Difference between revisions of "Probability Seminar"

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(Thursday, March 26, Ji Oon Lee, KAIST)
(November 21, 2019, Tung Nguyen, UW Madison)
 
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__NOTOC__
 
__NOTOC__
  
= Spring 2015 =
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= 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]
  
<!-- [[File:probsem.jpg]] -->
+
</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, January 15, [http://www.stat.berkeley.edu/~racz/ Miklos Racz], [http://statistics.berkeley.edu/ UC-Berkeley Stats] ==
 
  
 +
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.
  
Title: Testing for high-dimensional geometry in random graphs
+
== September 19, 2019, [http://math.columbia.edu/~xuanw  Xuan Wu], Columbia University==
  
Abstract: I will talk about a random geometric graph model, where connections between vertices depend on distances between latent d-dimensional labels; we are particularly interested in the high-dimensional case when d is large. Upon observing a graph, we want to tell if it was generated from this geometric model, or from an Erdos-Renyi random graph. We show that there exists a computationally efficient procedure to do this which is almost optimal (in an information-theoretic sense). The key insight is based on a new statistic which we call "signed triangles". To prove optimality we use a bound on the total variation distance between Wishart matrices and the Gaussian Orthogonal Ensemble. This is joint work with Sebastien Bubeck, Jian Ding, and Ronen Eldan.
+
'''A Gibbs resampling method for discrete log-gamma line ensemble.'''
  
== Thursday, January 22, No Seminar  ==
+
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.
  
== Thursday, January 29, [http://www.math.umn.edu/~arnab/ Arnab Sen], [http://www.math.umn.edu/ University of Minnesota]  ==
+
== October 10, 2019, NO SEMINAR - [https://sites.math.northwestern.edu/mwp/ Midwest Probability Colloquium] ==
  
Title: '''Double Roots of Random Littlewood Polynomials'''
+
== October 17, 2019, [https://www.usna.edu/Users/math/hottovy/index.php Scott Hottovy], USNA ==
  
Abstract:
+
''' Simplified dynamics for noisy systems with delays.'''
We consider random polynomials whose coefficients are independent and uniform on {-1,1}. We will show that the probability that such a polynomial of degree n has a double root is o(n^{-2}) when n+1 is not divisible by 4 and is of the order n^{-2} otherwise. We will also discuss extensions to random polynomials with more general coefficient distributions.
 
  
This is joint work with Ron Peled and Ofer Zeitouni.
+
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.
  
== Thursday, February 5, No seminar this week  ==
+
== October 24, 2019, [https://math.temple.edu/~brider/ Brian Rider], Temple University ==
  
== Thursday, February 12, No Seminar this week==
+
'''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==
== Wednesday, <span style="color:red">February 11</span>, [http://www.math.wisc.edu/~stechmann/ Sam Stechmann], [http://www.math.wisc.edu/ UW-Madison==
 
  
<span style="color:red">Please note the unusual time and room.
+
'''Shift invariance for the six-vertex model and directed polymers.'''
</span>
 
  
 +
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: Stochastic Models for Rainfall: Extreme Events and Critical Phenomena
+
== November 7, 2019, [https://people.kth.se/~tobergg/ 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.  
Abstract:
 
In recent years, tropical rainfall statistics have been shown to conform to paradigms of critical phenomena and statistical physics. In this talk, stochastic models will be presented as prototypes for understanding the atmospheric dynamics that leads to these statistics and extreme events. Key nonlinear ingredients in the models include either stochastic jump processes or thresholds (Heaviside functions). First, both exact solutions and simple numerics are used to verify that a suite of observed rainfall statistics is reproduced by the models, including power-law distributions and long-range correlations. Second, we prove that a stochastic trigger, which is a time-evolving indicator of whether it is raining or not, will converge to a deterministic threshold in an appropriate limit. Finally, we discuss the connections among these rainfall models, stochastic PDEs, and traditional models for critical phenomena.
 
--->
 
 
 
== Thursday, February 19, [http://www.math.purdue.edu/people/bio/guo297 Xiaoqin Guo], [http://www.math.purdue.edu/ Purdue]  ==
 
 
 
Title: Quenched invariance principle for random walks in time-dependent random environment
 
 
 
Abstract: In this talk we discuss random walks in a time-dependent zero-drift random environment in <math>Z^d</math>. We prove a quenched invariance principle under an appropriate moment condition. The proof is based on the use of a maximum principle for parabolic difference operators. This is a joint work with Jean-Dominique Deuschel and Alejandro Ramirez.
 
 
 
== Thursday, February 26, [http://wwwf.imperial.ac.uk/~dcrisan/ Dan Crisan], [http://www.imperial.ac.uk/natural-sciences/departments/mathematics/ Imperial College London]  ==
 
 
 
Title: '''Smoothness properties of randomly perturbed semigroups with application to nonlinear filtering'''
 
 
 
Abstract:
 
In this talk I will discuss sharp gradient bounds for perturbed diffusion semigroups. In contrast with existing results, the perturbation is here random and the bounds obtained are pathwise. Our approach builds on the classical work of Kusuoka and Stroock and extends their program developed for the heat semi-group to solutions of stochastic partial differential equations. The work is motivated by and applied to nonlinear filtering. The analysis allows us to derive pathwise gradient bounds for the un-normalised conditional distribution of a partially observed signal. The estimates we derive have sharp small time asymptotics
 
 
   
 
   
This is joint work with Terry Lyons (Oxford) and Christian Literrer (Ecole Polytechnique) and is based on the paper
+
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.
 
 
D Crisan, C Litterer, T Lyons, Kusuoka–Stroock gradient bounds for the solution of the filtering equation, Journal of Functional Analysis, 2105
 
 
 
== Wednesday, <span style="color:red">March 4</span>, [http://www.math.wisc.edu/~stechmann/ Sam Stechmann], [http://www.math.wisc.edu/ UW-Madison], <span style="color:red"> 2:25pm Van Vleck B113</span>  ==
 
 
 
<span style="color:red">Please note the unusual time and room.
 
</span>
 
 
 
 
 
Title: Stochastic Models for Rainfall: Extreme Events and Critical Phenomena
 
 
 
 
 
Abstract:
 
In recent years, tropical rainfall statistics have been shown to conform to paradigms of critical phenomena and statistical physics. In this talk, stochastic models will be presented as prototypes for understanding the atmospheric dynamics that leads to these statistics and extreme events. Key nonlinear ingredients in the models include either stochastic jump processes or thresholds (Heaviside functions). First, both exact solutions and simple numerics are used to verify that a suite of observed rainfall statistics is reproduced by the models, including power-law distributions and long-range correlations. Second, we prove that a stochastic trigger, which is a time-evolving indicator of whether it is raining or not, will converge to a deterministic threshold in an appropriate limit. Finally, we discuss the connections among these rainfall models, stochastic PDEs, and traditional models for critical phenomena.
 
 
 
== Thursday, March 12, [http://www.ima.umn.edu/~ohadfeld/Website/index.html Ohad Feldheim], [http://www.ima.umn.edu/ IMA]  ==
 
 
 
 
 
Title: '''The 3-states AF-Potts model in high dimension'''
 
 
 
Abstract:
 
<!--
 
Take a bounded odd domain of the bipartite graph $\mathbb{Z}^d$. Color the boundary of the set by $0$, then
 
color the rest of the domain at random with the colors $\{0,\dots,q-1\}$, penalizing every
 
configuration with proportion to the number of improper edges at a given rate $\beta>0$ (the "inverse temperature").
 
Q: "What is the structure of such a coloring?"
 
 
 
This model is called the $q$-states Potts antiferromagnet(AF), a classical spin glass model in statistical mechanics.
 
The $2$-states case is the famous Ising model which is relatively well understood.
 
The $3$-states case in high dimension has been studies for $\beta=\infty$,
 
when the model reduces to a uniformly chosen proper three coloring of the domain.
 
Several words, by Galvin, Kahn, Peled, Randall and Sorkin established the structure of the model
 
showing long-range correlations and phase coexistence. In this work, we generalize this result to positive temperature,
 
showing that for large enough $\beta$ (low enough temperature)
 
the rigid structure persists. This is the first rigorous result for $\beta<\infty$.
 
 
 
In the talk, assuming no acquaintance with the model, we shall give the physical background, introduce all the
 
relevant definitions and shed some light on how such results are proved using only combinatorial methods.
 
Joint work with Yinon Spinka.
 
-->
 
Take a bounded odd domain of the bipartite graph <math>\mathbb{Z}^d</math>.    Color the
 
boundary of the set by <math>0</math>, then
 
color the rest of the domain at random with the colors <math>\{0,\dots,q-1\}</math>,
 
penalizing every
 
configuration with proportion to the number of improper edges at a given rate
 
<math>\beta>0</math> (the "inverse temperature").
 
Q: "What is the structure of such a coloring?"
 
 
 
This model is called the <math>q</math>-states Potts antiferromagnet(AF), a    classical spin
 
glass model in statistical mechanics.
 
The <math>2</math>-states case is the famous Ising model which is relatively    well
 
understood.
 
The <math>3</math>-states case in high dimension has been studies for          <math>\beta=\infty</math>,
 
when the model reduces to a uniformly chosen proper three coloring of the
 
domain.
 
Several words, by Galvin, Kahn, Peled, Randall and Sorkin established the
 
structure of the model
 
showing long-range correlations and phase coexistence. In this work, we
 
generalize this result to positive temperature,
 
showing that for large enough <math>\beta</math> (low enough temperature)
 
the rigid structure persists. This is the first rigorous result for
 
<math>\beta<\infty</math>.
 
 
 
In the talk, assuming no acquaintance with the model, we shall give the
 
physical background, introduce all the
 
relevant definitions and shed some light on how such results are proved using
 
only combinatorial methods.
 
Joint work with Yinon Spinka.
 
 
 
== Thursday, March 19,  [http://www.cmc.edu/pages/faculty/MHuber/ Mark Huber], [http://www.cmc.edu/math/ Claremont McKenna Math]  ==
 
 
 
Title: Understanding relative error in Monte Carlo simulations
 
 
 
Abstract:  The problem of estimating the probability <math>p</math> of heads on an unfair coin has been around for centuries, and has inspired numerous advances in probability such as the Strong Law of Large Numbers and the Central Limit Theorem.  In this talk, I'll consider a new twist:  given an estimate <math>\hat p</math>, suppose we want to understand the behavior of the relative error <math>(\hat p - p)/p</math>.  In classic estimators, the values that the relative error can take on depends on the value of <math>p</math>.  I will present a new estimate with the remarkable property that the distribution of the relative error does not depend in any way on the value of <math>p</math>.  Moreover, this new estimate is very fast:  it takes a number of coin flips that is very close to the theoretical minimum.  Time permitting, I will also discuss new ways to use concentration results for estimating the mean of random variables where normal approximations do not apply.
 
 
 
== Thursday, March 26, [http://mathsci.kaist.ac.kr/~jioon/ Ji Oon Lee], [http://www.kaist.edu/html/en/index.html KAIST]  ==
 
 
 
Title: Tracy-Widom Distribution for Sample Covariance Matrices with General Population
 
 
 
Abstract:
 
Consider the sample covariance matrix <math>(\Sigma^{1/2} X)(\Sigma^{1/2} X)^*</math>, where the sample <math>X</math> is an <math>M \times N</math>  random matrix whose entries are real independent random variables with variance <math>1/N</math> and <math>\Sigma</math> is an  <math>M \times M</math> positive-definite deterministic diagonal matrix. We show that the fluctuation of its rescaled largest eigenvalue is given by the type-1 Tracy-Widom distribution. This is a joint work with Kevin Schnelli.
 
 
 
== Thursday, April 2, No Seminar, Spring Break  ==
 
 
 
 
 
 
 
 
 
== Thursday, April 9, [http://www.math.wisc.edu/~emrah/ Elnur Emrah], [http://www.math.wisc.edu/ UW-Madison]  ==
 
 
 
Title: TBA
 
 
 
Abstract:
 
 
 
 
 
== Thursday, April 16, TBA  ==
 
 
 
Title: TBA
 
 
 
Abstract:
 
 
 
== Thursday, April 23, [http://people.math.osu.edu/nguyen.1261/ Hoi Nguyen], [http://math.osu.edu/ Ohio State University]  ==
 
 
 
Title: TBA
 
 
 
Abstract:
 
 
 
== Thursday, April 30, TBA  ==
 
 
 
Title: TBA
 
 
 
Abstract:
 
 
 
 
 
== Thursday, May 7, TBA  ==
 
 
 
Title: TBA
 
 
 
Abstract:
 
 
 
 
 
 
 
 
 
 
 
 
 
<!--
 
== Thursday, December 11, TBA  ==
 
 
 
Title: TBA
 
 
 
Abstract:
 
-->
 
 
 
 
 
 
 
<!--
 
 
 
== Thursday, September 11, <span style="color:red">Van Vleck B105,</span> [http://www.math.wisc.edu/~mmwood/ Melanie Matchett Wood], UW-Madison ==
 
 
 
Please note the non-standard room.
 
 
 
Title: '''The distribution of sandpile groups of random graphs'''
 
 
 
Abstract:<br>
 
The sandpile group is an abelian group associated to a graph, given as
 
the cokernel of the graph Laplacian.  An Erdős–Rényi random graph
 
then gives some distribution of random abelian groups.  We will give
 
an introduction to various models of random finite abelian groups
 
arising in number theory and the connections to the distribution
 
conjectured by Payne et. al. for sandpile groups.  We will talk about
 
the moments of random finite abelian groups, and how in practice these
 
are often more accessible than the distributions themselves, but
 
frustratingly are not a priori guaranteed to determine the
 
distribution.  In this case however, we have found the moments of the
 
sandpile groups of random graphs, and proved they determine the
 
measure, and have proven Payne's conjecture.
 
 
 
== Thursday, September 18, [http://www.math.purdue.edu/~peterson/ Jonathon Peterson], [http://www.math.purdue.edu/ Purdue University]  ==
 
 
 
Title: '''Hydrodynamic limits for directed traps and systems of independent RWRE'''
 
 
 
Abstract:
 
 
 
We study the evolution of a system of independent random walks in a common random environment (RWRE). Previously a hydrodynamic limit was proved in the case where the environment is such that the random walks are ballistic (i.e., transient with non-zero speed <math>v_0 \neq 0)</math>. In this case it was shown that the asymptotic particle density is simply translated deterministically by the speed $v_0$. In this talk we will consider the more difficult case of RWRE that are transient but with $v_0=0$. Under the appropriate space-time scaling, we prove a hydrodynamic limit for the system of random walks. The statement of the hydrodynamic limit that we prove is non-standard in that the evolution of the asymptotic particle density is given by the solution of a random rather than a deterministic PDE. The randomness in the PDE comes from the fact that under the hydrodynamic scaling the effect of the environment does not ``average out'' and so the specific instance of the environment chosen actually matters.
 
 
 
The proof of the hydrodynamic limit for the system of RWRE will be accomplished by coupling the system of RWRE with a simpler model of a system of particles in an environment of ``directed traps.'' This talk is based on joint work with Milton Jara.
 
 
 
== Thursday, September 25, [http://math.colorado.edu/~seor3821/ Sean O'Rourke],  [http://www.colorado.edu/math/ University of Colorado Boulder]  ==
 
 
 
Title: '''Singular values and vectors under random perturbation'''
 
 
 
Abstract:
 
Computing the singular values and singular vectors of a large matrix is a basic task in high dimensional data analysis with many applications in computer science and statistics. In practice, however, data is often perturbed by noise. A natural question is the following. How much does a small perturbation to the matrix change the singular values and vectors?
 
 
 
Classical (deterministic) theorems, such as those by Davis-Kahan, Wedin, and Weyl, give tight estimates for the worst-case scenario. In this talk, I will consider the case when the perturbation is random. In this setting, better estimates can be achieved when our matrix has low rank.  This talk is based on joint work with Van Vu and Ke Wang.
 
 
 
== Thursday, October 2, [http://www.math.wisc.edu/~jyin/jun-yin.html Jun Yin], [http://www.math.wisc.edu/ UW-Madison]  ==
 
 
 
Title: '''Anisotropic local laws for random matrices'''
 
 
 
Abstract:
 
In this talk, we introduce a new method of deriving  local laws of random matrices.  As applications, we will show the local laws  and some universality results on general sample covariance matrices: TXX^*T^* (where $T$ is non-square deterministic matrix),  and deformed Wigner matrix: H+A (where A is deterministic symmetric matrix). Note: here $TT^*$ and $A$ could be full rank matrices.
 
 
 
== Thursday, October 9, No seminar due to [http://www.math.northwestern.edu/mwp/ Midwest Probability Colloquium]  ==
 
 
 
No seminar due to [http://www.math.northwestern.edu/mwp/ Midwest Probability Colloquium].
 
 
 
 
 
== Thursday, October 16, [http://www.math.utah.edu/~firas/ Firas Rassoul-Agha], [http://www.math.utah.edu/ University of Utah]==
 
 
 
Title: '''The growth model: Busemann functions, shape, geodesics, and other stories'''
 
 
 
Abstract:
 
We consider the directed last-passage percolation model on the planar integer lattice with nearest-neighbor steps and general i.i.d. weights on the vertices, outside the class of exactly solvable models. Stationary cocycles are constructed for this percolation model from queueing fixed points. These cocycles define solutions to variational formulas that characterize limit shapes and yield new results for Busemann functions, geodesics and the competition interface.  This is joint work with Nicos Georgiou and Timo Seppalainen.
 
 
 
 
 
== Thursday, November 6, Vadim Gorin, [http://www-math.mit.edu/people/profile.php?pid=1415 MIT]  ==
 
 
 
Title: '''Multilevel Dyson Brownian Motion and its edge limits.'''
 
 
 
Abstract: The GUE Tracy-Widom distribution is known to govern the large-time asymptotics for a variety of
 
interacting particle systems on one side, and the asymptotic behavior for largest eigenvalues of
 
random Hermitian matrices on the other side. In my talk I will explain some reasons for this
 
connection between two seemingly unrelated classes of stochastic systems, and how this relation can
 
be extended to general beta random matrices. A multilevel extension of the Dyson Brownian Motion
 
will be the central object in the discussion.
 
 
 
(Based on joint papers with Misha Shkolnikov.)
 
 
 
==<span style="color:red"> Friday</span>, November 7, [http://tchumley.public.iastate.edu/ Tim Chumley], [http://www.math.iastate.edu/ Iowa State University] ==
 
 
 
<span style="color:darkgreen">Please note the unusual day.</span>
 
 
 
Title: '''Random billiards and diffusion'''
 
 
 
Abstract: We introduce a class of random dynamical systems derived from billiard maps and study a certain Markov chain derived from them. We then discuss the interplay between the billiard geometry and stochastic properties of the random system.  The main results presented investigate the affect of billiard geometry on a diffusion process obtained from an appropriate scaling limit of the Markov chain.
 
 
 
== Thursday, November 13, [http://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], [http://www.math.wisc.edu/ UW-Madison]==
 
 
 
Title: '''Variational formulas for directed polymer and percolation models'''
 
 
 
Abstract:
 
Explicit formulas for subadditive limits of polymer and percolation models in probability and statistical mechanics have been difficult to find. We describe variational formulas for these limits and their connections with other features of the models such as Busemann functions and Kardar-Parisi-Zhang (KPZ) fluctuation exponents.
 
 
 
 
 
 
 
== <span style="color:red">Monday</span>, December 1,  [http://www.ma.utexas.edu/users/jneeman/index.html Joe Neeman], [http://www.ma.utexas.edu/ UT-Austin], <span style="color:red">4pm, Room B239 Van Vleck Hall</span>==
 
 
 
<span style="color:darkgreen">Please note the unusual time and room.</span>
 
 
 
Title: '''Some phase transitions in the stochastic block model'''
 
  
Abstract: The stochastic block model is a random graph model that was originally 30 years ago to study community detection in networks. To generate a random graph from this model, begin with two classes of vertices and then connect each pair of vertices independently at random, with probability p if they are in the same class and probability q otherwise. Some questions come to mind: can we reconstruct the classes if we only observe the graph? What if we only want to partially reconstruct the classes? How different is this model from an Erdos-Renyi graph anyway? The answers to these questions depend on p and q, and we will say exactly how.
+
== November 14, 2019, [https://math.mit.edu/directory/profile.php?pid=2076 Benjamin Landon], MIT ==
 +
'''Universality of extremal eigenvalue statistics of random matrices'''
  
== Thursday, December 4, Arjun Krishnan, [http://www.fields.utoronto.ca/ Fields Institute] ==
+
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.
  
Title: '''Variational formula for the time-constant of first-passage percolation'''
+
== November 21, 2019, Tung Nguyen, UW Madison ==
  
Abstract:
+
'''Prevalence of deficiency zero reaction networks under an Erdos-Renyi framework
Consider first-passage percolation with positive, stationary-ergodic
+
'''
weights on the square lattice in d-dimensions. Let <math>T(x)</math> be the
 
first-passage time from the origin to <math>x</math> in <math>Z^d</math>. The convergence of
 
<math>T([nx])/n</math> to the time constant as <math>n</math> tends to infinity is a consequence
 
of the subadditive ergodic theorem. This convergence can be viewed as
 
a problem of homogenization for a discrete Hamilton-Jacobi-Bellman
 
(HJB) equation. By borrowing several tools from the continuum theory
 
of stochastic homogenization for HJB equations, we derive an exact
 
variational formula (duality principle) for the time-constant. Under a
 
symmetry assumption, we will use the variational formula to construct
 
an explicit iteration that produces the limit shape.
 
  
 +
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]]
 
[[Past Seminars]]

Latest revision as of 09:17, 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