Difference between revisions of "Probability Seminar"

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[[Probability | Back to Probability Group]]
  
= Spring 2022 =
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= Fall 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>  
  
We usually end for questions at 3:20 PM.
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We usually end for questions at 3:20 PM.
  
 
[https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM LINK. Valid only for online seminars.]
 
[https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM LINK. Valid only for online seminars.]
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== February 3, 2022, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://zhipengliu.ku.edu/ Zhipeng Liu] (University of Kansas)    ==
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== September 22, 2022, in person: [https://sites.google.com/site/pierreyvesgl/home Pierre Yves Gaudreau Lamarre] (University of Chicago)    ==
  
'''One-point distribution of the geodesic in directed last passage percolation'''
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'''Moments of the Parabolic Anderson Model with Asymptotically Singular Noise'''
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The Parabolic Anderson Model (PAM) is a stochastic partial differential equation that describes the time-evolution of particle system with the following dynamics: Each particle in the system undergoes a diffusion in space, and as they are moving through space, the particles can either multiply or get killed at a rate that depends on a random environment.
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One of the fundamental problems in the theory of the PAM is to understand its behavior at large times. More specifically, the solution of the PAM at large times tends to be intermittent, meaning that most of the particles concentrate in small regions where the environment is most favorable for particle multiplication.
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In this talk, we discuss a new technique to study intermittency in the PAM with a singular random environment. In short, the technique consists of approximating the singular PAM with a regularized version that becomes increasingly singular as time goes to infinity.
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This talk is based on a joint work with Promit Ghosal and Yuchen Liao.
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== September 29, 2022, in person: Christian Gorski (Northwestern University)    ==
  
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.
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'''Strict monotonicity for first passage percolation on graphs of polynomial growth and quasi-trees'''
  
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.
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I'll present strict monotonicity results for first passage percolation (FPP) on bounded degree graphs which either have strict polynomial growth (uniform upper and lower volume growth bounds of the same polynomial degree) or are quasi-isometric to a tree; the case of the standard Cayley graph of Z^d is due to van den Berg and Kesten (1993). Roughly speaking, if we use two different weight distributions to perform FPP on a fixed graph, and one of the distributions is "larger" than the other and "subcritical" in some appropriate sense, then the expected passage times with respect to that distribution exceed those of the other distribution by an amount proportional to the graph distance.
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If "larger" here refers to stochastic domination of measures, this result is closely related to "absolute continuity with respect to the expected empirical measure," that is, the fact that long geodesics "use all possible weights". If "larger" here refers to variability (another ordering on measures), then a strict monotonicity theorem holds if and only if the graph also satisfies a condition we call "admitting detours". I intend to sketch the proof of absolute continuity, and, if time allows, give some indication of the difficulties that arise when proving strict monotonicity with respect to variability.
  
== February 10, 2022, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://jscalvert.github.io/ Jacob Calvert] (U.C. Berkeley)   ==
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== October 6, 2022, in person: [https://danielslonim.github.io/ Daniel Slonim] (University of Virginia)   ==  
  
'''Harmonic activation and transport'''
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'''Random Walks in (Dirichlet) Random Environments with Jumps on Z'''
  
Models of Laplacian growth, such as diffusion-limited aggregation (DLA), describe interfaces which move in proportion to harmonic measure. I will introduce a model, called harmonic activation and transport (HAT), in which a finite subset of Z^2 is rearranged according to harmonic measure. HAT exhibits a phenomenon called collapse, whereby the diameter of the set is reduced to its logarithm over a number of steps comparable to this logarithm. I will describe how collapse can be used to prove the existence of the stationary distribution of HAT, which is supported on a class of sets viewed up to translation. Lastly, I will discuss the problem of quantifying the least positive harmonic measure as a function of set cardinality, which arises in the study of HAT, and a partial resolution of which rules out predictions about DLA from the physics literature. Based on joint work with Shirshendu Ganguly and Alan Hammond.
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We introduce the model of random walks in random environments (RWRE), which are random Markov chains on the integer lattice. These random walks are well understood in the nearest-neighbor, one-dimensional case due to reversibility of almost every Markov chain. For example, directional transience and limiting speed can be characterized in terms of simple expectations involving the transition probabilities at a single site. The reversibility is lost, however, if we go up to higher dimensions or relax the nearest-neighbor assumption by allowing jumps, and therefore much less is known in these models. Despite this non-reversibility, certain special cases have proven to be more tractable. Random Walks in Dirichlet environments (RWDE), where the transition probability vectors are drawn according to a Dirichlet distribution, have been fruitfully studied in the nearest-neighbor, higher dimensional setting. We look at RWDE in one dimension with jumps and characterize when the walk is ballistic: that is, when it has non-zero limiting velocity. It turns out that in this model, there are two factors which can cause a directionally transient walk to have zero limiting speed: finite trapping and large-scale backtracking. Finite trapping involves finite subsets of the graph where the walk is liable to get trapped for a long time. It is a highly local phenomenon that depends heavily on the structure of the underlying graph. Large-scale backtracking is a more global and one-dimensional phenomenon. The two operate "independently" in the sense that either can occur with or without the other. Moreover, if neither factor on its own is enough to cause zero speed, then the walk is ballistic, so the two factors cannot conspire together to slow a walk down to zero speed if neither is sufficient to do so on its own. This appearance of two independent factors affecting ballisticity is a new feature not seen in any previously studied RWRE models.  
  
== February 17, 2022, in person: [https://sites.math.northwestern.edu/~kivimae/ Pax Kivimae] (Northwestern University)  ==  
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== October 13, 2022, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://www.maths.univ-evry.fr/pages_perso/loukianova/ Dasha Loukianova] (Université d'Évry Val d'Essonne)  ==
  
'''The Ground-State Energy and Concentration of Complexity in Spherical Bipartite Models'''
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'''"In law" ergodic theorem for the environment viewed from Sinaï's walk'''
  
Bipartite spin glass models have been gaining popularity in the study of glassy systems with distinct interacting species. Recently, the annealed complexity of the pure spherical bipartite model was obtained by B. McKenna. In this talk, I will explain how to show that the low-lying complexity actually concentrates around this value, and how from this one can obtain a formula for the ground-state energy.
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For Sinaï's walk <math>\scriptsize(X_k)</math> we show that the empirical measure of the environment seen from the particle converges in law to some random measure. This limit measure is explicitly given in terms of the infinite valley, which construction goes back to Golosov.  As a consequence an "in law" ergodic theorem holds for additive functionals of the environment's chain. When the limit in this theorem is deterministic, it holds in probability. This allows some extensions to the recurrent case of the ballistic "environment's method" dating back to Kozlov and Molchanov. In particular, we show an LLN and a mixed CLT for the sums <math>\scriptsize\sum_{k=1}^nf(\Delta X_k)</math> where <math>\scriptsize f</math> is bounded and
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depending on the steps <math>\scriptsize\Delta X_k:=X_{k+1}-X_k</math>.
  
== February 24, 2022, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [http://math.uchicago.edu/~lbenigni/ Lucas Benigni] (University of Chicago)   ==  
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== October 20, 2022, '''4pm, VV911''', in person: [https://tavarelab.cancerdynamics.columbia.edu/ Simon Tavaré] (Columbia University) ==
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''Note the unusual time and room!''
  
'''Optimal delocalization for generalized Wigner matrices'''
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'''An introduction to counts-of-counts data'''
  
We consider eigenvector statistics of large symmetric random matrices. When the matrix entries are sampled from independent Gaussian random variables, eigenvectors are uniformly distributed on the sphere and numerous properties can be computed exactly. In particular, we can bound their extremal coordinates with high probability. There has been an extensive amount of work on generalizing such a result, known as delocalization, to more general entry distributions. After giving a brief overview of the previous results going in this direction, we present an optimal delocalization result for matrices with sub-exponential entries for all eigenvectors. The proof is based on the dynamical method introduced by Erdos-Yau, an analysis of high moments of eigenvectors as well as new level repulsion estimates which will be presented during the talk. This is based on a joint work with P. Lopatto.
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Counts-of-counts data arise in many areas of biology and medicine, and have been studied by statisticians since the 1940s. One of the first examples, discussed by R. A. Fisher and collaborators in 1943 [1], concerns estimation of the number of unobserved species based on summary counts of the number of species observed once, twice, … in a sample of specimens. The data are summarized by the numbers ''C<sub>1</sub>, C<sub>2</sub>, …'' of species represented once, twice, … in a sample of size
  
== March 3, 2022, in person: [https://math.wisc.edu/staff/keating-david/ David Keating] (UW-Madison)  ==
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''N = C<sub>1</sub> + 2 C<sub>2</sub> + 3 C<sub>3</sub> + <sup>….</sup>''  containing ''S = C<sub>1</sub> + C<sub>2</sub> + <sup>…</sup>'' species; the vector ''C ='' ''(C<sub>1</sub>, C<sub>2</sub>, …)'' gives the counts-of-counts. Other examples include the frequencies of the distinct alleles in a human genetics sample, the counts of distinct variants of the SARS-CoV-2 S protein obtained from consensus sequencing experiments, counts of sizes of components in certain combinatorial structures [2], and counts of the numbers of SNVs arising in one cell, two cells, … in a cancer sequencing experiment.
  
'''$k$-tilings of the Aztec diamond'''
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In this talk I will outline some of the stochastic models used to model the distribution of ''C,'' and some of the inferential issues that come from estimating the parameters of these models. I will touch on the celebrated Ewens Sampling Formula [3] and Fisher’s multiple sampling problem concerning the variance expected between values of ''S'' in samples taken from the same population [3]. Variants of birth-death-immigration processes can be used, for example when different variants grow at different rates. Some of these models are mechanistic in spirit, others more statistical. For example, a non-mechanistic model is useful for describing the arrival of covid sequences at a database. Sequences arrive one at a time, and are either a new variant, or a copy of a variant that has appeared before. The classical Yule process with immigration provides a starting point to model this process, as I will illustrate.
  
We study $k$-tilings ($k$-tuples of domino tilings) of the Aztec diamond of rank $m$.  We assign a weight to each $k$-tiling, depending on the number of vertical dominos and also on the number of "interactions" between the different tilings.  We will compute the generating polynomials of the $k$-tilings by relating them to an integrable colored vertex model.  We will then prove some combinatorial results about $k$-tilings in certain limits of the interaction strength.
 
  
== March 10, 2022, in person: [https://qiangwu2.github.io/martingale/ Qiang Wu] (University of Illinois Urbana-Champaign)  ==
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''References''
  
'''Mean field spin glass models under weak external field'''
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[1] Fisher RA, Corbet AS & Williams CB. J Animal Ecology, 12, 1943
  
We study the fluctuation and limiting distribution of free energy in mean-field spin glass models with Ising spins under weak external fields. We prove that at high temperature, there are three regimes concerning the strength of external field $h \approx \rho N^{-\alpha}$ with $\rho,\alpha\in (0,\infty)$. In the super-critical regime $\alpha < 1/4$, the variance of the log-partition function is $\approx N^{1-4\alpha}$. In the critical regime $\alpha = 1/4$, the fluctuation is of constant order but depends on $\rho$. Whereas, in the sub-critical regime $\alpha>1/4$, the variance is $\Theta(1)$ and does not depend on $\rho$. We explicitly express the asymptotic mean and variance in all three regimes and prove Gaussian central limit theorems. Our proofs mainly follow two approaches. One utilizes quadratic coupling and Guerra's interpolation scheme for Gaussian disorder, extending to many other spin glass models. However, this approach can prove the CLT only at very high temperatures. The other one is a cluster-based approach for general symmetric disorders, first used in the seminal work of Aizenman, Lebowitz, and Ruelle (Comm. Math. Phys. 112 (1987), no. 1, 3-20) for the zero external field case. It was believed that this approach does not work if the external field is present. We show that if the external field is present but not too strong, it still works with a new cluster structure. In particular, we prove the CLT up to the critical temperature in the Sherrington-Kirkpatrick (SK) model when $\alpha \ge 1/4$. We further address the generality of this cluster-based approach. Specifically, we give similar results for the multi-species SK model and diluted SK model. Based on joint work with Partha S. Dey.
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[2] Arratia R, Barbour AD & Tavaré S. ''Logarithmic Combinatorial Structures,'' EMS, 2002
  
== March 24, 2022, in person: [http://math.columbia.edu/~sayan/ Sayan Das] (Columbia University)  ==
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[3] Ewens WJ. Theoret Popul Biol, 3, 1972
  
'''Path properties of the KPZ Equation and related polymers'''
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[4] Da Silva P, Jamshidpey A, McCullagh P & Tavaré S. Bernoulli Journal, in press, 2022 (online)
  
The KPZ equation is a fundamental stochastic PDE that can be viewed as the log-partition function of continuum directed random polymer (CDRP). In this talk, we will first focus on the fractal properties of the tall peaks of the KPZ equation. This is based on separate joint works with Li-Cheng Tsai and Promit Ghosal. In the second part of the talk, we will study the KPZ equation through the lens of polymers. In particular, we will discuss localization aspects of CDRP that will shed light on certain properties of the KPZ equation such as ergodicity and limiting Bessel behaviors around the maximum. This is based on joint work with Weitao Zhu.
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== October 27, 2022, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://www-users.cse.umn.edu/~arnab/ Arnab Sen] (University of Minnesota, Twin Cities) ==
  
== March 31, 2022, in person: [http://willperkins.org/ Will Perkins] (University of Illinois Chicago)  ==
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'''Maximum weight matching on sparse graphs'''
  
'''Potential-weighted connective constants and uniqueness of Gibbs measures'''
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We consider the maximum weight matching of a finite bounded degree graph whose edges have i.i.d. random weights. It is natural to ask whether the weight of the maximum weight matching follows a central limit theorem. We obtain an affirmative answer to the above question in the case when the weight distribution is exponential and the graphs are locally tree-like. The key component of the proof involves a cavity analysis on arbitrary bounded degree trees which yields a correlation decay for the maximum weight matching. The central limit theorem holds if we take the underlying graph to be also random with i.i.d. degree distribution (configuration model).
  
Classical gases (or Gibbs point processes) are models of gases or fluids, with particles interacting in the continuum via a density against background Poisson processes.  The major questions about these models are about phase transitions, and while these models have been studied in mathematics for over 70 years, some of the most fundamental questions remain open. After giving some background on these models, I will describe a new method for proving absence of phase transition (uniqueness of infinite volume Gibbs measures) in the low-density regime of processes interacting via a repulsive pair potential.  The method involves defining a new quantity, the "potential-weighted connective constant", and is motivated by techniques from computer science.  Joint work with Marcus Michelen.
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This is joint work with Wai-Kit Lam.
  
== April 7, 2022, in person and on [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://sites.google.com/view/eliza-oreilly/home Eliza O'Reilly] (Caltech)  ==  
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== November 3, 2022, in person: [https://www.ias.edu/scholars/sky-yang-cao Sky Cao] (Institute for Advanced Study)  ==  
  
'''Stochastic Geometry for Machine Learning'''
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'''Exponential decay of correlations in finite gauge group lattice gauge theories'''
  
The Mondrian process is a stochastic process that produces a recursive partition of space with random axis-aligned cuts. It has been used in machine learning to build random forests and Laplace kernel approximations.  The construction allows for efficient online algorithms, but the restriction to axis-aligned cuts does not capture dependencies between features. By viewing the Mondrian as a special case of the stable under iterated (STIT) process in stochastic geometry, we resolve open questions about the generalization of cut directions. We utilize the theory of stationary random tessellations to show that STIT random forests achieve minimax rates for Lipschitz and $C^2$ functions and STIT random features approximate a large class of stationary kernels. This work opens many new questions at the intersection of stochastic geometry and machine learning. Based on joint work with Ngoc Mai Tran.
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Lattice gauge theories with finite gauge groups are statistical mechanical models, very much akin to the Ising model, but with some twists. In this talk, I will describe how to show exponential decay of correlations for these models at low temperatures. This is based on joint work with Arka Adhikari.
  
== April 14, 2022, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://cmps.ok.ubc.ca/about/contact/eric-foxall/ Eric Foxall] (UBC-Okanagan)  ==  
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== November 10, 2022, in person: [https://ifds.info/david-clancy/ David Clancy] (UW-Madison)  ==  
  
'''Bifurcation theory of density-dependent Markov chains, well-mixed stochastic population models, and diffusively perturbed dynamical systems'''
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'''Component Sizes of the degree corrected stochastic blockmodel'''
  
Abstract: A common choice for modeling a finite, interacting population of individuals is to specify a variable system size parameter $N$, and to otherwise assume that interactions are well-mixed. When the model is cast in continuous time, has finitely many types, and is otherwise fairly simple, the vector $X$ of population sizes is a density-dependent Markov chain (DDMC), and the population density vector $\frac{X}{N}$ can be roughly seen as a diffusive perturbation of a deterministic system, with noise parameter $\varepsilon = 1/\sqrt{N}$. The law of large numbers and central limit theorem of these models has been known since the work of Thomas Kurtz in the 1970s. Here, we study bifurcations of parametrized models of this type, as a function of $\varepsilon$. We introduce the notion of limit scales to understand the shape of fluctuations in and around the bifurcation point, and give a three-step framework for finding them. The result is an enhanced bifurcation diagram that encodes this information with the help of a few well-chosen functions. We develop the theory in detail for one-dimensional models and illustrate it for simple bifurcation types including transcritical, saddle-node, pitchfork, and the Hopf bifurcation.
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The stochastic blockmodel (SBM) is a simple probabilistic model for graphs which exhibit clustering and is used to test algorithms for detecting these clusters. Each vertex is assigned a type ''i = 1, 2, ..., m'' and edges are included independently with probability depending on the types of the two incident vertices. The degree corrected SBM (DCSBM) exhibits similar clustering behavior but allows for inhomogeneous degree distributions. The sizes of connected components for these graph models are not well understood unless ''m = 1'' or the SBM is a random bipartite graph. We show that under fairly general conditions, the asymptotic sizes of connected components in the DCSBM can be precisely described in terms of a multiparameter and multidimensional random field. Not only that, but we describe the asymptotic proportion of vertices of each type in each of the macroscopic connected components. This talk is based on joint work with Vitalii Konarovskyi and Vlada Limic.
  
== April 21, 2022, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: Hugo Falconet (NYU)  ==  
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== November 17, 2022, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://sites.google.com/site/leandroprpimentel/ Leandro Pimentel] (Federal University of Rio de Janeiro)  ==  
  
'''Metric growth dynamics in Liouville quantum gravity'''
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'''Integration by Parts and the KPZ Two-Point Function'''
  
Abstract:  Liouville quantum gravity (LQG) is a canonical model of random geometry. Associated with the planar Gaussian free field, this geometry with conformal symmetries was introduced in the physics literature by Polyakov in the 80’s and is conjectured to describe the scaling limit of random planar maps. In this talk, I will introduce LQG as a metric measure space and discuss recent results on the associated metric growth dynamics. The primary focus will be on the dynamics of the trace of the free field on the boundary of growing LQG balls.  Based on a joint work with Julien Dubédat.
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In this talk we will consider two models within Kardar-Parisi-Zhang (KPZ) universality class, and apply the integration by parts formula from Malliavin calculus to establish a key relation between the two-point correlation function, the polymer end-point distribution and the second derivative of the variance of the associated height function. Besides that, we will further develop an adaptation of Malliavin-Stein method that quantifies asymptotic independence with respect to the initial data.  
  
== April 28, 2022, in person: [https://www.ias.edu/scholars/amol-aggarwal Amol Aggarwal] (Columbia/IAS)  ==  
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== December 1, 2022, in person: [https://cims.nyu.edu/~ajd594/ Alex Dunlap] (Courant Institute)  ==  
  
'''ASEP Speed Process'''
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'''The nonlinear stochastic heat equation in the critical dimension'''
  
We consider the asymmetric simple exclusion process, started under step initial data, with a single second class particle at the origin. We show that the trajectory of the second class particle almost surely follows a line, whose slope is a uniform random variable in [-1, 1]. The proof is based on a combination of probabilistic couplings and effective hydrodynamic bounds arising from the ASEP's solvability under specific choices of initial data. This is joint work with Ivan Corwin and Promit Ghosal.
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I will discuss a two-dimensional stochastic heat equation with a nonlinear noise strength, and consider a limit in which the correlation length of the noise is taken to 0 but the noise is attenuated by a logarithmic factor. The limiting pointwise statistics can be related to a stochastic differential equation in which the diffusivity solves a PDE somewhat reminiscent of the porous medium equation. This relationship is established through the theory of forward-backward SDEs. I will also explain several cases in which the PDE can be solved explicitly, some of which correspond to known probabilistic models. This talk will be based on current joint work with Cole Graham and older joint work with Yu Gu.
  
== May 5, 2022, in person: [https://people.math.gatech.edu/~dharper40/ David Harper] (Georgia Tech)  ==  
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== December 8, 2022, in person: [https://sites.northwestern.edu/juliagaudio/ Julia Gaudio] (Northwestern University)  ==  
  
'''Dynamical critical $2d$ first-passage percolation'''
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'''Finding Communities in Networks'''
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Networks are used to represent physical, biological, and social systems. Many networks exhibit community structure, meaning that there are two or more groups of nodes which are densely connected. Identifying these communities gives valuable insights about the latent features of the nodes. Community detection has been used in a wide array of applications including online advertising, recommender systems (e.g., Netflix), webpage sorting, fraud detection, and neurobiology.
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I will present my work on efficient algorithms for community detection in three contexts. <br>
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(1) Censored networks: How can we identify communities when some connectivity information is missing? <br>
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(2) Higher-order networks: Beyond pairwise relationships <br>
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(3) Multiple correlated networks: How can we effectively combine data from multiple networks? <br>
 
   
 
   
In first-passage percolation (FPP), we let $\tau_v$ be i.i.d. nonnegative weights on the vertices of a graph and study the weight of the minimal path between distant vertices. If $F$ is the distribution function of $\tau_v$, there are different regimes: if $F(0)$ is small, this weight typically grows like a linear function of the distance, and when $F(0)$ is large, the weight is typically of order one. In between these is the critical regime in which the weight can diverge, but does so sublinearly. This talk will consider a dynamical version of critical FPP on the triangular lattice where vertices resample their weights according to independent rate-one Poisson processes. We will discuss results which show that if sum of $F^{-1}(1/2+1/2^k)$ diverges, then a.s. there are exceptional times at which the weight grows atypically, but if sum of $k^{7/8} F^{-1}(1/2+1/2^k)$ converges, then a.s. there are no such times. Furthermore, in the former case, we compute the Hausdorff and Minkowski dimensions of the exceptional set and show that they can be but need not be equal. These results show a wider range of dynamical behavior than one sees in subcritical (usual) FPP. This is a joint work with M. Damron, J. Hanson, W.-K. Lam.
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Joint work with: Souvik Dhara, Nirmit Joshi, Elchanan Mossel, Miklós Rácz, Colin Sandon, and Anirudh Sridhar
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[[Past Seminars]]
 
[[Past Seminars]]

Latest revision as of 23:32, 27 November 2022

Back to Probability Group

Fall 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.


September 22, 2022, in person: Pierre Yves Gaudreau Lamarre (University of Chicago)

Moments of the Parabolic Anderson Model with Asymptotically Singular Noise

The Parabolic Anderson Model (PAM) is a stochastic partial differential equation that describes the time-evolution of particle system with the following dynamics: Each particle in the system undergoes a diffusion in space, and as they are moving through space, the particles can either multiply or get killed at a rate that depends on a random environment.

One of the fundamental problems in the theory of the PAM is to understand its behavior at large times. More specifically, the solution of the PAM at large times tends to be intermittent, meaning that most of the particles concentrate in small regions where the environment is most favorable for particle multiplication.

In this talk, we discuss a new technique to study intermittency in the PAM with a singular random environment. In short, the technique consists of approximating the singular PAM with a regularized version that becomes increasingly singular as time goes to infinity.

This talk is based on a joint work with Promit Ghosal and Yuchen Liao.

September 29, 2022, in person: Christian Gorski (Northwestern University)

Strict monotonicity for first passage percolation on graphs of polynomial growth and quasi-trees

I'll present strict monotonicity results for first passage percolation (FPP) on bounded degree graphs which either have strict polynomial growth (uniform upper and lower volume growth bounds of the same polynomial degree) or are quasi-isometric to a tree; the case of the standard Cayley graph of Z^d is due to van den Berg and Kesten (1993). Roughly speaking, if we use two different weight distributions to perform FPP on a fixed graph, and one of the distributions is "larger" than the other and "subcritical" in some appropriate sense, then the expected passage times with respect to that distribution exceed those of the other distribution by an amount proportional to the graph distance. If "larger" here refers to stochastic domination of measures, this result is closely related to "absolute continuity with respect to the expected empirical measure," that is, the fact that long geodesics "use all possible weights". If "larger" here refers to variability (another ordering on measures), then a strict monotonicity theorem holds if and only if the graph also satisfies a condition we call "admitting detours". I intend to sketch the proof of absolute continuity, and, if time allows, give some indication of the difficulties that arise when proving strict monotonicity with respect to variability.

October 6, 2022, in person: Daniel Slonim (University of Virginia)

Random Walks in (Dirichlet) Random Environments with Jumps on Z

We introduce the model of random walks in random environments (RWRE), which are random Markov chains on the integer lattice. These random walks are well understood in the nearest-neighbor, one-dimensional case due to reversibility of almost every Markov chain. For example, directional transience and limiting speed can be characterized in terms of simple expectations involving the transition probabilities at a single site. The reversibility is lost, however, if we go up to higher dimensions or relax the nearest-neighbor assumption by allowing jumps, and therefore much less is known in these models. Despite this non-reversibility, certain special cases have proven to be more tractable. Random Walks in Dirichlet environments (RWDE), where the transition probability vectors are drawn according to a Dirichlet distribution, have been fruitfully studied in the nearest-neighbor, higher dimensional setting. We look at RWDE in one dimension with jumps and characterize when the walk is ballistic: that is, when it has non-zero limiting velocity. It turns out that in this model, there are two factors which can cause a directionally transient walk to have zero limiting speed: finite trapping and large-scale backtracking. Finite trapping involves finite subsets of the graph where the walk is liable to get trapped for a long time. It is a highly local phenomenon that depends heavily on the structure of the underlying graph. Large-scale backtracking is a more global and one-dimensional phenomenon. The two operate "independently" in the sense that either can occur with or without the other. Moreover, if neither factor on its own is enough to cause zero speed, then the walk is ballistic, so the two factors cannot conspire together to slow a walk down to zero speed if neither is sufficient to do so on its own. This appearance of two independent factors affecting ballisticity is a new feature not seen in any previously studied RWRE models.

October 13, 2022, ZOOM: Dasha Loukianova (Université d'Évry Val d'Essonne)

"In law" ergodic theorem for the environment viewed from Sinaï's walk

For Sinaï's walk [math]\displaystyle{ \scriptsize(X_k) }[/math] we show that the empirical measure of the environment seen from the particle converges in law to some random measure. This limit measure is explicitly given in terms of the infinite valley, which construction goes back to Golosov. As a consequence an "in law" ergodic theorem holds for additive functionals of the environment's chain. When the limit in this theorem is deterministic, it holds in probability. This allows some extensions to the recurrent case of the ballistic "environment's method" dating back to Kozlov and Molchanov. In particular, we show an LLN and a mixed CLT for the sums [math]\displaystyle{ \scriptsize\sum_{k=1}^nf(\Delta X_k) }[/math] where [math]\displaystyle{ \scriptsize f }[/math] is bounded and depending on the steps [math]\displaystyle{ \scriptsize\Delta X_k:=X_{k+1}-X_k }[/math].

October 20, 2022, 4pm, VV911, in person: Simon Tavaré (Columbia University)

Note the unusual time and room!

An introduction to counts-of-counts data

Counts-of-counts data arise in many areas of biology and medicine, and have been studied by statisticians since the 1940s. One of the first examples, discussed by R. A. Fisher and collaborators in 1943 [1], concerns estimation of the number of unobserved species based on summary counts of the number of species observed once, twice, … in a sample of specimens. The data are summarized by the numbers C1, C2, … of species represented once, twice, … in a sample of size

N = C1 + 2 C2 + 3 C3 + ….  containing S = C1 + C2 + species; the vector C = (C1, C2, …) gives the counts-of-counts. Other examples include the frequencies of the distinct alleles in a human genetics sample, the counts of distinct variants of the SARS-CoV-2 S protein obtained from consensus sequencing experiments, counts of sizes of components in certain combinatorial structures [2], and counts of the numbers of SNVs arising in one cell, two cells, … in a cancer sequencing experiment.

In this talk I will outline some of the stochastic models used to model the distribution of C, and some of the inferential issues that come from estimating the parameters of these models. I will touch on the celebrated Ewens Sampling Formula [3] and Fisher’s multiple sampling problem concerning the variance expected between values of S in samples taken from the same population [3]. Variants of birth-death-immigration processes can be used, for example when different variants grow at different rates. Some of these models are mechanistic in spirit, others more statistical. For example, a non-mechanistic model is useful for describing the arrival of covid sequences at a database. Sequences arrive one at a time, and are either a new variant, or a copy of a variant that has appeared before. The classical Yule process with immigration provides a starting point to model this process, as I will illustrate.


References

[1] Fisher RA, Corbet AS & Williams CB. J Animal Ecology, 12, 1943

[2] Arratia R, Barbour AD & Tavaré S. Logarithmic Combinatorial Structures, EMS, 2002

[3] Ewens WJ. Theoret Popul Biol, 3, 1972

[4] Da Silva P, Jamshidpey A, McCullagh P & Tavaré S. Bernoulli Journal, in press, 2022 (online)

October 27, 2022, ZOOM: Arnab Sen (University of Minnesota, Twin Cities)

Maximum weight matching on sparse graphs

We consider the maximum weight matching of a finite bounded degree graph whose edges have i.i.d. random weights. It is natural to ask whether the weight of the maximum weight matching follows a central limit theorem. We obtain an affirmative answer to the above question in the case when the weight distribution is exponential and the graphs are locally tree-like. The key component of the proof involves a cavity analysis on arbitrary bounded degree trees which yields a correlation decay for the maximum weight matching. The central limit theorem holds if we take the underlying graph to be also random with i.i.d. degree distribution (configuration model).

This is joint work with Wai-Kit Lam.

November 3, 2022, in person: Sky Cao (Institute for Advanced Study)

Exponential decay of correlations in finite gauge group lattice gauge theories

Lattice gauge theories with finite gauge groups are statistical mechanical models, very much akin to the Ising model, but with some twists. In this talk, I will describe how to show exponential decay of correlations for these models at low temperatures. This is based on joint work with Arka Adhikari.

November 10, 2022, in person: David Clancy (UW-Madison)

Component Sizes of the degree corrected stochastic blockmodel

The stochastic blockmodel (SBM) is a simple probabilistic model for graphs which exhibit clustering and is used to test algorithms for detecting these clusters. Each vertex is assigned a type i = 1, 2, ..., m and edges are included independently with probability depending on the types of the two incident vertices. The degree corrected SBM (DCSBM) exhibits similar clustering behavior but allows for inhomogeneous degree distributions. The sizes of connected components for these graph models are not well understood unless m = 1 or the SBM is a random bipartite graph. We show that under fairly general conditions, the asymptotic sizes of connected components in the DCSBM can be precisely described in terms of a multiparameter and multidimensional random field. Not only that, but we describe the asymptotic proportion of vertices of each type in each of the macroscopic connected components. This talk is based on joint work with Vitalii Konarovskyi and Vlada Limic.

November 17, 2022, ZOOM: Leandro Pimentel (Federal University of Rio de Janeiro)

Integration by Parts and the KPZ Two-Point Function

In this talk we will consider two models within Kardar-Parisi-Zhang (KPZ) universality class, and apply the integration by parts formula from Malliavin calculus to establish a key relation between the two-point correlation function, the polymer end-point distribution and the second derivative of the variance of the associated height function. Besides that, we will further develop an adaptation of Malliavin-Stein method that quantifies asymptotic independence with respect to the initial data.

December 1, 2022, in person: Alex Dunlap (Courant Institute)

The nonlinear stochastic heat equation in the critical dimension

I will discuss a two-dimensional stochastic heat equation with a nonlinear noise strength, and consider a limit in which the correlation length of the noise is taken to 0 but the noise is attenuated by a logarithmic factor. The limiting pointwise statistics can be related to a stochastic differential equation in which the diffusivity solves a PDE somewhat reminiscent of the porous medium equation. This relationship is established through the theory of forward-backward SDEs. I will also explain several cases in which the PDE can be solved explicitly, some of which correspond to known probabilistic models. This talk will be based on current joint work with Cole Graham and older joint work with Yu Gu.

December 8, 2022, in person: Julia Gaudio (Northwestern University)

Finding Communities in Networks

Networks are used to represent physical, biological, and social systems. Many networks exhibit community structure, meaning that there are two or more groups of nodes which are densely connected. Identifying these communities gives valuable insights about the latent features of the nodes. Community detection has been used in a wide array of applications including online advertising, recommender systems (e.g., Netflix), webpage sorting, fraud detection, and neurobiology.

I will present my work on efficient algorithms for community detection in three contexts.
(1) Censored networks: How can we identify communities when some connectivity information is missing?
(2) Higher-order networks: Beyond pairwise relationships
(3) Multiple correlated networks: How can we effectively combine data from multiple networks?

Joint work with: Souvik Dhara, Nirmit Joshi, Elchanan Mossel, Miklós Rácz, Colin Sandon, and Anirudh Sridhar


Past Seminars