Philip Matchett Wood : Random doubly stochastic tridiagonal matrices
- Probability ( 105 Views )Let $T_n$ be the compact convex set of tridiagonal doubly stochastic matrices. These arise naturally as birth and death chains with a uniform stationary distribution. One can think of a typical matrix $T_n$ as one chosen uniformly at random, and this talk will present a simple algorithm to sample uniformly in $T_n$. Once we have our hands on a 'typical' element of $T_n$, there are many natural questions to ask: What are the eigenvalues? What is the mixing time? What is the distribution of the entries? This talk will explore these and other questions, with a focus on whether a random element of $T_n$ exhibits a cutoff in its approach to stationarity. Joint work with Persi Diaconis.
Eric Foxall : Social contact processes and the partner model.
- Probability ( 110 Views )We consider a model of infection spread on the complete graph on N vertices. Edges are dynamic, modelling the formation and breakup of non-permanent monogamous partnerships, and the infection can spread only along active edges. We identify a basic reproduction number \(R_0\) such that the infection dies off in \(O(\log N)\) time when \(R_0\)<1, and survives for at least \(e^{cN}\) time when \(R_0\)>1 and a positive fraction of vertices are initially infectious. We also identify a unique endemic state that exists when \(R_0\)>1, and show it is metastable. When \(R_0\)=1, with considerably more effort we can show the infection survives on the order of \(N^{1/2}\) amount of time.
Amir Dembo : Factor models on locally tree-like graphs
- Probability ( 103 Views )Consider factor (graphical) models on sparse graph sequences that converge locally to a random tree T. Using a novel interpolation scheme we prove existence of limiting free energy density under uniqueness of relevant Gibbs measures for the factor model on T. We demonstrate this for Potts and independent sets models and further characterize this limit via large-deviations type minimization problem and provide an explicit formula for its solution, as the Bethe free energy for a suitable fixed point of the belief propagation recursions on T (thereby rigorously generalize heuristic calculations by statistical physicists using ``replica'' or ``cavity'' methods). This talk is based on a joint work with Andrea Montanari and Nike Sun.
Jasmine Foo : Accumulation and spread of advantageous mutations in a spatially structured tissue
- Probability ( 105 Views )I will discuss a stochastic model of mutation accumulation and spread in a spatially-structured population. This situation arises in a variety of ecological and biological problems, including the process of cancer initiation from healthy tissue. Cancer arises via the accumulation of mutations to the genetic code. Although many mathematical models of cancer initiation have assumed `perfect mixing' or spatial homogeneity, solid tumors often initiate from tissues with well-regulated spatial architecture and dynamics. Here, we study a stochastic model to investigate the temporal dynamics and patterns of mutation accumulation (i.e. how they depend on system parameters such as mutation rate, population size, and selective fitness advantage of mutations). Joint work with R. Durrett (Duke) and K. Leder (Minnesota).
Kevin McGoff : Gibbs posteriors and thermodynamics, Gibbs posterior convergence and the thermodynamic formalism
- Probability ( 134 Views )We consider a Bayesian framework for making inferences about dynamical systems from ergodic observations. The proposed Bayesian procedure is based on the Gibbs posterior, a decision theoretic generalization of standard Bayesian inference. We place a prior over a model class consisting of a parametrized family of Gibbs measures on a mixing shift of finite type. This model class generalizes (hidden) Markov chain models by allowing for long range dependencies, including Markov chains of arbitrarily large orders. We characterize the asymptotic behavior of the Gibbs posterior distribution on the parameter space as the number of observations tends to infinity. In particular, we define a limiting variational problem over the space of joinings of the model system with the observed system, and we show that the Gibbs posterior distributions concentrate around the solution set of this variational problem. In the case of properly specified models our convergence results may be used to establish posterior consistency. This work establishes tight connections between Gibbs posterior inference and the thermodynamic formalism, which may inspire new proof techniques in the study of Bayesian posterior consistency for dependent processes.
Elena Kosygina : Excited random walks
- Probability ( 111 Views )The idea behind excited random walks (ERWs), roughly speaking, is to take a well-known underlying process (such as, for example, simple symmetric random walk (SSRW)) and modify its transition probabilities for the "first few" visits to every site of the state space. These modifications can be deterministic or random. The resulting process is not markovian, and its properties can be very different from those of the underlying process. I shall give a short review of some of the known results for ERW (with SSRW as underlying process) on the d-dimensional integer lattice and then concentrate on a specific model for d=1. For this model we can give a complete picture including functional limit theorems.
Leonid Bogachev : Gaussian fluctuations for Plancherel partitions
- Probability ( 115 Views )The limit shape of Young diagrams under the Plancherel measure was found by Vershik & Kerov (1977) and Logan & Shepp (1977). We obtain a central limit theorem for fluctuations of Young diagrams in the bulk of the partition 'spectrum'. More specifically, under a suitable (logarithmic) normalization, the corresponding random process converges (in the FDD sense) to a Gaussian process with independent values. We also discuss a link with an earlier result by Kerov (1993) on the convergence to a generalized Gaussian process. The proof is based on poissonization of the Plancherel measure and an application of a general central limit theorem for determinantal point processes. (Joint work with Zhonggen Su.) (see more details hear.
Kevin McGoff : An introduction to thermodynamic formalism in ergodic theory through (counter)examples
- Probability ( 115 Views )The goal of this talk is to give a self-contained introduction to some aspects of the thermodynamic formalism in ergodic theory that should be accessible to probabilists. In particular, the talk will focus on equilibrium states and Gibbs measures on the Z^d lattice. We'll present some basic examples in the theory, as well as some recent results that are joint with Christopher Hoffman.
Michael Damron : A simplified proof of the relation between scaling exponents in first passage percolation
- Probability ( 117 Views )In first passage percolation, we place i.i.d. non-negative weights on the nearest-neighbor edges of Z^d and study the induced random metric. A long-standing conjecture gives a relation between two "scaling exponents": one describes the variance of the distance between two points and the other describes the transversal fluctuations of optimizing paths between the same points. In a recent breakthrough work, Sourav Chatterjee proved this conjecture using a strong definition of the exponents. I will discuss work I just completed with Tuca Auffinger, in which we introduce a new and intuitive idea that replaces Chatterjee's main argument and gives an alternative proof of the relation. One advantage of our argument is that it does not require a non-trivial technical assumption of Chatterjee on the weight distribution.
Yu Gu : Scaling limits of random fluctuations in homogenization of divergence form operators
- Probability ( 168 Views )Recently, quantitative stochastic homogenization of operators in divergence form has witnessed important progress. Our goal is to go beyond the error bound to analyze statistical fluctuations around the homogenized limit. We prove a pointwise two-scale expansion and a large scale central limit theorem for the solution. The approach is probabilistic. The main ingredients include the Kipnis-Varadhan method applied to symmetric diffusion in random environment, a quantitative martingale central limit theorem, the Helffer-Sjostrand covariance representation and Stein's method. This is joint work with Jean-Christophe Mourrat.
Jonathan Weare : Stratification of Markov processes for rare event simulation
- Probability ( 97 Views )I will discuss an ensemble sampling scheme based on a decomposition of the target average of interest into subproblems that are each individually easier to solve and can be solved in parallel. The most basic version of the scheme computes averages with respect to a given density and is a generalization of the Umbrella Sampling method for the calculation of free energies. For equilibrium versions of the scheme we have developed error bounds that reveal that the existing understanding of umbrella sampling is incomplete and potentially misleading. We demonstrate that the improvement from umbrella sampling over direct simulation can be dramatic in certain regimes. Our bounds are motivated by new perturbation bounds for Markov Chains that we recently established and that are substantially more detailed than existing perturbation bounds for Markov chains. I will also briefly outline a ``trajectory stratification technique that extends the basic umbrella sampling philosophy to the calculation of dynamic averages with respect a given Markov process. The scheme is capable of computing very general dynamic averages and offers a natural way to parallelize in both time and space.
James Gleeson : Determinants of meme popularity
- Probability ( 140 Views )We will describe and analyze some models of the spread of information on Twitter. The competition between memes fro the limited resource of user attention leads to critical branching processes, and resulting heavy tailed distributions for meme popularity.
Asaf Nachmias : The Alexander-Orbach Conjecture Holds in High Dimensions
- Probability ( 124 Views )It is known that the simple random walk on the unique infinite cluster of supercritical percolation on Z^d diffuses in the same way it does on the original lattice. In critical percolation, however, the behavior of the random walk changes drastically. The infinite incipient cluster (IIC) of percolation on Z^d can be thought of as the critical percolation cluster conditioned on being infinite. Alexander and Orbach (1982) conjectured that the spectral dimension of the IIC is 4/3. This means that the probability of an n-step random walk to return to its starting point scales like n^{-2/3} (in particular, the walk is recurrent). In this work we prove this conjecture when d>18; that is, where the lace-expansion estimates hold. Joint work with Gady Kozma.
Dan Lacker : Probabilistic limit theory for mean field games
- Probability ( 111 Views )Mean field game theory describes continuum limits of symmetric large-population games. These games can often be seen as competitive extensions of classical models of interacting particle systems, where the particles are now "controlled state process" (with application-specific interpretation, such as position, income, wealth, etc.). The coupled optimization problems faced by each process are typically resolved by Nash equilibrium, and there is a large and growing literature on solvability problems (both theoretical and computational). On the other hand, relatively little is known on how to rigorously pass from a finite population to a continuum, especially for dynamic stochastic games. The basic question is: Given for each N a Nash equilibrium for the N-player game, do the equilibria (more precisely, the empirical distributions of state processes) converge as N tends to infinity? This talk is an overview of the known probabilistic limit theorems in this context (law of large numbers, fluctuations, and large deviations), the ideas behind them, and some open problems.
Jeremy Quastel : The effect of noise on KPP traveling fronts
- Probability ( 143 Views )It was noticed experimentally in the late 90's that the speeds of traveling fronts in microscopic systems approximating the KPP equation converge unusually slowly to their continuum values. Brunet and Derrida made a very precise conjecture for the basic model equation, which is the KPP equation perturbed by white noise. We will explain the conjecture and sketch the main ideas of the proof. This is joint work with Carl Mueller and Leonid Mytnik.
Oliver R Diaz : Long wave expansions for water waves over random bottom
- Probability ( 100 Views )We introduce a technique, based on perturbation theory for Hamiltonian PDEs, to derive the asymptotic equations of the motion of a free surface of a fluid over a rough bottom (one dimension). The rough bottom is described by a realization of a stationary mixing process which varies on short length scales. We show that the problem in this case does not fully homogenize, and random effects are as important as dispersive and nonlinear phenomena in the scaling regime. We will explain how these technique can be generalized to higher dimensions
Robin PEMANTLE : Analytic Combinatorics in Several Variables Subtitle: estimating coefficients of multivariate rational power series
- Probability ( 92 Views )The analytic framework for estimating coefficients of a generating function is the same in many variables as in one variable: evaluate Cauchy's integral by manipulating the contour into a "standard" position. That being said, the geometry when dealing with several complex variables can be much more complicated. This talk, drawing on the recent book (with Mark Wilson) of the same title, surveys analytic methods for extracting asymptotics from multivariate generating functions. I will try to give an idea of the main pieces of the puzzle. In particular, I will try to explain in pictures the roles of Morse theory, complex algebraic geometry and hyperbolicity in the asymptotic evaluation of integrals.
Shankar Bhamidi : Scaling limits of random graph models at criticality: Universality and the basin of attraction of the Erd\H{o}s-R\enyi random graph
- Probability ( 94 Views )Over the last few years a wide array of random graph models have been postulated to understand properties of empirically observed networks. Most of these models come with a parameter t (usually related to edge density) and a (model dependent) critical time t_c which specifies when a giant component emerges. There is evidence to support that for a wide class of models, under moment conditions, the nature of this emergence is universal and looks like the classical Erdos-Renyi random graph, in the sense of the critical scaling window and (a) the sizes of the components in this window (all maximal component sizes scaling like n^{2/3}) and (b) the structure of components (rescaled by n^{-1/3}) converge to random fractals related to the continuum random tree. Till date, (a) has been proven for a number of models using different techniques while (b) has been proven for only two models, the classical \erdos random graph and the rank-1 inhomogeneous random graph. The aim of this paper is to develop a general program for proving such results. The program requires three main ingredients: (i) in the critical scaling window, components merge approximately like the multiplicative coalescent (ii) scaling exponents of susceptibility functions in the barely subcritical regime are the same as the Erdos-Renyi random graph and (iii) macroscopic averaging of expected distances between random points in the same component in the barely subcritical regime. We show that these apply to a number of fundamental random graph models including the configuration model, inhomogeneous random graphs modulated via a finite kernel and bounded size rules. Thus these models all belong to the domain of attraction of the classical Erdos-Renyi random graph. As a by product we also get the first known results for component sizes at criticality for a general class of inhomogeneous random graphs. This is joint work with Xuan Wang, Sanchayan Sen and Nicolas Broutin.
Tobias Johnson : Galton-Watson fixed points, tree automata, and interpretations
- Probability ( 108 Views )Consider a set of trees such that a tree belongs to the set if and only if at least two of its root child subtrees do. One example is the set of trees that contain an infinite binary tree starting at the root. Another example is the empty set. Are there any other sets satisfying this property other than trivial modifications of these? I'll demonstrate that the answer is no, in the sense that any other such set of trees differs from one of these by a negligible set under a Galton-Watson measure on trees, resolving an open question of Joel Spencer's. This follows from a theorem that allows us to answer questions of this sort in general. All of this is part of a bigger project to understand the logic of Galton-Watson trees, which I'll tell you more about. Joint work with Moumanti Podder and Fiona Skerman.
Sebastien Roch : Cascade Processes in Social Networks
- Probability ( 152 Views )Social networks are often represented by directed graphs where the nodes are individuals and the edges indicate a form of social relationship. A simple way to model the diffusion of ideas, innovative behavior, or word-of-mouth effects on such a graph is to consider a stochastic process of ``infection'': each node becomes infected once an activation function of the set of its infected neighbors crosses a random threshold value. I will prove a conjecture of Kempe, Kleinberg, and Tardos which roughly states that if such a process is ``locally'' submodular then it must be ``globally'' submodular on average. The significance of this result is that it leads to a good algorithmic solution to the problem of maximizing the spread of influence in the network--a problem known in data mining as "viral marketing"'. This is joint work with Elchanan Mossel.