Probability and Mathematical Physics Seminar

Synopsis:

We discuss the scaling limit of various connection probabilities in two-dimensional critical percolation.

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Synopsis:

We consider supercritical bond percolation onZ^d and study the chemical distance, i.e., the graph distance on the infinite cluster. It is well-known that there exists a deterministic constant μ(x) such that the chemical distance D(0,nx) between two connected points 0 and nx grows like nμ(x). We prove the existence of the rate function for the upper tail large deviation event {D(0,nx)>nμ(x)(1+ϵ),0↔nx} for d>=3.  

Joint work with Shuta Nakajima.

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Synopsis:

Lattice gauge theories with finite gauge groups are statistical mechanical models, very much akin to the Ising model, but with some twists. In recent years there have been various works studying the behavior of Wilson loop observables (which are the basic observables associated with these models) in various regimes. In my talk I will survey these results and give some proof ideas. 

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We study the number of equilibria (stationary points) in a non-relaxational analog of the gradient flow for the spherical p-spin model. The asymptotics for the expected number of equilibria in this model were recently computed by Fyodorov, followed by a similar computation for the expected number of stable equilibria by Garcia. These results suggest a critical threshold in terms of the strength of the non-relaxational term, above which the system would transition from having an abundance of stable equilibria, to having none at all. This mirrors the transition from relative to absolute instability found in the generalized May-Wigner model.

We confirm this picture by showing that for $p>9$ the number of equilibria, as well as the number of stable equilibria, concentrate around their respective averages, generalizing recent results of Subag and Zeitouni in the relaxational case. This confirms the existence of the above transition, as well as shows that the fraction of equilibria which are stable coincides with its annealed variant. 

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Abstract:

Selberg’s central limit theorem gives that the logarithm of the Riemann zeta function taken at a uniformly drawn height in $[T, 2T]$ behaves as a complex centered Gaussian random variable with variance $\log\log T$.  A natural question is to investigate how far the Gaussian decay persists.  We present results on the right tail for the real part of the logarithm, where the absolute value of zeta is `unusually large’, on the scale of the exponential of the variance. Our proof employs a recursive scheme of Arguin, Bourgade and Radziwi{\l}{\l} to inductively work with the logarithm of zeta, interpreted as a random walk. The result is in agreement with the corresponding (known) random matrix result, under the usual dictionary, and has a number of nice corollaries. This work is joint with Louis-Pierre Arguin. 

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In recent years there has been a lot of progress in studying the extrema of logarithmically correlated random fields. In the talk I will review some of the results in the area, and then discuss an extension to log-correlated Gaussian fields in an environment that is itself random. One key example is the two-dimensional Gaussian free field on a supercritical bond percolation cluster. In order to study such models, one needs to combine tools from quantitative stochastic homogenization and classical probabilistic estimates for branching structures.

Based on joint work with Ofer Zeitouni.

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Synopsis:

Bandit problems with linear or concave reward have been extensively studied, but relatively few works have studied bandits with non-concave reward. In this talk, we consider a large family of bandit problems where the unknown underlying reward function is non-concave, including the low-rank generalized linear bandit problems and two-layer neural network with polynomial activation bandit problem. For the low-rank generalized linear bandit problem, we provide a minimax-optimal algorithm in the dimension, refuting both conjectures in (Lu et al. 2021) and (Jun et al. 2019). Our algorithms are based on a unified zeroth-order optimization paradigm that applies in great generality and attains optimal rates in several structured polynomial settings (in the dimension). We further demonstrate the applicability of our algorithms in RL in the generative model setting, resulting in improved sample complexity over prior approaches. Finally, we show that the standard optimistic algorithms (e.g., UCB) are sub-optimal by dimension factors. In the neural net setting (with polynomial activation functions) with noiseless reward, we provide a bandit algorithm with sample complexity equal to the intrinsic algebraic dimension. Again, we show that optimistic approaches have worse sample complexity, polynomial in the extrinsic dimension (which could be exponentially worse in the polynomial degree).

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The Probability and the City seminar is a joint meeting of the probability seminars at Courant and at Columbia, held twice every semester (hosted once at each institution).
The second meeting of Fall 2022 will feature:

• Evita Nestoridi (Princeton)
• Tom Trogdon (University of Washington)

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Synopsis:

Conference in honor of Ofer Zeitouni's 60th birthday

9:00 AM      Sylvia Serfaty
9:50 AM      Break
10:00 AM    Elliot Paquette
10:50 AM    Break
11:10 AM    Rami Atar
12:00 PM    Lunch break
2:00 PM      Eliran Subag
2:50 PM      Break
3:10 PM      Amir Dembo

https://cims.nyu.edu/conferences/zeitouni

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Synopsis:

Conference in honor of Ofer Zeitouni's 60th birthday

9:00 AM      Andrea Montanari
9:50 AM      Break
10:00 AM    Percy Deift
10:50 AM    Break
11:10 AM    Elena Kosygina / Jon Peterson
12:00 PM    Lunch break
2:00 PM      Scott Armstrong
2:50 PM      Break
3:10 PM      Stefano Olla
4:00 PM      Break
4:10 PM      Fanny Augeri

https://cims.nyu.edu/conferences/zeitouni

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Synopsis:

Conference in honor of Ofer Zeitouni's 60th birthday

9:00 AM      Alice Guionnet
9:50 AM      Break
10:00 AM    Erwin Bolthausen
10:50 AM    Break
11:10 AM    Takashi Kumagai
12:00 PM    Lunch break
2:00 PM      Lenya Ryzhik
2:50 PM      Break
3:10 PM      Kavita Ramanan
4:00 PM      Break
4:10 PM      Mark Rudelson

https://cims.nyu.edu/conferences/zeitouni

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Synopsis:

Conference in honor of Ofer Zeitouni's 60th birthday

9:00 AM      Gerard Ben Arous
9:50 AM      Break
10:00 AM    S.R.Srinivasa Varadhan
10:50 AM    Break
11:10 AM    Ivan Corwin
12:00 PM    Lunch break
2:00 PM      Gady Kozma
2:50 PM      Break
3:10 PM      Nina Gantert
4:00 PM      Break
4:10 PM      Nike Sun

https://cims.nyu.edu/conferences/zeitouni

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Synopsis:

We consider a general system of interacting random loops which includes several models of interest, such as the spin O(N) model, the double dimer model, random lattice permutations, and is related to the loop O(N) model and to the interacting Bose gas in discrete space. We present an overview on these models, introduce some of the main open questions about the size and the geometry of the loops, and present some recent results about the occurrence of macroscopic loops in dimensions d > 2 as the inverse temperature is large enough. More precisely, we consider arbitrary large boxes and prove that, given any two vertices whose distance is proportional to the diameter of the box, the probability of observing a loop visiting both is uniformly positive. 

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A distinct covering system of congruences is a list of congruences \[ a_i \bmod m_i, \qquad i = 1, 2, ..., k \] whose union is the integers. Erd\H{o}s asked if the least modulus $m_1$ of a distinct covering system of congruences can be arbitrarily large (the minimum modulus problem for covering systems, $1000) and if there exist distinct covering systems of congruences all of whose moduli are odd (the odd problem for covering systems, $25). I'll discuss my proof of a negative answer to the minimum modulus problem, and a quantitative refinement with Pace Nielsen that proves that any distinct covering system of congruences has a modulus divisible by either 2 or 3. The proofs use the probabilistic method and in particular use a sequence of pseudorandom probability measures adapted to the covering process. Time permitting, I may briefly discuss a reformulation of our method due to Balister, Bollob\'{a}s, Morris, Sahasrabudhe and Tiba which solves a conjecture of Shinzel (any distinct covering system of congruences has one modulus that divides another) and gives a negative answer to the square-free version of the odd problem.

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Synopsis:

An important open question is to show that the height function associated with the square ice model (i.e., planar six vertex model with uniform weights), or equivalently the uniform graph homeomorphisms, converges to a continuum Gaussian free field In the scaling limit, I will review some recent results about this model, including that the single point height function, upon renormalization, converges to a Gaussian random variable. 

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Synopsis:

The Probability and the City seminar is a joint meeting of the probability seminars at Courant and at Columbia, held twice every semester (hosted once at each institution).
The first meeting of Fall 2022 will feature:

• Nina Holden (Courant Institute, NYU)
  Conformal welding in Liouville quantum gravity: recent results and applications
  
  Abstract:
  
  Liouville quantum gravity (LQG) is a natural model for a random fractal surface with origin in the physics literature. A powerful tool in the study of LQG is conformal welding, where multiple LQG surfaces are combined into a single LQG surface. The interfaces between the original LQG surfaces are typically described by variants of the random fractal curves known as Schramm-Loewner evolutions (SLE). We will present a few recent conformal welding results for LQG surfaces and their applications, which range from SLE and LQG to planar maps and random permutations. Based on joint works with Ang and Sun, with Lehmkuehler, and with Borga, Sun and Yu.
   
• Ron Peled (Tel-Aviv University)
  Random packings and liquid crystals
  
  Abstract:
  Let T be a subset of R^d, such as a ball, a cube or a cylinder, and consider all possibilities for packing translates of T, perhaps with its rotations, in some bounded domain in R^d. What does a typical packing of this sort look like? One mathematical formalization of this question is to fix the density of the packing and sample uniformly among all possible packings with this density. Discrete versions of the question may be formulated on lattice graphs.
  The question arises naturally in the sciences, where T may be thought of as a molecule and its packing is related to the spatial arrangement of molecules of a material under given conditions. In some cases, the material forms a liquid crystal - states of matter which are, in a sense, between liquids and crystals. 
  I will review ideas from this topic, mentioning some of the predictions and the mathematical progress. Time permitting, I will elaborate on a recent result, joint with Daniel Hadas, on the structure of high-density packings of 2x2 squares with centers on the square lattice.
  The talk is meant to be accessible to a general mathematical audience.

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Synopsis:

The Liouville quantum gravity measure is a properly renormalized exponential of the 2d GFF. In this talk, I will explain how it appears as a limit of natural random matrix dynamics: if (U_t) is a Brownian motion on the unitary group at equilibrium, then the measures $|det(U_t - e^{i theta}|^gamma dt dtheta$ converge to the 2d LQG measure with parameter $gamma$, in the limit of large dimension. This extends results from Webb, Nikula and Saksman for fixed time. The proof relies on a new method for Fisher-Hartwig asymptotics of Toeplitz determinants with real symbols, which extends to multi-time settings. In particular, I will explain how to obtain multi-time loop equations by stochastic analysis on Lie groups.
This is based on a joint work with Paul Bourgade.

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