We show that a limiting empirical spectral distribution LSD exists, and via local weak convergence of associated graphs, the LSD corresponds to the spectral measure associated to the root of a graph which is formed by connecting infinitely many Poisson weighted infinite trees using a backbone structure of special edges. One example covered are matrices with i.
In this case, the LSD is a semi-circle law. Increasingly, it plays an important role in phylogenetics where it can be used to model the joint evolution of a large number of genes across multiple species. Motivated by information-theoretic questions, I will present a recent probabilistic analysis of the multispecies coalescent which establishes fundamental limits on the inference of this model from molecular sequence data.
No biology background is required. Friday, January 16th: Steve Lalley - Univ. Title: Nash Equilibria for a Quadratic Voting Game Abstract: Voters making a binary decision purchase votes from a centralized clearing house, paying the square of the number of votes purchased.
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The utilities of the voters are assumed to arise by random sampling from a probability distribution F with compact support; each voter knows her own utility, but not those of the other voters, although she does know the sampling distribution F. Nash equilibria for this game are described. Title: Brownian motion on spaces with varying dimension Abstract: The model can be picturized as the random movement of an insect on the ground with a pole standing on it.
That is, part of the state space has dimension 2, and the other part of the state space has dimension 1. We show that the behavior of this process switches between 1-dimensional and 2-dimensional, which depends on both the time and the positions of the points. An open ongoing project will also be introduced: Can we approximate such a process by random walks?
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The main results of this talk are based on my joint work with Zhen-Qing Chen. Friday, February 13th: James R. Lee - University of Washington. Title: Regularization under diffusion and Talagrand's convolution conjecture Abstract: It is a well-known phenomenon that functions on Gaussian space become smoother under the Ornstein-Uhlenbeck semigroup. Ball, Barthe, Bednorz, Oleszkiewicz, and Wolff proved that this holds in fixed dimensions. We resolve Talagrand's conjecture conjecture positively with no dimension dependence.
The key insight is to study a subset of Gaussian space at various granularities by approaching it as "efficiently" as possible. To this end, we employ an Ito process that arose in the context of optimal control theory. Efficiency is measured by the average "work" required to couple the approach process to a Brownian motion. Here we address the Gaussian limiting case. This is joint work with Ronen Eldan. Friday, February 20th: Philippe Sosoe - Harvard. No more precise lower bound has been given so far. Conditional on the existence of an open crossing of a box of side length n, there is a distinguished open path which can be characterized in terms of arm exponents: the lowest open path crossing the box.
This clearly gives an upper bound for the shortest path. Following a question of Kesten and Zhang, we compare the length of shortest circuit in an annulus to that of the innermost circuit defined analogously to the lowest crossing. I will explain how to show that the ratio of the expected length of the shortest circuit to the expected length of the innermost crossing tends to zero as the size of the annulus grows.
Joint work with Jack Hanson and Michael Damron. Abstract: The directed polymer model at intermediate disorder regime was introduced by Alberts-Khanin-Quastel This result was obtained under the assumption that the disorder variables posses exponential moments, but its universality was also conjectured under the assumption of six moments. We show that this conjecture is valid and we further extend it by exhibiting classes of different universal limiting behaviors in the case of less than six moments.
Based on joint work with Nikos Zygouras. Title: Universality in spin glasses Abstract: This talk is concerned about some universal properties of the Parisi solution in spin glass models. We will show universality of chaos phenomena and ultrametricity in the mixed p-spin model under mild moment assumptions on the environment. We will explain that the results also extend to quenched self-averaging of some physical observables in the mixed p-spin model as well as in different spin glass models including the Edwards-Anderson model and the random field Ising model.
Fall Seminars Friday, Oct 3rd Speical time: ! Title: Displacement convexity of entropy and curvature in discrete settings Abstract: Inspired by exciting developments in optimal transport and Riemannian geometry due to the work of Lott-Villani and Sturm , several independent groups have formulated a discrete notion of curvature in graphs and finite Markov chains.
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I will describe some of these approaches briefly, and mention some related open problems of potential independent interest. Friday, Oct 10th: No seminar. I will tell you exactly which ones work, and will describe colorings with these properties. No knowledge of advanced probability is needed to follow the lecture. There are several connections with combinatorics, but again, no specialized knowledge is needed.
This is joint work with A. Title: Stationary Eden Model on amenable groups Abstract: We consider stationary versions of the Eden model, on a product of a Cayley graph G of an amenable group and positive integers. The process results in a collection of disjoint trees rooted at G, each of which consists of geodesic paths in a corresponding first passage percolation model on the product graph.
Under weak assumptions on the weight distribution and by relying on ergodic theorems, we prove that almost surely all trees are finite. This generalizes certain known results on the two-type Richardson model, in particular of Deijfen and Haggstrom on the Euclidean lattice. This is a joint work with Eviatar Procaccia.
Title: Random tilings and Hurwitz numbers Abstract: This talk is about random tilings of a special class of planar domains, which I like to call "sawtooth domains. Consequently, many observables can be expressed in terms of special functions of representation-theoretic origin. I will explain how this fact allows one to prove that tiles along a slice fluctuate like the eigenvalues of a Gaussian random matrix. This line of research culminated in the work of Baik-Deift-Johansson who related this length to the Tracy-Widom distribution.
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We study the length of the LIS and LDS of random permutations drawn from the Mallows measure, introduced by Mallows in connection with ranking problems in statistics. We determine the typical order of magnitude of the LIS and LDS, large deviation bounds for these lengths and a law of large numbers for the LIS for various regimes of the parameter q.
This is joint work with Ron Peled. The main assumptions are that the sequence is not diffusive the variance does not grow linearly and that it has a weak dependence structure. Various examples, including first and last passage percolation, bin packing, and longest common subsequence fall into this class. This is joint work with Michael Damron and Jack Hanson. Friday, Dec 5th: Brent M. Werness -University of Washington. Title: Hierarchical approximations to the Gaussian free field and fast simulation of Schramm-Loewner evolutions Abstract: The Schramm--Loewner evolutions SLE are a family of stochastic processes which describe the scaling limits of curves which occur in two-dimensional critical statistical physics models.
SLEs have had found great success in this task, greatly enhancing our understanding of the geometry of these curves. Despite this, it is rather difficult to produce large, high-fidelity simulations of the process due to the significant correlation between segments of the simulated curve. The standard simulation method works by discretizing the construction of SLE through the Loewner ODE which provides a quadratic time algorithm in the length of the curve.
Recent work of Sheffield and Miller has provided an alternate description of SLE, where the curve generated is taken to be a flow line of the vector field obtained by exponentiating a Gaussian free field. In this talk, I will describe a new hierarchical method of approximately sampling a Gaussian free field, and show how this allows us to more efficiently simulate an SLE curve. Additionally, we will briefly discuss questions of the computational complexity of simulating SLE which arise naturally from this work.
Title: Limited choice and randomness in the evolution of networks Abstract: The last few years have seen an explosion in network models describing the evolution of real world networks. I will describe ongoing work in understanding such dynamic network models, their connections to classical constructs such as the standard multiplicative coalescent and applications of these simple models in fitting retweet networks in Twitter.
It is well-known that a diffusion which is allowed to travel only in these directions is smooth, in the sense that its transition probability measure is absolutely continuous with respect to the volume measure and has a strictly positive smooth density. Smoothness results of this kind in infinite dimensions are typically not known, the first obstruction being the lack of an infinite-dimensional volume measure.
Mathematics Colloquium, pm Eckhart Title: Free probability and random matrices; from isomorphisms to universality Abstract: Free probability is a probability theory for non-commutative variables introduced by Voiculescu about thirty years ago. It is equipped with a notion of freeness very similar to independence. It is a natural framework to study the limit of random matrices with size going to infinity.
In this talk, we will discuss these connections and how they can be used to adapt ideas from classical probability theory to operator algebra and random matrices.
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We will in particular focus on how to adapt classical ideas on transport maps following Monge and Ampere to construct isomorphisms between algebras and prove universality in matrix models. This talk is based on joint works with F. Bekerman, Y. Dabrowski, A. Figalli and D.
Title: Strict Convexity of the Parisi Functional Abstract: Spin glasses are magnetic systems exhibiting both quenched disorder and frustration, and have often been cited as examples of "complex systems. This talk will cover recent progress that shed more light in the mysterious and beautiful solution proposed 30 years ago by G. We will focus on properties of the free energy of the famous Sherrington-Kirkpatrick model and we will explain a recent proof of the strict convexity of the Parisi functional.
Based on a joint work with Wei-Kuo Chen. Friday, May 16th -- Double Talk!