Past Probability Seminars Spring 2013

Thursday, January 31, Bret Larget, UW-Madison

Title: Approximate conditional independence of separated subtrees and phylogenetic inference

Abstract: Bayesian methods to reconstruct evolutionary trees from aligned DNA sequence data from different species depend on Markov chain Monte Carlo sampling of phylogenetic trees from a posterior distribution. The probabilities of tree topologies are typically estimated with the simple relative frequencies of the trees in the sample. When the posterior distribution is spread thinly over a very large number of trees, the simple relative frequencies from finite samples are often inaccurate estimates of the posterior probabilities for many trees. We present a new method for estimating the posterior distribution on the space of trees from samples based on the approximation of conditional independence between subtrees given their separation by an edge in the tree. This approximation procedure effectively spreads the estimated posterior distribution from the sampled trees to the larger set of trees that contain clades (sets of species in subtrees) that have been sampled, even if the full tree is not part of the sample. The approximation is shown to be accurate for many data sets and is theoretically justified. We also explore a consequence of this result that may lead to substantial increases in computational efficiency for sampling trees from posterior distributions. Finally, we present an open problem to compare rates of convergence between the simple relative frequency approach and the approximation approach.

Thursday, February 14, Jean-Luc Thiffeault, UW-Madison

Title: Biomixing and large deviations

Abstract: As fish, micro-organisms, or other bodies move through a fluid, they stir their surroundings. This can be beneficial to some fish, since the plankton they eat depends on a well-stirred medium to feed on nutrients. Bacterial colonies also stir their environment, and this is even more crucial for them since at small scales there is no turbulence to help mixing. I will discuss a simple model of the stirring action of moving bodies through a fluid. An attempt will be made to explain existing data on the displacements of small particles, which exhibits probability densities with exponential tails. A large-deviation approach helps to explain some of the data, but mysteries remain.

Tuesday, March 5, 2:30pm VV B341, Janosch Ortmann, University of Toronto

Title: Product-form Invariant Measures for Brownian Motion with Drift Satisfying a Skew-symmetry Type Condition

Abstract: Motivated by recent developments on positive-temperature polymer models we propose a generalisation of reflected Brownian motion (RBM) in a polyhedral domain. Our process is obtained by replacing the singular drift on the boundary by a continuous one which depends, via a potential U, on the position of the process relative to the domain. We show that our generalised process has an invariant measure in product form, under a certain skew-symmetry condition that is independent of the choice of potential. Applications include TASEP-like particle systems, generalisations of Brownian motion with rank-dependent drift and diffusions connected to the generalised Pitman transform.

Thursday, March 14, Brian Rider, Temple University

Title: Universality for the stochastic Airy operator

Abstract: The stochastic Airy operator (SAO) has the form second derivative plus shifted white noise potential. Its reason for being is that it describes the Tracy-Widom laws extended to "general beta" (from the original beta=1,2,4 laws tied to real, complex, and quaternion symmetries). More to the point, SAO is known to be the operator limit for certain random tridiagonal matrices which realize, for example, log-gas distributions on the line with quadratic potential (the "beta Hermite ensembles"), scaled to the edge of their spectrum. Here we show that SAO characterizes edge universality for a more general class of log-gases, defined by more general polynomial potentials beyond the quadratic case. Joint work with M. Krishnapur and B. Virag.

Thursday, March 21, Timo Seppalainen (UW Madison)

Title: Limits of ratios of partition functions for the log-gamma polymer

Abstract: For the model known as the directed polymer in a random medium, the definition of weak disorder is that normalized partition functions converge to a positive limit. In strong disorder this limit vanishes. In the log-gamma polymer we can show that ratios of point-to-point and point-to-line partition functions converge to gamma-distributed limits. One consequence of this is that the quenched polymer measure converges to a random walk in a correlated random environment. This RWRE can be regarded as a positive temperature analogue of the competition interface of last-passage percolation, or the second class particle.

Thursday, April 11, Kevin Lin, University of Arizona

Title: Stimulus-response reliability of dynamical networks

Abstract: A network of dynamical systems (e.g., neurons) driven by a fluctuating time-dependent signal is said to be reliable if, upon repeated presentations of the same signal, it gives essentially the same response each time. As a system's degree of reliability may constrain its ability to encode and transmit information, a natural question is how network conditions affect reliability; this question is of interest in e.g. computational neuroscience. In this talk, I will report on a body of work aimed at discovering network conditions and dynamical mechanisms that affect the reliability of networks, within a class of idealized neural network models. I will discuss a general condition for reliability, and survey some specific mechanisms for reliable and unreliable behavior in concrete models.

Tuesday, April 16, 2:30pm, VV B341 Lea Popovic, Concordia University

Title: Stochastically induced bistability in chemical reaction systems

Abstract: We study a stochastic two-species interacting population system, in which species interact within each compartment according to some nonlinear dynamics. In addition we have another mechanism (e.g. migration between compartments, or splitting of compartments) which yield unbiased perturbative changes to species amounts. If each compartment has a large but bounded capacity, then certain combination of these two mechanisms can lead to stochastically induced bistability. In fact, depending on the relative rates between the mechanisms, there are two ways in which bistability can occur, with distinct signatures. This problem is motivated by dynamics of certain biochemical processes such as gene expression, where the numbers of species interacting are small enough that the randomness inherent in chemical reaction processes can no longer be ignored. This is joint work with J. McSweeney.

Thursday, April 18, Lee DeVille, University of Illinois

Title: Emergent metastability for dynamical systems on networks

Abstract: We will consider stochastic dynamical systems defined on networks that exhibit the phenomenon of collective metastability---by this we mean network dynamics where none of the individual nodes' dynamics are metastable, but the configuration is metastable in its collective behavior. We will concentrate on the case of SDE with small white noise for concreteness. We also present some specific results relating to stochastic perturbations of the Kuramoto system of coupled nonlinear oscillators. Along the way, we show that there is a non-standard spectral problem that appears naturally, and that the important features of this spectral problem are determined by a certain homology group.

Thursday, April 25, Fraydoun Rezakhanlou, UC - Berkeley

Title: Stochastic Poincare--Birkhoff Theorem

Abstract: The Poincare--Birkhoff Theorem , also called Poincare Last Geometric Theorem, is a landmark result in area preserving dynamics. It was formulated by Poincare based on his investigations in celestial mechanics. The theorem may be easily stated: a periodic twist map of an infinite closed strip, or closed annulus, has at least two geometrically distinct fixed points. V.I. Arnold realized that the correct generalization to higher dimensions concerned symplectic maps, not volume preserving maps. He then formulated the higher dimensional analogue of the Poincare--Birkhoff Theorem: the Arnold Conjecture. A parallel generalization of classical the Poincare--Birkhoff Theorem is to investigate whether it holds in the stochastic setting. That is, maps are now stochastic with respect to some probability measure. In this talk, I discuss a variant of the Poincare--Birkhoff Theorem for stationary area preserving dynamics, and hopefully opens the way to a stochastic Arnold Conjecture. (Joint work with Alvaro Pelayo.)

Wednesday, May 1, VV B115, Bálint Vető, University of Bonn

Title: Stationary Solution of the 1D KPZ Equation

Abstract: The KPZ equation is believed to describe a variety of surface growth phenomena that appear naturally, e.g. crystal growth, facet boundaries, solidification fronts, paper wetting or burning fronts. In the recent years, serious efforts were made to describe the solution with different types of initial data. In the present work, we derive an explicit solution for the equation with stationary, i.e. two-sided Brownian motion initial condition. Our approach to the solution for the KPZ equation is via its representation as the free energy of a certain directed random polymer model. By providing integral formulas for the action of Macdonald difference operators, we characterize explicitly the free energy of another polymer model by giving a Fredholm determinant formula which is suitable for asymptotic analysis. In the large time limit of the solution, we recover the distribution obtained for the limiting fluctuations of the height function of the stationary totally asymmetric simple exclusion process (TASEP).