Noncommutative Majorization Principles and Grothendieck's Inequality

Speaker: 

Steven Heilman

Institution: 

UCLA

Time: 

Tuesday, November 29, 2016 - 11:00pm to 11:50pm

Host: 

Location: 

RH 306

The seminal invariance principle of Mossel-O'Donnell-Oleszkiewicz implies the following. Suppose we have a multilinear polynomial Q, all of whose partial derivatives are small. Then the distribution of Q on i.i.d. uniform {-1,1} inputs is close to the distribution of Q on i.i.d. standard Gaussian inputs. The case that Q is a linear function recovers the Berry-Esseen Central Limit Theorem. In this way, the invariance principle is a nonlinear version of the Central Limit Theorem. We prove the following version of one of the two inequalities of the invariance principle, which we call a majorization principle. Suppose we have a multilinear polynomial Q with matrix coefficients, all of whose partial derivatives are small. Then, for any even K>1, the Kth moment of Q on i.i.d. uniform {-1,1} inputs is larger than the Kth moment of Q on (carefully chosen) random matrix inputs, minus a small number. The exact statement must be phrased carefully in order to avoid being false. Time permitting, we discuss applications of this result to anti-concentration, and to computational hardness for the noncommutative Grothendieck inequality. (joint with Thomas Vidick) (

Loop erased random walk, uniform spanning tree and bi-Laplacian Gaussian field in the critical dimension.

Speaker: 

Wei Wu

Institution: 

CIMS

Time: 

Monday, October 24, 2016 - 11:00am to 11:50am

Host: 

Location: 

NS2 1201

Critical lattice models are believed to converge to a free field in the scaling limit, at or above their critical dimension.  This has been (partially) established for Ising and Phi^4 models for d \geq 4. We describe a simple spin model from uniform spanning forests in $\Z^d$ whose critical dimension is 4 and prove that the scaling limit is the bi-Laplacian Gaussian field for $d\ge 4$. At dimension 4, there is a logarithmic correction for the spin-spin correlation and the bi-Laplacian Gaussian field is a log correlated field. The proof also improves the known mean field picture of LERW in d=4, by showing that the renormalized escape probability (and arm events) of 4D LERW converge to some "continuum escaping probability". Based on joint works with Greg Lawler and Xin Sun.

Compactness and Large Deviations

Speaker: 

Chiranjib Mukherjee

Institution: 

CIMS

Time: 

Tuesday, October 25, 2016 - 11:00am to 11:50am

Host: 

Location: 

RH 306

COMPACTNESS AND LARGE DEVIATIONS

In a reasonable topological space, large deviation estimates essentially deal
with probabilities of events that are asymptotically (exponentially) small,
and in a certain sense, quantify the rate of these decaying probabilities.
In such estimates, upper bounds for such small probabilities often require 
compactness of the ambient space, which is often absent in problems arising in
statistical mechanics (for example, distributions of local times of
Brownian motion in the full space R^d). Motivated by such a problem, we
present a robust theory of “translation-invariant compactification”
of probability measures in R^d. Thanks to an inherent shift-invariance of
the underlying problem, we are able to apply this abstract theory painlessly
and solve a long standing problem in statistical mechanics, the mean-field
polaron problem.

This talk is based on joint works with S. R. S. Varadhan (New York), as
well as with Erwin Bolthausen(Zurich)and Wolfgang Koenig (Berlin).

Universality for the Toda algorithm to compute the eigenvalues of a random matrix.

Speaker: 

Tom Trogdon

Institution: 

UCI

Time: 

Tuesday, October 11, 2016 - 11:00am to 11:50am

Host: 

Location: 

RH 3066

 

 

The Toda lattice, beyond being a completely integrable dynamical system, has many important properties.  Classically, the Toda flow is seen as acting on a specific class of bi-infinite Jacobi matrices.  Depending on the boundary conditions imposed for finite matrices, it is well known that the flow can be used as an eigenvalue algorithm. It was noticed by P. Deift, G. Menon and C. Pfrang that the fluctuations in the time it takes to compute eigenvalues of a random symmetric matrix with the Toda, QR and matrix sign algorithms are universal. In this talk, I will present a proof of such universality for the Toda algorithm.  I will also discuss empirical and rigorous results for other algorithms from numerical analysis.  This is joint work with P. Deift.

A two scale proof of the Eyring-Kramers formula (joint work with Andre Schlichting)

Speaker: 

Georg Menz

Institution: 

UCLA

Time: 

Tuesday, October 4, 2016 - 11:00pm to 11:50pm

Host: 

Location: 

RH 306

We consider a drift-diffusion process on a smooth potential landscape

with small noise. We give a new proof of the Eyring-Kramers formula

which asymptotically characterizes the spectral gap of the generator of

the diffusion. The proof is based on a refinement of the two-scale

approach introduced by Grunewald, Otto, Villani, and Westdickenberg and

of the mean-difference estimate introduced by Chafai and Malrieu. The

new proof exploits the idea that the process has two natural

time-scales: a fast time-scale resulting from the fast convergence to a

metastable state, and a slow time-scale resulting from exponentially

long waiting times of jumps between metastable states. A nice feature

of the argument is that it can be used to deduce an asymptotic formula

for the log-Sobolev constant, which was previously unknown.

Obliquely reflected Brownian motion

Speaker: 

Zhen-qing Chen

Institution: 

University of Washington

Time: 

Tuesday, May 17, 2016 - 1:00pm to 1:50pm

Host: 

Location: 

RH 306

Boundary theory for one-dimensional diffusions is now well understood. Boundary theory for multi-dimensional diffusions is much richer and remains to be better understood. In this talk, we will be concerned with the construction and characterization of obliquely reflected Brownian motions in all bounded simply connected planar domains, including non-smooth domains,
with general reflection vector field on the boundary.  
We show that the family of all obliquely reflected Brownian motions in a given domain can be characterized in two different ways, either by the field of angles of oblique reflection on the boundary or by the stationary distribution and the rate of rotation of the process about a reference point in the domain. We further show that Brownian motion with darning and excursion reflected Brownian motion can be obtained as a limit of obliquely reflected Brownian motions.

Based on joint work with K. Burdzy, D. Marshall and K. Ramanan.

On the % of zeros of Riemann zeta-Function on the critical line.

Speaker: 

Nicolas Martinez Robles

Institution: 

Univ. Illinois

Time: 

Tuesday, April 19, 2016 - 11:00am to 11:50am

Host: 

Location: 

RH 306

Abstract: We will review the techniques used to prove that a positive proportion of the zeros of the Riemann zeta-Function lie on the critical line Re(s)=1/2. The famous Riemann hypothesis states that all the zeros lie there. We will then discuss the mollifiers that allow us to show that > 41% of zeros are critical. This is joint work with A. Roy and A. Zaharescu.

Coupling for Brownian Motion with Redistribution

Speaker: 

Iddo Ben-Ari

Institution: 

University of Connecticut

Time: 

Tuesday, April 5, 2016 - 11:00am to 12:00pm

Location: 

RH 306

We consider a model of Brownian motion on a bounded interval which upon exiting the interval is being redistributed back  into the interval according to a probability measure depending on the exit point, then starting afresh, repeating the above mechanism indefinitely.  It is not hard to show that the process is exponentially ergodic, although characterizing the rate of convergence is non-trivial. In this talk, after providing a general overview of the probabilistic method of coupling and its applications,  I’ll show how to study the ergodicity for the model through coupling, how it leads to an  intuitive and geometric explanation for  the rates of convergence previously obtained analytically, other insights, and more questions. The talk will be accessible to general mathematical audience. 

Reminiscing on partition function zeros and the Lee-Yang circle theorem.

Speaker: 

Marek Biskup

Institution: 

UCLA

Time: 

Tuesday, March 29, 2016 - 11:00am to 12:00pm

Host: 

Location: 

RH 306

I will review some (actually quite old) results by C. Borgs, J.T. Chayes, R. Kotecky and myself concerning the partition function zeros of the Ising model. The focus will be on the fact that, for specific boundary conditions, the zeros lie (in a suitable representation) on the unit circle. I will explain (1) the classic proof of the Lee-Yang circle theorem and (2) how one can nail the positions of the zeros up to exponentially small errors in the system size for the periodic boundary conditions. I may find time to explain how one uses this result to prove the so called Griffiths singularities in site-diluted Ising model.

A Boundedness Trichotomy for the Stochastic Heat Equation

Speaker: 

Davar Khoshnevisan

Institution: 

University of Utah

Time: 

Tuesday, January 26, 2016 - 11:00am to 12:00pm

Host: 

Location: 

RH 306

  We consider the stochastic heat equation with a multiplicative space-time white noise forcing term under standard "intermitency conditions.” The main byproduct of this talk is that, under mild regularity hypotheses, the a.s.-boundedness of the solution$x\mapsto u(t\,,x)$ can be characterized generically by the decay rate, at $\pm\infty$, of the initial function $u_0$. More specifically, we prove that there are 3 generic boundedness regimes, depending on the numerical value of $\Lambda:=\lim_{|x|\to\infty} \vert\log u_0(x)\vert/(\log|x|)^{2/3}$.

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