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Student Probability

Organizer: Jiyun Park

Past Events

Apr
19
Date4:00 PM
Location
383N
Speaker
Andrew Lin

Abstract

First- and last-passage percolation are models in which weights are placed on the edges of a graph (usually Z^d), and where the objects of interest are the shortest (respectively longest) paths from the origin to various points on the graph. I'll describe both models and describe…

Mar
08
Date4:00 PM
Location
383N
Speaker
Michael Howes

Abstract: The set of all n by n orthogonal (or unitary) matrices form a compact topological group. This means that orthogonal matrices have a uniform distribution called Haar measure. A natural question is, what do typical orthogonal or unitary matrices “look like”? We will answer two version of…

Mar
01
Date4:00 PM
Location
383N
Speaker
Andrew Lin

Abstract: This quarter, we've explored the empirical distribution of the eigenvalues for general Wigner matrices, showing convergence in distribution to the semicircle law for Hermitian matrices. As Christian described, the special case with Gaussian entries allows us to say more and get an…

Feb
23
Date4:00 PM
Location
383N
Speaker
Jiyun Park

Abstract: The circular law states that the spectral measure of a square matrix with i.i.d. entries of mean zero must converge to the uniform distribution on the unit disk in the complex plane. This result is analogous to the semicircular law for Wigner matrices, but the spectral instability of…

Feb
16
Date4:00 PM
Location
383N
Speaker
Alexandra Stavrianidi (Stanford)

Abstract: In 1980, Voiculescu introduced free probability theory, which lets us study non-commutative random variables, such as random matrices. In particular, in the free probability CLT the Gaussian limit is replaced by a semi-circular limit, which implies the semi-circular law. So far,…

Feb
02
Date4:00 PM
Location
383N
Speaker
Christian Serio

Abstract: I will introduce the GOE (orthogonal) and GUE (unitary) Gaussian ensembles, which are special Wigner matrices with Gaussian entries leading to nice symmetries. The main result will be the Ginibre formula for the density of the eigenvalues of these ensembles. There are several ways to…

Jan
26
Date4:00 PM
Location
383N
Speaker
Milo Marsden

Abstract: We prove the convergence in distribution of the empirical law of the eigenvalues of a Wigner matrix to the semicircle distribution. This is a classical result dating to the founding of the subject. The proof is by the moment method - specifically the convergence of the random…

Jan
19
Date4:00 PM
Location
383N
Speaker
Nathan Tung

Abstract: We will use the random sign matrix model to examine methods for bounding the probability that the least singular value is small. This quantity is relevant in many (and in some cases the only known) methods to establish limiting spectral laws of all sorts of random matrix models. It's…

Nov
17
Date4:00 PM
Location
383N
Speaker
Shengtong Zhang

Abstract:

I will be talking about the paper Improved Spin-Wave Estimate for Wilson Loops in U(1) Lattice Gauge Theory by Garban and Sepúlveda. One of their main results shows that lattice Yang-Mills theory with G = U(1) decouples into a "gradient spin wave" and a "…

Nov
10
Date4:00 PM
Location
383N
Speaker
Daniel Kim (Stanford)

Abstract: 

Continuing from last week, we will use cluster expansion toprove the main theorem of [Cao20] for finite abelian gauge groups witha one-dimensional unitary representation. I will first explain wherethe expression e^{-ell r_beta (1 - A_beta)} comes from and why weintuitively…