Difference between revisions of "Probability Seminar"

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__NOTOC__
 
__NOTOC__
  
= Fall 2016 =
+
= Fall 2019 =
  
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted.  
+
<b>Thursdays in 901 Van Vleck Hall at 2:30 PM</b>, unless otherwise noted.  
 +
<b>We  usually end for questions at 3:20 PM.</b>
  
<b>
+
If you would like to sign up for the email list to receive seminar announcements then please send an email to  
If you would like to sign up for the email list to receive seminar announcements then please send an email to join-probsem@lists.wisc.edu.
+
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]
</b>
 
  
 +
 +
== September 12, 2019, [https://perso.univ-rennes1.fr/victor.kleptsyn/ Victor Kleptsyn], CNRS and University of Rennes 1 ==
 +
'''Furstenberg theorem: now with a parameter!'''
  
== Thursday, September 8, Daniele Cappelletti, [http://www.math.wisc.edu UW-Madison] ==
+
The classical Furstenberg theorem describes the (almost sure) behaviour of a random product of independent matrices; their norms turn out to grow exponentially. In our joint work with A. Gorodetski, we study what happens if the random matrices depend on an additional parameter.
Title: '''Reaction networks: comparison between deterministic and stochastic models'''
+
It turns out that in this new situation, the conclusion changes. Namely, under some conditions, there almost surely exists a (random) "exceptional" set on parameters where the lower limit for the Lyapunov exponent vanishes.
 +
Our results are related to the Anderson localization in dimension one, providing a purely dynamical viewpoint on its proof. I will also speak about some generalizations and related open questions.
  
Abstract: Mathematical models for chemical reaction networks are widely used in biochemistry, as well as in other fields. The original aim of the models is to predict the dynamics of a collection of reactants that undergo chemical transformations. There exist two standard modeling regimes: a deterministic and a stochastic one. These regimes are chosen case by case in accordance to what is believed to be more appropriate. It is natural to wonder whether the dynamics of the two different models are linked, and whether properties of one model can shed light on the behavior of the other one. Some connections between the two modelling regimes have been known for forty years, and new ones have been pointed out recently. However, many open questions remain, and the issue is still largely unexplored.
+
== September 19, 2019, [http://math.columbia.edu/~xuanw  Xuan Wu], Columbia University==
  
== <span style="color:red"> Friday</span>, September 16, <span style="color:red"> 11 am </span> [http://www.baruch.cuny.edu/math/elenak/ Elena Kosygina], [http://www.baruch.cuny.edu/ Baruch College] and the [http://www.gc.cuny.edu/Page-Elements/Academics-Research-Centers-Initiatives/Doctoral-Programs/Mathematics CUNY Graduate Center] ==
+
'''A Gibbs resampling method for discrete log-gamma line ensemble.'''
  
<div style="width:320px;height:50px;border:5px solid black">
+
In this talk we will construct the discrete log-gamma line ensemble, which is assocaited with inverse gamma polymer model. This log-gamma line ensemble enjoys a random walk Gibbs resampling invariance that follows from the integrable nature of the inverse gamma polymer model via geometric RSK correspondance. By exploiting such resampling invariance, we show the tightness of this log-gamma line ensemble under weak noise scaling. Furthermore, a Gibbs property, as enjoyed by KPZ line ensemble, holds for all subsequential limits.
<b><span style="color:red"> Please note the unusual day and time </span></b>
 
</div>
 
  
The talk will be in Van Vleck 910 as usual.
+
== October 10, 2019, NO SEMINAR - [https://sites.math.northwestern.edu/mwp/ Midwest Probability Colloquium] ==
  
Title: '''Homogenization of viscous Hamilton-Jacobi equations: a remark and an application.'''
+
== October 17, 2019, [https://www.usna.edu/Users/math/hottovy/index.php Scott Hottovy], USNA ==
  
Abstract: It has been pointed out in the seminal work of P.-L. Lions, G. Papanicolaou, and S.R.S. Varadhan that for the first order
+
''' Simplified dynamics for noisy systems with delays.'''
Hamilton-Jacobi (HJ) equation, homogenization starting with affine initial data should imply homogenization for general uniformly
 
continuous initial data. The argument utilized the properties of the HJ semi-group, in particular, the finite speed of propagation. The
 
last property is lost for viscous HJ equations. We remark that the above mentioned implication holds under natural conditions for both
 
viscous and non-viscous Hamilton-Jacobi equations. As an application of our result, we show homogenization in a stationary ergodic  setting for a special class of viscous HJ equations with a non-convex Hamiltonian in one space dimension.
 
This is a joint work with Andrea Davini, Sapienza Università di Roma.
 
  
== Thursday, September 22[http://www.math.wisc.edu/~pmwood/ Philip Matchett Wood], [https://www.math.wisc.edu/ UW-Madison] ==
+
Many biological and physical systems include some type of random noise with a temporal delay. For example, many sperm cells travel in a random motion where their velocity changes according to a chemical signal. This chemotaxis is transmitted through a delay in the system. That is, the sperm notices chemical gradients after a certain time has elapsed. In this case, the delay causes the sperm to aggregate around the egg. In this talk I will consider a general stochastic differential delay equation (SDDE) with state-dependent colored noises and derive its limit as the time delays and the correlation times of the noises go to zero. The analysis leads to a much simpler Stochastic Differential Equation to study. The work is motivated by an experiment involving an electrical circuit with noisy, delayed feedback. The main methods used in the proof are a theorem about convergence of solutions of stochastic differential equations by Kurtz and Protter and a maximal inequality for sums of a stationary sequence of random variables by Peligrad and Utev.
Title:  '''Low-degree factors of random polynomials'''
 
  
Abstract: We study the probability that a monic polynomial with integer coefficients has a low-degree factor over the integers.
+
== October 24, 2019, [https://math.temple.edu/~brider/ Brian Rider], Temple University ==
It is known that certain models are very likely to produce random polynomials that are irreducible, and our project
 
can be viewed as part of a general program of testing whether this is a universal behavior exhibited by many random
 
polynomial models. Interestingly, though the question comes from algebra and number theory, we primarily use tools
 
from combinatorics, including additive combinatorics, and probability theory. We prove for a variety of models that it
 
is very unlikely for a random polynomial with integer coefficients to have a low-degree factor—suggesting that this is, in
 
fact, a universal behavior. For example, we show that the characteristic polynomial of random matrix with independent
 
+1 or −1 entries is very unlikely to have a factor of degree up to <math>n^{1/2-\epsilon}</math>. Joint work with Sean O’Rourke. The talk will also discuss joint work with UW-Madison
 
undergraduates Christian Borst, Evan Boyd, Claire Brekken, and Samantha Solberg, who were supported
 
by NSF grant DMS-1301690 and co-supervised by Melanie Matchett Wood.
 
  
== Thursday, September 29, [http://www.artsci.uc.edu/departments/math/fac_staff.html?eid=najnudjh&thecomp=uceprof Joseph Najnudel],  [http://www.artsci.uc.edu/departments/math.html University of Cincinnati]==
+
'''A general beta crossover ensemble'''
Title:  '''On the maximum of the characteristic polynomial of the Circular Beta Ensemble'''
 
  
In this talk, we present our result on the extremal values of (the logarithm of) the characteristic polynomial of a random unitary matrix whose spectrum is distributed according to the Circular Beta Ensemble. Using different techniques, it gives an improvement and a generalization of the previous recent results by Arguin, Belius, Bourgade on the one hand, and Paquette, Zeitouni on the other hand. They recently treated the CUE case, which corresponds to beta equal to 2.
+
I'll describe an operator limit for a family of general beta ensembles which exhibit a double-scaling. In particular, a free parameter in the system provides for a crossover between the more well-known  "soft" and "hard"  edge point processes.  This new limit operator takes as input the Riccati diffusion associated with the Stochastic Airy Operator. I like to suggest that this hints at a hierarchy of random operators analogous to the Painlevé hierarchy observed at the level of correlation functions for double-scaling ensembles most widely studied at beta = 2. Full disclosure: the result remains partially conjectural due to an unresolved  uniqueness question, but I’ll provide lots of evidence to convince you we have the right answer. Joint work with Jose Ramírez (Univ. Costa Rica).
  
== Thursday, October 6, No Seminar ==
+
== October 31, 2019, Vadim Gorin, UW Madison==
  
== Thursday, October 13, No Seminar due to [http://sites.math.northwestern.edu/mwp/ Midwest Probability Colloquium] ==
+
'''Shift invariance for the six-vertex model and directed polymers.'''
For details, see [http://sites.math.northwestern.edu/mwp/ Midwest Probability Colloquium].
 
  
== Thursday, October 20, [http://www.math.harvard.edu/people/index.html Amol Aggarwal], [http://www.math.harvard.edu/ Harvard] ==
+
I will explain a recently discovered mysterious property in a variety of stochastic systems ranging  from the six-vertex model and to the directed polymers, last passage percolation, Kardar-Parisi-Zhang equation, and Airy sheet. Vaguely speaking, the property says that the multi-point joint distributions are unchanged when some (but not necessarily all!) points of observations are shifted. The property leads to explicit computations for the previously inaccessible joint distributions in all these settings.
Title: Current Fluctuations of the Stationary ASEP and Six-Vertex Model
 
  
Abstract: We consider the following three models from statistical mechanics: the asymmetric simple exclusion process, the stochastic six-vertex model, and the ferroelectric symmetric six-vertex model. It had been predicted by the physics communities for some time that the limiting behavior for these models, run under certain classes of translation-invariant (stationary) boundary data, are governed by the large-time statistics of the stationary Kardar-Parisi-Zhang (KPZ) equation. The purpose of this talk is to explain these predictions in more detail and survey some of our recent work that verifies them.
+
== November 7, 2019, [https://people.kth.se/~tobergg/ Tomas Berggren], KTH Stockholm ==
 +
'''Domino tilings of the Aztec diamond with doubly periodic weightings​'''
  
== Thursday, October 27, [http://www.math.wisc.edu/~hung/ Hung Tran], [http://www.math.wisc.edu/ UW-Madison] ==
+
This talk will be centered around domino tilings of the Aztec diamond with doubly periodic weightings. In particular asymptotic results of the $ 2 \times k $-periodic Aztec diamond will be discussed, both in the macroscopic and microscopic scale. The macroscopic picture is described using a close connection to a Riemann surface. For instance, the number of smooth regions (also called gas regions) is the same as the genus of the mentioned Riemann surface.  
 +
 +
The starting point of the asymptotic analysis is a non-intersecting path formulation and a double integral formula for the correlation kernel. The proof of this double integral formula is based on joint work with M. Duits, which will be discuss briefly if time permits.
  
Title: '''Homogenization of non-convex Hamilton-Jacobi equations'''
+
== November 14, 2019, [https://math.mit.edu/directory/profile.php?pid=2076 Benjamin Landon], MIT ==
 +
'''Universality of extremal eigenvalue statistics of random matrices'''
  
Abstract: I will describe why it is hard to do homogenization for non-convex Hamilton-Jacobi equations and explain some recent results in this direction. I will also make a very brief connection to first passage percolation and address some challenging questions which appear in both directions. This is based on joint work with Qian and Yu.
+
The past decade has seen significant progress on the understanding of universality of various eigenvalue statistics of random matrix theory.  However, the behavior of certain ``extremal'' or ``critical'' observables is not fully understood.  Towards the former, we discuss progress  on the universality of the largest gap between consecutive eigenvalues.  With regards to the latter, we discuss the central limit theorem for the eigenvalue counting function, which can be viewed as a linear spectral statistic with critical regularity and has logarithmically growing variance.
  
== Thursday, November 3, Alejandro deAcosta, [http://math.case.edu/ Case-Western Reserve] ==
+
== November 21, 2019, Tung Nguyen, UW Madison ==
Title:  '''Large deviations for irreducible Markov chains with general state space'''
 
  
Abstract:
+
== November 28, 2019, Thanksgiving (no seminar) ==
We study the large deviation principle for the empirical measure of general irreducible Markov chains in the tao topology for a broad class of initial distributions.  The roles of several rate functions, including the rate function based on the convergence parameter of the transform kernel and the Donsker-Varadhan rate function, are clarified.
 
 
 
== Thursday, November 17, [https://sites.google.com/a/wisc.edu/louisfan/home Louis Fan], [https://www.math.wisc.edu/ UW-Madison] ==
 
Title:  TBA
 
 
 
== Thursday, November 24, No Seminar due to Thanksgiving ==
 
 
 
== Thursday, December 1, TBA, TBA ==
 
Title:  TBA
 
 
 
== Thursday, December 8, TBA, TBA ==
 
Title:  TBA
 
 
 
== Thursday, December 15, TBA, TBA ==
 
Title:  TBA
 
 
 
 
 
 
 
 
 
 
 
<!--
 
 
 
== Thursday, January 28, [http://faculty.virginia.edu/petrov/ Leonid Petrov], [http://www.math.virginia.edu/ University of Virginia] ==
 
 
 
Title: '''The quantum integrable particle system on the line'''
 
 
 
I will discuss the higher spin six vertex model - an interacting  particle
 
system on the discrete 1d line in the Kardar--Parisi--Zhang universality
 
class. Observables of this system admit explicit contour integral expressions
 
which degenerate  to many known formulas of such type for other integrable
 
systems on the line in the KPZ class, including stochastic six vertex model,
 
ASEP, various <math>q</math>-TASEPs, and associated zero range processes. The structure
 
of the higher spin six vertex model (leading to contour integral formulas for
 
observables) is based on Cauchy summation identities for certain symmetric
 
rational functions, which in turn can be traced back to the sl2 Yang--Baxter
 
equation. This framework allows to also include space and spin inhomogeneities
 
into the picture, which leads to new particle systems with unusual phase
 
transitions.
 
 
 
== Thursday, February 4, [http://homepages.math.uic.edu/~nenciu/Site/Contact.html Inina Nenciu], [http://www.math.uic.edu/ UIC], Joint Probability and Analysis Seminar ==
 
 
 
Title: '''On some concrete criteria for quantum and stochastic confinement'''
 
 
 
Abstract: In this talk we will present several recent results on criteria ensuring the confinement of a quantum or a stochastic particle to a bounded domain in <math>\mathbb{R}^n</math>. These criteria are given in terms of explicit growth and/or decay rates for the diffusion matrix and the drift potential close to the boundary of the domain. As an application of the general method, we will discuss several cases, including some where the background Riemannian manifold (induced by the diffusion matrix) is geodesically incomplete. These results are part of an ongoing joint project with G. Nenciu (IMAR, Bucharest, Romania).
 
 
 
== <span style="color:green">Friday, February 5</span>, [http://www.math.ku.dk/~d.cappelletti/index.html Daniele Cappelletti], [http://www.math.ku.dk/ Copenhagen University], speaks in the [http://www.math.wisc.edu/wiki/index.php/Applied/ACMS Applied Math Seminar], <span style="color:green">2:25pm in Room 901 </span>==
 
 
 
'''Note:''' Daniele Cappelletti is speaking in the [http://www.math.wisc.edu/wiki/index.php/Applied/ACMS Applied Math Seminar], but his research on stochastic reaction networks uses probability theory and is related to work of our own [http://www.math.wisc.edu/~anderson/ David Anderson].
 
 
 
Title: '''Deterministic and Stochastic Reaction Networks'''
 
 
 
Abstract:  Mathematical models of biochemical reaction networks are of great interest for the analysis of experimental data and theoretical biochemistry. Moreover, such models can be applied in a broader framework than that provided by biology. The classical deterministic model of a reaction network is a system of ordinary differential equations, and the standard stochastic model is a continuous-time Markov chain. A relationship between the dynamics of the two models can be found for compact time intervals, while the asymptotic behaviours of the two models may differ greatly. I will give an overview of these problems and show some recent development.
 
 
 
 
 
== Thursday, February 25, [http://www.princeton.edu/~rvan/ Ramon van Handel], [http://orfe.princeton.edu/ ORFE] and [http://www.pacm.princeton.edu/ PACM, Princeton] ==
 
 
 
Title: '''The norm of structured random matrices'''
 
 
 
Abstract: Understanding the spectral norm of random matrices is a problem
 
of basic interest in several areas of pure and applied mathematics. While
 
the spectral norm of classical random matrix models is well understood,
 
existing methods almost always fail to be sharp in the presence of
 
nontrivial structure. In this talk, I will discuss new bounds on the norm
 
of random matrices with independent entries that are sharp under mild
 
conditions. These bounds shed significant light on the nature of the
 
problem, and make it possible to easily address otherwise nontrivial
 
phenomena such as the phase transition of the spectral edge of random band
 
matrices. I will also discuss some conjectures whose resolution would
 
complete our understanding of the underlying probabilistic mechanisms.
 
 
 
== Thursday,  March 3, [http://www.math.wisc.edu/~janjigia/ Chris Janjigian], [http://www.math.wisc.edu/ UW-Madison] ==
 
 
 
Title: '''Large deviations for certain inhomogeneous corner growth models'''
 
 
 
Abstract:
 
The corner growth model is a classical model of growth in the plane and is connected to other familiar models such as directed last passage percolation and the TASEP through various geometric maps. In the case that the waiting times are i.i.d. with exponential or geometric marginals, the model is well understood: the shape function can be computed exactly, the fluctuations around the shape function are known to be given by the Tracy-Widom GUE distribution, and large deviation principles corresponding to this limit have been derived.
 
 
 
This talk considers the large deviation properties of a generalization of the classical model in which the rates of the exponential are drawn randomly in an appropriate way. We will discuss some exact computations of rate functions in the quenched and annealed versions of the model, along with some interesting properties of large deviations in this model. (Based on joint work with Elnur Emrah.)
 
 
 
== Thursday,  March 10, [http://www.math.wisc.edu/~jyin/jun-yin.html Jun Yin], [http://www.math.wisc.edu/ UW-Madison] ==
 
 
 
Title: '''Delocalization and Universality of band matrices.'''
 
 
 
Abstract: in this talk we introduce our new work on band matrices, whose eigenvectors and eigenvalues are  widely believed  to have the same asymptotic behaviors as those of Wigner matrices.
 
We proved that this conjecture is true as long as the bandwidth is wide enough.
 
 
 
== Thursday,  March 17, [http://www.math.wisc.edu/~roch/ Sebastien Roch], [http://www.math.wisc.edu/ UW-Madison] ==
 
 
 
 
 
Title: '''Recovering the Treelike Trend of Evolution Despite Extensive Lateral Genetic Transfer'''
 
 
 
Abstract
 
Reconstructing the tree of life from molecular sequences is a fundamental problem in computational
 
biology. Modern data sets often contain large numbers of genes. That can complicate the reconstruction because different genes often undergo different evolutionary histories. This is the case in particular in the presence of lateral genetic transfer (LGT), where a gene is inherited from a distant species rather than an immediate ancestor. Such an event produces a gene tree which is distinct from (but related to) the species phylogeny. In this talk I will sketch recent results showing that, under a natural stochastic model of LGT, the species phylogeny can be reconstructed from gene trees despite surprisingly high rates of LGT.
 
 
 
== Thursday,  March 24, No Seminar, Spring Break ==
 
 
 
== Thursday,  March 31, [http://www.ssc.wisc.edu/~whs/ Bill Sandholm], [http://www.econ.wisc.edu/ Economics, UW-Madison] ==
 
 
 
Title: '''A Sample Path Large Deviation Principle for a Class of Population Processes'''
 
 
 
Abstract:  We establish a sample path large deviation principle for sequences of Markov chains arising in game theory and other applications. As the state spaces of these Markov chains are discrete grids in the simplex, our analysis must account for the fact that the processes run on a set with a boundary. A key step in the analysis establishes joint continuity properties of the state-dependent Cramer transform L(·,·), the running cost appearing in the large deviation principle rate function.
 
 
 
[http://www.ssc.wisc.edu/~whs/research/ldp.pdf paper preprint]
 
 
 
== Thursday,  April 7, No Seminar ==
 
 
 
== Thursday,  April 14, [https://www.math.wisc.edu/~jessica/ Jessica Lin], [https://www.math.wisc.edu/~jessica/ UW-Madison], Joint with [https://www.math.wisc.edu/wiki/index.php/PDE_Geometric_Analysis_seminar PDE Geometric Analysis seminar] ==
 
 
 
Title: '''Optimal Quantitative Error Estimates in Stochastic Homogenization for Elliptic Equations in Nondivergence Form'''
 
 
 
Abstract: I will present optimal quantitative error estimates in the
 
stochastic homogenization for uniformly elliptic equations in
 
nondivergence form. From the point of view of probability theory,
 
stochastic homogenization is equivalent to identifying a quenched
 
invariance principle for random walks in a balanced random
 
environment. Under strong independence assumptions on the environment,
 
the main argument relies on establishing an exponential version of the
 
Efron-Stein inequality. As an artifact of the optimal error estimates,
 
we obtain a regularity theory down to microscopic scale, which implies
 
estimates on the local integrability of the invariant measure
 
associated to the process. This talk is based on joint work with Scott
 
Armstrong.
 
 
 
== Thursday,  April 21, [http://www.cims.nyu.edu/~bourgade/ Paul Bourgade], [https://www.cims.nyu.edu/ Courant Institute, NYU] ==
 
 
 
Title: '''Freezing and extremes of random unitary matrices'''
 
 
 
Abstract: A conjecture of Fyodorov, Hiary & Keating states that the maxima of the characteristic polynomial of random unitary matrices behave like the maxima of a specific class of Gaussian fields, the log-correlated Gaussian fields. We will outline the proof of the conjecture for the leading order of the maximum, and a freezing of the free energy related to the matrix model. This talk is based on a joint work with Louis-Pierre Arguin and David Belius.
 
 
 
== Thursday,  April 28, [http://www.ime.unicamp.br/~nancy/ Nancy Garcia], [http://www.ime.unicamp.br/conteudo/departamento-estatistica Statistics], [http://www.ime.unicamp.br/ IMECC], [http://www.unicamp.br/unicamp/ UNICAMP, Brazil] ==
 
 
 
Title: '''Rumor processes on <math>\mathbb{N}</math> and discrete renewal processe'''
 
 
 
Abstract: We study two rumor processes on the positive integers, the dynamics of which are related to an SI epidemic model with long range transmission. Start with one spreader at site <math>0</math> and ignorants situated at some other sites of <math>\mathbb{N}</math>. The spreaders transmit the information within a random distance on their right.  Depending on the initial distribution of the ignorants, we obtain  probability of survival, information on the distribution of the range of the rumor and limit theorems for the proportion of spreaders. The key step of our approach is to relate this model to the house-of-cards.
 
 
 
== Thursday,  May 5, [http://math.arizona.edu/~dianeholcomb/ Diane Holcomb], [http://math.arizona.edu/ University of Arizona]  ==
 
 
 
 
 
Title: '''Local limits of Dyson's Brownian Motion at multiple times'''
 
 
 
Abstract: Dyson's Brownian Motion may be thought of as a generalization of  Brownian Motion to the matrix setting. We  can study the eigenvalues of a Dyson's Brownian motion at multiple times. The resulting object has different "color" points corresponding to the eigenvalues at different times. Similar to a single time, the correlation functions of the process may be described in terms of determinantal formulas. We study the local behavior of the eigenvalues as we take the dimension of the associated matrix to infinity. The resulting limiting process in the bulk is again determinantal and is described with an "extended sine kernel." This work aims to give an alternate description of the limiting process in terms of the counting function. In this seminar I will go over the the description and methods for finding such a limit. This is work in progress and is joint with Elliot Paquette (Weizmann Institute).
 
 
 
-->
 
 
 
== ==
 
  
  
 +
==December 5, 2019 ==
  
 
[[Past Seminars]]
 
[[Past Seminars]]

Latest revision as of 16:10, 6 November 2019


Fall 2019

Thursdays in 901 Van Vleck Hall at 2:30 PM, unless otherwise noted. We usually end for questions at 3:20 PM.

If you would like to sign up for the email list to receive seminar announcements then please send an email to join-probsem@lists.wisc.edu


September 12, 2019, Victor Kleptsyn, CNRS and University of Rennes 1

Furstenberg theorem: now with a parameter!

The classical Furstenberg theorem describes the (almost sure) behaviour of a random product of independent matrices; their norms turn out to grow exponentially. In our joint work with A. Gorodetski, we study what happens if the random matrices depend on an additional parameter. It turns out that in this new situation, the conclusion changes. Namely, under some conditions, there almost surely exists a (random) "exceptional" set on parameters where the lower limit for the Lyapunov exponent vanishes. Our results are related to the Anderson localization in dimension one, providing a purely dynamical viewpoint on its proof. I will also speak about some generalizations and related open questions.

September 19, 2019, Xuan Wu, Columbia University

A Gibbs resampling method for discrete log-gamma line ensemble.

In this talk we will construct the discrete log-gamma line ensemble, which is assocaited with inverse gamma polymer model. This log-gamma line ensemble enjoys a random walk Gibbs resampling invariance that follows from the integrable nature of the inverse gamma polymer model via geometric RSK correspondance. By exploiting such resampling invariance, we show the tightness of this log-gamma line ensemble under weak noise scaling. Furthermore, a Gibbs property, as enjoyed by KPZ line ensemble, holds for all subsequential limits.

October 10, 2019, NO SEMINAR - Midwest Probability Colloquium

October 17, 2019, Scott Hottovy, USNA

Simplified dynamics for noisy systems with delays.

Many biological and physical systems include some type of random noise with a temporal delay. For example, many sperm cells travel in a random motion where their velocity changes according to a chemical signal. This chemotaxis is transmitted through a delay in the system. That is, the sperm notices chemical gradients after a certain time has elapsed. In this case, the delay causes the sperm to aggregate around the egg. In this talk I will consider a general stochastic differential delay equation (SDDE) with state-dependent colored noises and derive its limit as the time delays and the correlation times of the noises go to zero. The analysis leads to a much simpler Stochastic Differential Equation to study. The work is motivated by an experiment involving an electrical circuit with noisy, delayed feedback. The main methods used in the proof are a theorem about convergence of solutions of stochastic differential equations by Kurtz and Protter and a maximal inequality for sums of a stationary sequence of random variables by Peligrad and Utev.

October 24, 2019, Brian Rider, Temple University

A general beta crossover ensemble

I'll describe an operator limit for a family of general beta ensembles which exhibit a double-scaling. In particular, a free parameter in the system provides for a crossover between the more well-known "soft" and "hard" edge point processes. This new limit operator takes as input the Riccati diffusion associated with the Stochastic Airy Operator. I like to suggest that this hints at a hierarchy of random operators analogous to the Painlevé hierarchy observed at the level of correlation functions for double-scaling ensembles most widely studied at beta = 2. Full disclosure: the result remains partially conjectural due to an unresolved uniqueness question, but I’ll provide lots of evidence to convince you we have the right answer. Joint work with Jose Ramírez (Univ. Costa Rica).

October 31, 2019, Vadim Gorin, UW Madison

Shift invariance for the six-vertex model and directed polymers.

I will explain a recently discovered mysterious property in a variety of stochastic systems ranging from the six-vertex model and to the directed polymers, last passage percolation, Kardar-Parisi-Zhang equation, and Airy sheet. Vaguely speaking, the property says that the multi-point joint distributions are unchanged when some (but not necessarily all!) points of observations are shifted. The property leads to explicit computations for the previously inaccessible joint distributions in all these settings.

November 7, 2019, Tomas Berggren, KTH Stockholm

Domino tilings of the Aztec diamond with doubly periodic weightings​

This talk will be centered around domino tilings of the Aztec diamond with doubly periodic weightings. In particular asymptotic results of the $ 2 \times k $-periodic Aztec diamond will be discussed, both in the macroscopic and microscopic scale. The macroscopic picture is described using a close connection to a Riemann surface. For instance, the number of smooth regions (also called gas regions) is the same as the genus of the mentioned Riemann surface.

The starting point of the asymptotic analysis is a non-intersecting path formulation and a double integral formula for the correlation kernel. The proof of this double integral formula is based on joint work with M. Duits, which will be discuss briefly if time permits.

November 14, 2019, Benjamin Landon, MIT

Universality of extremal eigenvalue statistics of random matrices

The past decade has seen significant progress on the understanding of universality of various eigenvalue statistics of random matrix theory. However, the behavior of certain ``extremal or ``critical observables is not fully understood. Towards the former, we discuss progress on the universality of the largest gap between consecutive eigenvalues. With regards to the latter, we discuss the central limit theorem for the eigenvalue counting function, which can be viewed as a linear spectral statistic with critical regularity and has logarithmically growing variance.

November 21, 2019, Tung Nguyen, UW Madison

November 28, 2019, Thanksgiving (no seminar)

December 5, 2019

Past Seminars