Past Probability Seminars Spring 2020: Difference between revisions

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== Fall 2010 ==
= Fall 2019 =


<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>


Thursdays in 901 Van Vleck Hall at 2:25 PM, unless otherwise noted. If you would like to receive announcements about upcoming seminars, please visit [https://www-old.cae.wisc.edu/mailman/listinfo/apseminar this page] to sign up for the email list.
If you would like to sign up for the email list to receive seminar announcements then please send an email to
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]


== September 12, 2019, [https://perso.univ-rennes1.fr/victor.kleptsyn/ Victor Kleptsyn], CNRS and University of Rennes 1 ==
'''Furstenberg theorem: now with a parameter!'''


[[Past Seminars]]
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, [http://math.columbia.edu/~xuanw  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 - [https://sites.math.northwestern.edu/mwp/ Midwest Probability Colloquium] ==
 
== October 17, 2019, [https://www.usna.edu/Users/math/hottovy/index.php 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, [https://math.temple.edu/~brider/ Brian Rider], Temple University ==


'''A general beta crossover ensemble'''


==Friday, September 3, 4PM B239 [http://www.math.wisc.edu/~seppalai/ Timo Seppäläinen (UW Madison)]  ([[Colloquia|Math Colloquium]])==
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).
:Title: '''Scaling exponents for a 1+1 dimensional directed polymer'''


:Abstract: Directed polymer in a random environment is a model from statistical physics that has been around for 25 years. It is a type of random walk that evolves in a random potential. This means that the walk lives in a random landscape, some parts of which are favorable and other parts unfavorable to the walk. The objective is to understand the behavior of the walk on large space and time scales.
== October 31, 2019, Vadim Gorin, UW Madison==
:I will begin the talk with simple random walk straight from undergraduate probability and explain what diffusive behavior of random walk means and how Brownian motion figures into the picture. The recent result of the talk concerns a particular 1+1 dimensional polymer model: the order of magnitude of the fluctuations of the polymer path is described by the exponent 2/3, in contrast with the exponent 1/2 of diffusive paths. Finding a rigorous proof of this exponent has been an open problem since the introduction of the model.


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


==Thursday, September 16, [http://www.math.wisc.edu/~moreno/ Gregorio Moreno Flores (UW - Madison)]==
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: '''Asymmetric directed polymers in random environments.'''


:Abstract: It is well known that the asymmetric last passage percolation problem can be approximated by a Brownian percolation model, which is itself related to the GUE random matrices. This allows to transfer many results about random matrices to the setting of asymmetric last passage percolation.<br>
== November 7, 2019, [https://people.kth.se/~tobergg/ Tomas Berggren], KTH Stockholm ==
:In this talk, we will introduce two different schemes to treat asymmetric directed polymers in random environments.
'''Domino tilings of the Aztec diamond with doubly periodic weightings​'''


==Thursday, September 30, [http://math.colorado.edu/~brider/ Brian Rider (University of Colorado at Boulder)]==
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.
:Title: '''Solvable two-charge models'''
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.


:Abstract: I'll describe recent progress on ensembles of random matrix type which can be viewed as having particles of two distinct "charges", subject to coulombic interaction.  The natural (and classic) example is Ginibre's  non-symmetric Gaussian matrix in which the particles (eigenvalues) live in the complex plane.  Taking this as a starting point and forcing the particles down to the line produces a family of ensembles which interpolate (though not in the way we might want) between the well studied Gaussian Orthogonal and Symplectic Ensembles.
== November 14, 2019, [https://math.mit.edu/directory/profile.php?pid=2076 Benjamin Landon], MIT ==
'''Universality of extremal eigenvalue statistics of random matrices'''


:Joint work with Christopher Sinclair and Yuan Xu (Univ. Oregon).
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, October 7, [http://www.math.wisc.edu/~valko/ Benedek Valko (UW - Madison)]==
== November 21, 2019, Tung Nguyen, UW Madison ==
:Title: '''Scaling limits of tridiagonal matrices'''


:Abstract: I will describe the point process limits of the spectrum for a certain class of tridiagonal matrices. The limiting point process can be defined through a coupled system of stochastic differential equations. I will discuss various applications of this description, e.g. eigenvalue repulsion, probability of large gaps and central limit theorem for the number of points in an interval.
'''Prevalence of deficiency zero reaction networks under an Erdos-Renyi framework
'''


Reaction network models, which are used to model many types of systems in biology, have grown dramatically in popularity over the past decade. This popularity has translated into a number of mathematical results that relate the topological features of the network to different qualitative behaviors of the associated dynamical system. One of the main topological features studied in the field is ''deficiency'' of a network. A reaction network which has strong connectivity in its connected components and a deficiency of zero is stable in both the deterministic and stochastic dynamical models.


Joint work with E. Kritchevski and B. Virag (Toronto).
This leads to the question: how prevalent are deficiency zero models among all such network models. In this talk, I will quantify the prevalence of deficiency zero networks among random reaction networks generated under an Erdos-Renyi framework. Specifically, with n being the number of species, I will uncover a threshold function r(n) such that the probability of the random network being deficiency zero converges to 1 if the edge probability p_n << r(n) and converges to 0 if p_n >> r(n).


==Thursday, October 14, [http://www.math.northwestern.edu/mwp/ MIDWEST PROBABILITY COLLOQUIUM], (no seminar)==
== November 28, 2019, Thanksgiving (no seminar) ==
<br>


==Thursday, October 21, [http://www.math.wisc.edu/~kuelbs/ Jim Kuelbs (UW - Madison)]==
:Title: '''TBA'''


==Thursday, October 28, [http://www.math.toronto.edu/alberts/ Tom Alberts (University of Toronto)]==
==December 5, 2019 ==
:Title: '''TBA'''


==Thursday, November 18, [http://www.math.wisc.edu/~jmiller/ Joseph S. Miller (UW - Madison)]==
[[Past Seminars]]
:Title: '''TBA'''

Revision as of 15:17, 20 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

Prevalence of deficiency zero reaction networks under an Erdos-Renyi framework

Reaction network models, which are used to model many types of systems in biology, have grown dramatically in popularity over the past decade. This popularity has translated into a number of mathematical results that relate the topological features of the network to different qualitative behaviors of the associated dynamical system. One of the main topological features studied in the field is deficiency of a network. A reaction network which has strong connectivity in its connected components and a deficiency of zero is stable in both the deterministic and stochastic dynamical models.

This leads to the question: how prevalent are deficiency zero models among all such network models. In this talk, I will quantify the prevalence of deficiency zero networks among random reaction networks generated under an Erdos-Renyi framework. Specifically, with n being the number of species, I will uncover a threshold function r(n) such that the probability of the random network being deficiency zero converges to 1 if the edge probability p_n << r(n) and converges to 0 if p_n >> r(n).

November 28, 2019, Thanksgiving (no seminar)

December 5, 2019

Past Seminars