Probability Seminar: Difference between revisions

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= Fall 2021 =
= Spring 2022 =


<b>Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b>  
<b>Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b>  
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If you would like to sign up for the email list to receive seminar announcements then please join [https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/probsem our group].
If you would like to sign up for the email list to receive seminar announcements then please join [https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/probsem our group].
== September 16, 2021, in person: [https://hanbaeklyu.com/ Hanbaek Lyu] (UW-Madison)  ==




'''Scaling limit of soliton statistics of a multicolor box-ball system'''
== February 3, 2022, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://zhipengliu.ku.edu/ Zhipeng Liu] (University of Kansas)    ==


The box-ball systems (BBS) are integrable cellular automata whose long-time behavior is characterized by the soliton solutions, and have rich connections to other integrable systems such as Korteweg-de Veris equation. Probabilistic analysis of BBS is an emerging topic in the field of integrable probability, which often reveals novel connection between the rich integrable structure of BBS and probabilistic phenomena such as phase transition and invariant measures. In this talk, we give an overview on the recent development in scaling limit theory of multicolor BBS with random initial configurations. Our analysis uses various methods such as modified Greene-Kleitman invariants for BBS, circular exclusion processes, Kerov–Kirillov–Reshetikhin bijection, combinatorial R, and Thermodynamic Bethe Ansatz.
'''One-point distribution of the geodesic in directed last passage percolation'''


== September 23, 2021, no seminar ==
In the recent twenty years, there has been a huge development in understanding the universal law behind a family of 2d random growth models, the so-called Kardar-Parisi-Zhang (KPZ) universality class. Especially, limiting distributions of the height functions are identified for a number of models in this class. Different from the height functions, the geodesics of these models are much less understood. There were studies on the qualitative properties of the geodesics in the KPZ universality class very recently,  but the precise limiting distributions of the geodesic locations remained unknown.


In this talk, we will discuss our recent results on the one-point distribution of the geodesic of a representative model in the KPZ universality class, the directed last passage percolation with iid exponential weights. We will give the explicit formula of the one-point distribution of the geodesic location joint with the last passage times, and its limit when the parameters go to infinity under the KPZ scaling. The limiting distribution is believed to be universal for all the models in the KPZ universality class. We will further give some applications of our formulas.


== September 30, 2021, in person: [https://mrusskikh.mit.edu/home Marianna Russkikh] (MIT)   ==
== February 17, 2022, in person: [https://sites.math.northwestern.edu/~kivimae/ Pax Kivimae] (Northwestern University)   ==  


'''Lozenge tilings and the Gaussian free field on a cylinder'''
'''TBA'''


We discuss new results on lozenge tilings on an infinite cylinder, which may be analyzed using the periodic Schur process introduced by Borodin. Under one variant of the $q^{vol}$ measure, corresponding to random cylindric partitions, the height function converges to a deterministic limit shape and fluctuations around it are given by the Gaussian free field in the conformal structure predicted by the Kenyon-Okounkov conjecture. Under another variant, corresponding to an unrestricted tiling model on the cylinder, the fluctuations are given by the same Gaussian free field with an additional discrete Gaussian shift component. Fluctuations of the latter type have been previously conjectured for tiling models on planar domains with holes.
== February 24, 2022, in person: [http://math.uchicago.edu/~lbenigni/ Lucas Benigni] (University of Chicago)  ==


== October 7, 2021, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://n.ethz.ch/~bdembin/home.html Barbara Dembin] (ETH Zurich)    ==
'''TBA'''


'''The time constant for Bernoulli percolation is Lipschitz continuous strictly above $p_c$'''
== March 3, 2022, in person: [https://math.wisc.edu/staff/keating-david/ David Keating] (UW-Madison)  ==


We consider the standard model of i.i.d. first passage percolation on $\mathbb Z^d$ given a distribution $G$ on $[0,+\infty]$ ($+\infty$ is allowed). When $G([0,+\infty))>p_c(d)$, it is known that the time constant $\mu_G$ exists. We are interested in the regularity properties of the map $G\mapsto\mu_G$. We study the specific case of distributions of the form $G_p=p\delta_1+(1-p)\delta_\infty$ for $p>p_c(d)$. In this case, the travel time between two points is equal to the length of the shortest path between the two points in a bond percolation of parameter $p$. We prove that the function $p\mapsto \mu_{G_p}$ is Lipschitz continuous on every interval $[p_0,1]$, where $p_0>p_c(d)$.
'''TBA'''
This is a joint work with Raphaël Cerf.


== October 14, 2021, <span style="color:red">UPDATED FORMAT: </span> [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://sites.google.com/wisc.edu/evan-sorensen Evan Sorensen] (UW-Madison) ==
== March 10, 2022, format TBD: [https://qiangwu2.github.io/martingale/ Qiang Wu] (University of Illinois Urbana-Champaign)   ==  


'''Busemann functions and semi-infinite geodesics in a semi-discrete space'''
'''TBA'''


In the last 10-15 years, Busemann functions have been a key tool for studying semi-infinite geodesics in planar first and last-passage percolation. We study Busemann functions in the semi-discrete Brownian last-passage percolation (BLPP) model and use these to derive geometric properties of the full collection of semi-infinite geodesics in BLPP. This includes a characterization of  uniqueness and coalescence of semi-infinite geodesics across all asymptotic directions. To deal with the uncountable set of points in BLPP, we develop new methods of proof and uncover new phenomena, compared to discrete models. For example, for each asymptotic direction, there exists a random countable set of initial points out of which there exist two semi-infinite geodesics in that direction. Further, there exists a random set of points, of Hausdorff dimension ½, out of which, for some random direction, there are two semi-infinite geodesics that split from the initial point and never come back together. We derive these results by studying variational problems for Brownian motion with drift.
== March 24, 2022, in person: [http://math.columbia.edu/~sayan/ Sayan Das] (Columbia University)   ==


== October 21, 2021, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [http://stat.columbia.edu/~sumitm/ Sumit Mukherjee] (Columbia)    ==
'''TBA'''


'''Fluctuations in Mean Field Ising Models'''
== March 31, 2022, in person: [http://willperkins.org/ Will Perkins] (University of Illinois Chicago)  ==


We study fluctuations of the magnetization (average of spins) in an Ising model on a sequence of "well-connected" approximately $d_n$ regular graphs on $n$ vertices. We show that if $d_n\gg n^{1/2}$,  then the fluctuations are universal, and same as that of the Curie–Weiss model, in the entire ferromagnetic parameter regime. We then give a counterexample to show that $d_n\gg n^{1/2}$ is actually tight, in the sense that the limiting distribution changes if $d_n\sim n^{1/2}$ except in the high temperature regime. By refining our argument, we show that in the high temperature regime universality holds for $d_n\gg n^{1/3}$. As a by-product of our proof technique, we prove rates of convergence, as well as exponential concentration for the sum of spins, and tight estimates for several statistics of interest.
'''TBA'''


This is based on joint work with Nabarun Deb at Columbia University.
== April 7, 2022, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://sites.google.com/view/eliza-oreilly/home Eliza O'Reilly] (Caltech)  ==


== October 28, 2021, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://www-users.cse.umn.edu/~wkchen/ Wei-Kuo Chen] (Minnesota)    ==
'''TBA'''
 
'''Grothendieck $L_p$ problem for Gaussian matrices'''
 
The Grothendieck $L_p$ problem is defined as an optimization problem that maximizes the quadratic form of a Gaussian matrix over the unit $L_p$ ball. The $p=2$ case corresponds to the top eigenvalue of the Gaussian Orthogonal Ensemble, while for $p=\infty$ this problem is known as the ground state energy of the Sherrington-Kirkpatrick mean-field spin glass model and its limit can be expressed by the famous Parisi formula. In this talk, I will describe the limit of this optimization problem for general $p$ and discuss some results on the behavior of the near optimizers along with some open problems. This is based on a joint work with Arnab Sen.
 
== November 4, 2021, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://mathematics.stanford.edu/people/mackenzie-simper Mackenzie Simper] (Stanford)  ==
 
'''Double Cosets, Mallows Measure, and a Transvections Markov Chain'''
 
If $G = GL_n(\mathbb{F}_q)$ and $B$ is the subgroup of lower triangular matrices, then the $B\backslash G/B$ double cosets are indexed by permutations $S_n$. This is the famous Bruhat decomposition, closely related to the LU decomposition of a matrix. The Markov chain on $G$ generated by random transvections – matrices which fix a hyperplane – induces a Markov chain on $S_n$ with the Mallows measure as stationary distribution. We characterize this process, study the mixing time, and discuss the connection with the number of pivoting steps needed in Gaussian elimination. This is joint work with Persi Diaconis and Arun Ram.
 
== November 11, 2021, in person: [http://web.lfzhang.com/ Lingfu Zhang] (Princeton)    ==
 
'''Shift-invariance of the colored TASEP and random sorting networks'''
 
In this talk, I will introduce a new shift-invariance property of the colored TASEP. It is in a similar spirit as recent results for the six-vertex models (by Borodin-Gorin-Wheeler and Galashin), but its proof is via non-algebraic arguments. This new shift-invariance is applied to prove a conjectured distributional equality between the classical exponential Last Passage Percolation model and the oriented swap process (OSP). The OSP is a model for a random sorting network, with N particles labeled $1,\dots,N$ performing successive adjacent swaps at random times until they reach the reverse configuration $N,\dots,1$. Our distributional equality implies new asymptotic results about the OSP, some of which are in connection with the Airy sheet.
 
== November 18, 2021, in person: [http://www.ilt.kharkov.ua/bvi/structure/depart_e/d24/mariya_shcherbina-cv.htm Mariya Shcherbina]: (Kharkov) ==
'''Supersymmetric approach to the deformed Ginibre ensemble.'''
 
We consider  non Hermitian random matrices of the form
$H=A+H_0$, where $A$ is a rather general $n\times n$ matrix (Hermitian
or non-Hermitian)
independent of $H_0$, and $H_0$ is a standard Ginibre  matrix.
It is known that under some reasonable conditions the limiting spectrum
of $H$ makes
some domain $D$ with a smooth boundary $\Gamma$. We apply the
supersymmetric approach
to study the behavior of the smallest singular value of the matrix
$(H-z)$  if $z\in D$
but $dist(z,\Gamma)\sim N^{-1/2}$.
 
== November 25, 2021, no seminar  ==
 
 
== December 2, 2021, in person: [http://math.uchicago.edu/~xuanw/ Xuan Wu] (Chicago)  ==
 
'''Scaling limits of the Laguerre unitary ensemble'''
 
Abstract: In this talk, we will discuss the LUE with a focus on the scaling limits. On the hard-edge side, we construct the $\alpha$-Bessel line ensemble for all $\alpha \in \mathbb{N}_0$. This is a novel Gibbsian line ensemble that enjoys the $\alpha$-squared Bessel Gibbs property. Moreover, all $\alpha$-Bessel line ensembles can be naturally coupled together in a Bessel field, which enjoys rich integrable structures. We will also talk about work in progress on the soft-edge side, where we expect to have the Airy field as the scaling limit. This talk is based on joint works with Lucas Benigni, Pei-Ken Hung, and Greg Lawler.
 
== December 9, 2021, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://www.maths.dur.ac.uk/users/sunil.chhita/ Sunil Chhita] (Durham)  ==
 
'''GOE Fluctuations for the maximum of the top path in ASMs'''
 
The six-vertex model is an important toy-model in statistical mechanics for two-dimensional ice with a natural parameter Δ. When Δ=0, the so-called free-fermion point, the model is in natural correspondence with domino tilings of the Aztec diamond. Although this model is integrable for all Δ, there has been very little progress in understanding its statistics in the scaling limit for other values. In this talk, we focus on the six-vertex model with domain wall boundary conditions at Δ=1/2, where it corresponds to alternating sign matrices (ASMs). We consider the level lines in a height function representation of ASMs. We report that the maximum of the topmost level line for a uniformly random ASMs has the GOE Tracy-Widom distribution after appropriate rescaling.  This talk is based on joint work with Arvind Ayyer and Kurt Johansson.




[[Past Seminars]]
[[Past Seminars]]

Revision as of 19:16, 25 January 2022


Spring 2022

Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom

We usually end for questions at 3:20 PM.

ZOOM LINK. Valid only for online seminars.

If you would like to sign up for the email list to receive seminar announcements then please join our group.


February 3, 2022, ZOOM: Zhipeng Liu (University of Kansas)

One-point distribution of the geodesic in directed last passage percolation

In the recent twenty years, there has been a huge development in understanding the universal law behind a family of 2d random growth models, the so-called Kardar-Parisi-Zhang (KPZ) universality class. Especially, limiting distributions of the height functions are identified for a number of models in this class. Different from the height functions, the geodesics of these models are much less understood. There were studies on the qualitative properties of the geodesics in the KPZ universality class very recently, but the precise limiting distributions of the geodesic locations remained unknown.

In this talk, we will discuss our recent results on the one-point distribution of the geodesic of a representative model in the KPZ universality class, the directed last passage percolation with iid exponential weights. We will give the explicit formula of the one-point distribution of the geodesic location joint with the last passage times, and its limit when the parameters go to infinity under the KPZ scaling. The limiting distribution is believed to be universal for all the models in the KPZ universality class. We will further give some applications of our formulas.

February 17, 2022, in person: Pax Kivimae (Northwestern University)

TBA

February 24, 2022, in person: Lucas Benigni (University of Chicago)

TBA

March 3, 2022, in person: David Keating (UW-Madison)

TBA

March 10, 2022, format TBD: Qiang Wu (University of Illinois Urbana-Champaign)

TBA

March 24, 2022, in person: Sayan Das (Columbia University)

TBA

March 31, 2022, in person: Will Perkins (University of Illinois Chicago)

TBA

April 7, 2022, ZOOM: Eliza O'Reilly (Caltech)

TBA


Past Seminars

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