critical value and the critical exponents for the magnetization close to it. We close by discussing recent results about the fast or slow mixing and metastability for annealed and quenched Ising models on random graphs.

  This talk is based on several joint works with Sander Dommers, Cristian Giardina, Claudio Giberti, Maria Luisa Prioriello, Takashi Kumagai and Hao Can.

  ---

  4:15-5:15 in 2-147:

  Josh Pfeffer (MIT)

  Understanding Liouville quantum gravity through two square subdivision models

  Abstract: In my talk I will discuss a general approach to better understand the geometry of Liouville quantum gravity (LQG). The idea, roughly speaking, is to partition the random surface into dyadic squares of roughly the same "LQG size". Based on this approach, I will introduce two different models of LQG that will provide answers to three questions in the field:

  1. Rigorously explain the so-called "DDK ansatz" by proving that, for a surface with metric tensor some regularized version of the LQG metric tensor e^ɣh (dx² + dy²), its law corresponds to sampling a surface with probability proportional to (det_ζ' Δ)^-c/2, with c the matter central charge
  2. Provide a heuristic picture of the geometry of LQG with matter central charge in the interval (1,25). (The geometry in this regime is mysterious even from a physics perspective.)
  3. Explain why many works in the physics literature may have missed the nontrivial conformal geometry of LQG with matter central charge in the interval (1,25) when they suggest (based on numerical simulations and heuristics) that LQG exhibits the macroscopic behavior of a continuum random tree in this phase.

  This talk is based on a joint work with Morris Ang, Minjae Park, and Scott Sheffield; and a joint work with Ewain Gwynne, Nina Holden, and Guillaume Remy.

• September 30

  Amir Dembo (Stanford)

  Averaging principle and shape theorem for growth with memory

  Abstract: We consider a family of random growth models in n-dimensional space. These models capture certain features expected to manifest at the mesoscopic level for certain self-interacting microscopic dynamics (such as once-reinforced random walk with strong reinforcement and origin-excited random walk). In a joint work with Pablo Groisman, Ruojun Huang and Vladas Sidoravicius, we establish for such models an averaging principle and deduce from it the convergence of the normalized domain boundary, to a limiting shape. The latter is expressed in terms of the invariant measure of an associated Markov chain.

• Friday, October 4

  CHARLES RIVERS LECTURES (Room: E51-115)

• October 7

  Souvik Dhara (MIT/Microsoft Research)

  Critical percolation on networks with heavy-tailed degrees

  Abstract: The talk concerns critical behavior of percolation on finite random networks with heavy-tailed degree distribution. In a seminal paper, Aldous (1997) identified the scaling limit for the component sizes in the critical window of phase transition for the Erdos-Renyi random graph. Subsequently, there has been a surge in the literature identifying two universality classes for the critical behavior depending on whether the asymptotic degree distribution has a finite or infinite third moment.

  In this talk, we will present a completely new universality class that arises in the context of degrees having infinite second moment. Specifically, the scaling limit of the rescaled component sizes is different from the general description of multiplicative coalescent given by Aldous and Limic (1998). Moreover, the study of critical behavior in this regime exhibits several surprising features that have never been observed in any other universality classes so far.

  This talk is based on joint works with Shankar Bhamidi, Remco van der Hofstad, Johan van Leeuwaarden.

• October 14

  Columbus Day/Canadian Thanksgiving: No seminar

• October 21

  ****CANCELLED****

  Eitan Bachmat (Ben-Gurion University)

  Probability, (Lorentzian) geometry and optimization in (low and) high dimension

• October 28

  Patrick Lopatto (Harvard)

  Spectral Statistics of Levy Matrices

  Abstract: Levy matrices are symmetric random matrices whose entries are independent α-stable laws. Such distributions have infinite variance, and when α <1, infinite mean. In the latter case these matrices are conjectured to exhibit a sharp transition from a delocalized regime at low energy to a localized regime at high energy, like the infamous Anderson model in mathematical physics. We discuss work establishing the existence of a delocalized regime with GOE eigenvalue statistics. Further, we characterize the eigenvector statistics in this regime and find they display novel, non-Gaussian behavior.

  These describe joint works with Amol Aggarwal, Jake Marcinek, and Horng-Tzer Yau.

• November 4

  **Special time and room** 3:00-4:00 in 2-139

  Kyle Luh (Harvard)

  Random Matrices over Finite Fields

  Abstract: We will survey some recent developments in this area and highlight several probabilistic tools that that have been successfully applied. One consequence is a notion of universality for several spectral statistics. This is joint work with Hoi Nguyen and Sean Meehan.

• November 11

  Veteran's Day: No seminar

• November 12

  Special date!! Tuesday, 4:15-5:15 in 2-131:

  Dan Romik (UC Davis)

  Random sorting networks and last passage percolation

  Abstract: The oriented swap process is a model for a random sorting network, in which N particles labeled 1,...,N arranged on the discrete lattice [1,N] start out in increasing order and then perform successive adjacent swaps at random times until they reach the reverse configuration N,...,1. In this talk, based on recent joint work with Elia Bisi, Fabio Cunden and Shane Gibbons, I will discuss several new exact distributional identities between a random vector encoding the termination time of the process and random vectors in the corner growth process, a well-known model for randomly growing Young diagrams, or equivalently what is known as last passage percolation. The main identity is still conjectural, and would imply a limiting Tracy-Widom GOE law for the termination time.

  The talk will include entertaining computer simulations and a demonstration of computer-assisted proofs.

• November 18

  Tselil Schramm (Harvard/MIT)

  The Threshold for SDP-Refutation of Random Regular NAE-3-SAT

  Abstract: Unlike its cousin 3SAT, the NAE-3SAT (not-all-equal-3SAT) problem has the property that spectral and semidefinite programming (SDP) algorithms can efficiently refute random instances when the constraint density is a large constant (with high probability). But do these methods work immediately above the "satisfiability threshold", or is there still a range of constraint densities for which random NAE-3SAT instances are unsatisfiable but hard to refute? In this talk I will describe a result in which we show that the latter situation prevails, at least in the context of random regular instances and SDP-based refutation (for basic SDPs). More precisely, whereas a random d-regular instance of NAE-3SAT is easily shown to be unsatisfiable (whp) once d is greater than or equal to 8, we establish the following sharp threshold result regarding efficient refutation: If d < 13.5 then the basic SDP, even augmented with triangle inequalities, fails to refute satisfiability (whp), if d > 13.5 then even the most basic spectral algorithm refutes satisfiability (whp).

  Based on joint work with Yash Deshpande, Andrea Montanari, Ryan O'Donnell and Subhabrata Sen.

• November 25

  Eliran Subag (NYU)

  Geometric TAP approach for spherical spin glasses

  Abstract: The celebrated Thouless-Anderson-Palmer approach suggests a way to relate the free energy of a mean-field spin glass model to the solutions of certain self-consistency equations for the local magnetizations. I will describe a new geometric approach to define free energy landscapes for general spherical mixed p-spin models and derive from them a generalized TAP representation for the free energy. I will then explain how these landscapes are related to the so-called pure states decomposition, ultrametricity property, and optimization of full-RSB models.

• December 2

  Miklos Racz (Princeton)

  Correlated randomly growing graphs

  Abstract: I will introduce a new model of correlated randomly growing graphs and discuss the questions of detecting correlation and estimating aspects of the correlated structure. The model is simple and starts with any model of randomly growing graphs, such as uniform attachment (UA) or preferential attachment (PA). Given such a model, a pair of graphs (G₁, G₂) is grown in two stages: until time t they are grown together (i.e., G₁ = G₂), after which they grow independently according to the underlying model.

  We show that whenever the seed graph has an influence in the underlying graph growth model - this has been shown for PA and UA trees and is conjectured to hold broadly - then correlation can be detected in this model, even if the graphs are grown together for just a single time step. We also give a general sufficient condition (which holds for PA and UA trees) under which detection is possible with probability going to 1 as t_* tends to infinity. Finally, we show for PA and UA trees that the amount of correlation, measured by t_*, can be estimated with vanishing relative error as t_* tends to infinity.

  This is based on joint work with Anirudh Sridhar.

• December 9

  Eyal Lubetzky (NYU)

  Maximum of 3D Ising interfaces

  Abstract: Consider the random surface separating the plus and minus phases, above and below the xy-plane, in the low temperature Ising model in dimension d>2. Dobrushin (1972) showed that if the inverse-temperature β is large enough then this interface is localized: it has O(1) height fluctuations above a fixed point, and its maximum height on a box of side length n is O_P(log n).

  We study the large deviations of the interface in Dobrushin’s setting, and derive a shape theorem for its "pillars" conditionally on reaching an atypically large height. We use this to obtain a law of large numbers for the maximum height M_n of the interface: M_n/ log n converges to c_β in probability, where c_β is given by a large deviation rate in infinite volume. Furthermore, the sequence (M_n - E[M_n])_n is tight, and even though this sequence does not converge, its subsequential limits satisfy uniform Gumbel tails bounds.

  Joint work with Reza Gheissari.