ABSTRACTS
May 05 @ Penn: Oren Louidor (Technion)
Aging in a logarithmically correlated potential
We consider a continuous time random walk on the box of side length N in Z^2, whose transition rates are governed by the discrete Gaussian free field h on the box with zero boundary conditions, acting as potential: At inverse temperature \beta, when at site x the walk waits an exponential time with mean \exp(\beta h_x) and then jumps to one of its neighbors chosen uniformly at random. This process can be used to model a diffusive particle in a random potential with logarithmic correlations or alternatively as Glauber dynamics for a spin-glass system with logarithmically correlated energy levels. We show that at any sub-critical temperature and at pre-equilibrium time scales, the walk exhibits aging. More precisely, for any \theta > 0 and suitable sequence of times (t_N), the probability that the walk at time t_N(1+\theta) is within O(1) of where it was at time t_N tends to a non-trivial constant as N \to \infty, whose value can be expressed in terms of the distribution function of the generalized arcsine law. This puts this process in the same aging universality class as many other spin-glass models, e.g. the random energy model. Joint work with Aser Cortines-Peixoto and Adela Svejda.
Apr 26 @ Penn: Josh Rosenberg (Penn)
The frog model with drift on R
This paper considers the following scenario. There is a Poisson process on R with intensity f where 0 \le f(x) \le infty for x \ge 0 and f(x)=0 for x \le 0. The "points" of the process represent sleeping frogs. In addition, there is one active frog initially located at the origin. At time t=0 this frog begins performing Brownian motion with leftward drift C (i.e. its motion is a random process of the form B_t-Ct). Any time an active frog arrives at a point where a sleeping frog is residing, the sleeping frog becomes active and begins performing Brownian motion with leftward drift C, that is independent of the motion of all of the other active frogs. This paper establishes sharp conditions on the intensity function f that determine whether the model is transient (meaning the probability that infinitely many frogs return to the origin is 0), or non-transient (meaning this probability is greater than 0).
Apr 19 @ Penn: Dan Jerison (Cornell)
Markov chain convergence via regeneration
How long does it take for a reversible Markov chain to converge to its stationary distribution? This talk discusses how to get explicit upper bounds on the time to stationarity by identifying a regenerative structure of the chain. I will demonstrate the flexibility of this approach by applying it in two very different cases: Markov chain Monte Carlo estimation on general state spaces, and finite birth and death chains. In the first case, an unusual perspective on the popular ``drift and minorization'' method leads to a simple bound that improves on existing convergence results. In the second case, a hidden connection between reversibility and monotonicity recovers sharp upper bounds on the cutoff window.
Apr 12 @ Penn: Zsolt Pajor-Gyulai (Courant)
Stochastic approach to anomalous diffusion in two dimensional, incompressible, periodic, cellular flows
It is a well known fact that velocity grandients in a flow change
the dispersion of a passive tracer. One clear manifestation of this
phenomenon is that in systems with homogenization type diffusive long
time/large scale behavior, the effective diffusivity often differs greatly
from the molecular one. An important aspect of these well known result is
that they are only valid on timescales much longer than the inverse
molecular diffusivity. We are interested in what happens on shorter
timescales (subhomogenization regimes) in a family of two-dimensional
incompressible periodic flows that consists only of pockets of
recirculations essentially acting as traps and infinite flowlines
separating these where significant transport is possible. Our approach is
to follow the random motion of a tracer particle and show that under
certain scaling it resembles a time-changed Brownian motions. This shows
that while the trajectories are still diffusive, the variance grows
differently than linear.
Apr 05 @ Penn: Boris Hanin (MIT)
Nodal Sets of Random Eigenfunctions of the Harmonic Oscillator
Random eigenfunctions of energy E for the isotropic harmonic oscillator in R^d have a U(d) symmetry and are in some ways analogous to random spherical harmonics of fixed degree on S^d, whose nodal sets have been the subject of many recent studies. However, there is a fundamentally new aspect to this ensemble, namely the existence of allowed and forbidden regions. In the allowed region, the Hermite functions behave like spherical harmonics, while in the forbidden region, Hermite functions are exponentially decaying and it is unclear to what extent they oscillate and have zeros.
The purpose of this talk is to present several results about the expected volume of the zero set of a random Hermite function in both the allowed and forbidden regions as well as in a shrinking tube around the caustic. The results are based on an explicit formula for the scaling limit around the caustic of the fixed energy spectral projector for the isotropic harmonic oscillator. This is joint work with Steve Zelditch and Peng Zhou.
Mar 29 @ Penn: John Pike (Cornell)
Random walks on abelian sandpiles
Given a simple connected graph $G=(V,E)$, the abelian sandpile Markov chain evolves by adding chips to random vertices and then stabilizing according to certain toppling rules. The recurrent states form an abelian group $\Gamma$, the sandpile group of $G$. I will discuss joint work with Dan Jerison and Lionel Levine in which we characterize the eigenvalues and eigenfunctions of the chain restricted to $\Gamma$ in terms of ``multiplicative harmonic functions'' on $V$. We show that the moduli of the eigenvalues are determined up to a constant factor by the lengths of vectors in an appropriate dual Laplacian lattice and use this observation to bound the mixing time of the sandpile chain in terms of the number of vertices and maximum vertex degree of $G$. We also derive a surprising inverse relationship between the spectral gap of the sandpile chain and that of simple random walk on $G$.
Mar 22 @ Temple: Christian Benes (CUNY)
The scaling limit of the loop-erased random walk Green's function
We show that the probability that a planar loop-erased random walk passes through a given edge in the interior of a lattice approximation of a simply connected domain converges, as the lattice spacing goes to zero, to a multiple of the SLE(2) Green's function. This is joint work with Greg Lawler and Fredrik Viklund.
Mar 15 @ Temple: Philippe Sosoe (Harvard)
The chemical distance in critical percolation
The chemical distance is the graph distance inside percolation clusters. In the
supercritical phase, this distance is known to be linear with exponential
probability, enabling a detailed study of processes like random walks on the
infinite cluster. By contrast, at the critical point, the distance is known to
be longer than Euclidean by some (unknown) power. I will discuss this and some
bounds on distance, as well as a result comparing the chemical distance to the
size of the lowest crossing. Joint work with Jack Hanson and Michael Damron.
Mar 01 @ Penn: Sumit Mukherjee (Columbia)
Mean field Ising models
In this talk we consider the asymptotics of the log partition function of an Ising model on a sequence of finite but growing graphs/matrices. We give a sufficient condition for the mean field prediction to the log partition function to be asymptotically tight, which in particular covers all regular graphs with degree going to infinity. We show via several examples that our condition is "almost necessary" as well.
As application of our result, we derive the asymptotics of the log partition function for approximately regular graphs, and bi-regular bi-partite graphs. We also re-derive asymptotics of the log partition function for a sequence of graphs convering in cut metric.
This is joint work with Anirban Basak from Duke University.
Feb 16 @ Temple: Yuri Bakhtin (Courant)
Burgers equation with random forcing
I will talk about the ergodic theory of randomly forced Burgers equation (a basic nonlinear evolution PDE related to fluid dynamics and growth models) in the noncompact setting. The basic objects are one-sided infinite minimizers of random action (in the inviscid case) and polymer measures on one-sided infinite trajectories (in the positive viscosity case). Joint work with Eric Cator, Kostya Khanin, Liying Li.
Feb 09 @ Penn: Nayantara Bhatnagar (Delaware)
Limit Theorems for Monotone Subsequences in Mallows Permutations
The longest increasing subsequence (LIS) of a uniformly random permutation is a well studied problem. Vershik-Kerov and Logan-Shepp first showed that asymptotically the typical length of the LIS is 2sqrt(n). This line of research culminated in the work of Baik-Deift-Johansson who related this length to the GUE Tracy-Widom distribution.
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. Under this measure, the probability of a permutation p in S_n is proportional to q^Inv(p) where q is a real parameter and Inv(p) is the number of inversions in p. 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. In the regime that q is constant, we make use of the regenerative structure of the permutation to prove a Gaussian CLT for the LIS.
This is based on joint work with Ron Peled and with Riddhi Basu.
Feb 02 @ Penn: Erik Slivken (UC Davis)
Bootstrap Percolation on the Hamming Torus
Bootstrap percolation on a graph is a simple to describe yet hard to analyze process on a graph. It begins with some initial configuration (open or closed) on the vertices. At each subsequent step a vertex may change from closed to open if enough of its neighbors are already open. For a random initial configuration where each vertex is open independently with probability p, how does the probability that eventually every vertex will be open change as p varies?
The large neighborhood size of the Hamming torus leads to a distinctly different flavor than previous results on the grid and hypercube. We will focus on Hamming tori with high dimension, giving a detailed description of the long term behavior of the process.
Jan 26 @ Penn: Vadim Gorin (MIT)
Largest eigenvalues in random matrix beta-ensembles: structures of the limit
Despite numerous articles devoted to its study, the universal scaling limit for the largest eigenvalues in general beta log-gases remains a mysterious object. I will present two new approaches to such edge scaling limits. The outcomes include a novel scaling limit for the differences between largest eigenvalues in submatrices and a Feynman-Kac type formula for the semigroup spanned by the Stochastic Airy Operator. (based on joint work with M.Shkolnikov)
Dec 01 @ Penn: Sivak Mkrtchyan (Rochester)
The entropy of Schur-Weyl measures
We will study local and global statistical properties of Young diagrams with respect to a Plancherel-type family of measures called Schur-Weyl measures and use the results to answer a question from asymptotic representation theory. More precisely, we will solve a variational problem to prove a limit-shape result for random Young diagrams with respect to the Schur-Weyl measures and apply the results to obtain logarithmic, order-sharp bounds for the dimensions of certain representations of finite symmetric groups.
Nov 17 @ Penn: Partha Dey (UIUC)
Longest increasing path within the critical strip
Consider a Poisson Point Process of intensity one in the two-dimensional square of side length $n$. In Baik-Deift-Johansson (1999), it was shown that the length of a longest increasing path (an increasing path that contains the most number of points) when properly centered and scaled converges to the Tracy-Widom distribution. Later Johansson (2000) showed that all maximal paths lie within the strip of width $n^{2/3+o(1)}$ around the diagonal with high probability. We consider the length $L(n,w)$ of longest increasing paths restricted to lie within a strip of width $w$ around the diagonal and show that when properly centered and scaled it converges to a Gaussian distribution whenever $w \ll n^{2/3}$. We also obtain tight bounds on the expectation and variance of $L(n,w)$ which involves application of BK inequality and approximation of the optimal restricted path by locally optimal unrestricted path. Based on joint work with Matthew Joseph and Ron Peled.
Nov 10 @ Penn: Charles Bordenave (Toulouse)
A new proof of Friedman’s second eigenvalue Theorem and its extensions
It was conjectured by Alon and proved by Friedman that a random d-regular graph has nearly the largest possible spectral gap, more precisely, the largest absolute value of the non-trivial eigenvalues of its adjacency matrix is at most 2 √ ( d − 1) + o(1) with probability tending to one as the size of the graph tends to infinity. We will discuss a new method to prove this statement and give some extensions to random lifts and related models.
Nov 03 @ Penn: Christian Gromoll (UVA)
Fluid limits and queueing policies
There are many different queueing policies discussed in the literature. They tend to be defined in model-specific ways that differ in format from one policy to another, each format suitable for the task at hand (e.g. steady-state derivation, scaling-limit theorem, or proof of some other property). The ad hoc nature of the policy definition often limits the scope of potentially quite general arguments. Moreover, because policies are defined variously, it's difficult to approach classification questions for which the answer presumably spans many policies.
In this talk I'll propose a definition of a general queueing policy and discuss exactly what I mean by "general". The setup makes it possible to frame questions about queues in terms of an arbitrary policy and, potentially, to classify policies according to the answer. In this vein, I'll discuss a few results and some ongoing work on proving fluid limit theorems for general policies.
Oct 27 @ Penn: Doug Rizzolo (U Delaware)
Random pattern-avoiding permutations
Abstract: In this talk we will discuss recent results on the structure of random pattern-avoiding permutations. We will focus a surprising connection between random permutations avoiding a fixed pattern of length three and Brownian excursion. For example, this connection lets us describe the shape of the graph of a random 231-avoiding permutation of {1,...,n} as n tends to infinity as well as the asymptotic distribution of fixed points in terms of Brownian excursion. Time permitting, we will discuss work in progress on permutations avoiding longer patterns. This talk is based on joint work with Christopher Hoffman and Erik Slivken.
Oct 20 @ Penn: Tai Melcher (UVA)
Smooth measures in infinite dimensions
A collection of vector fields on a manifold satisfies H\"{o}rmander's condition if any two points can be connected by a path whose tangent vectors lie in the given collection. 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. We will discuss some smoothness results for diffusions in a particular class of infinite-dimensional spaces. This is based on joint work with Fabrice Baudoin, Daniel Dobbs, Bruce Driver, Nate Eldredge, and Masha Gordina.
Oct 06 @ Penn: Leonid Petrov (UVA)
Bethe Ansatz and interacting particle systems
I will describe recent advances in bringing a circle of ideas and techniques around Bethe ansatz and Yang–Baxter relation under the probabilistic roof, which provides new examples of stochastic interacting particle systems, and techniques to solve them. In particular, I plan to discuss a new particle dynamics in continuous inhomogeneous medium with features resembling traffic models, as well as queuing systems. This system has phase transitions (discontinuities in the limit shape) and Tracy-Widom fluctuations (even at the point of the phase transition).
Sep 29 @ Temple: David Belius (Courant)
Branching in log-correlated random fields
This talk will discuss how log-correlated random fields show up in diverse settings, including the study of cover times and random matrix theory. This is explained by the presence of an underlying approximate branching structure in each of the models. I will describe the most basic model of the log-correlated class, namely Branching Random Walk (BRW), where the branching structure is explicit, and explain how to adapt ideas developed in the context of BRW to models where the branching structure is not immediately obvious.
Sep 24 @ Penn: Steven Heilman (UCLA)
Strong Contraction and Influences in Tail Spaces
We study contraction under a Markov semi-group and influence bounds for functions all of whose low level Fourier coefficients vanish. This study is motivated by the explicit construction of 3-regular expander graphs of Mendel and Naor, though our results have no direct implication for the construction of expander graphs. In the positive direction we prove an L_{p} Poincar\'{e} inequality and moment decay estimates for mean 0 functions and for all 1 \less p \less \infty, proving the degree one case of a conjecture of Mendel and Naor as well as the general degree case of the conjecture when restricted to Boolean functions. In the negative direction, we answer negatively two questions of Hatami and Kalai concerning extensions of the Kahn-Kalai-Linial and Harper Theorems to tail spaces. For example, we construct a function $f\colon\{-1,1\}^{n}\to\{-1,1\}$ whose Fourier coefficients vanish up to level $c \log n$, with all influences bounded by $C \log n/n$ for some constants $0\lessc,C\less \infty$. That is, the Kahn-Kalai-Linial Theorem cannot be improved, even if we assume that the first $c\log n$ Fourier coefficients of the function vanish. This implies there is a phase transition in the largest guaranteed influence of functions $f\colon\{-1,1\}^{n}\to\{-1,1\}$, which occurs when the first $g(n)\log n$ Fourier coefficients vanish and $g(n)\to\infty$ as $n\to\infty$ or $g(n)$ is bounded as $n\to\infty$.. joint with Elchanan Mossel and Krzysztof Oleszkiewicz
Sep 15 @ Penn: Toby Johnson (USC)
The frog model on trees
Imagine that every vertex of a graph contains a sleeping frog. At time 0, the frog at some designated vertex wakes up and begins a simple random walk. When it lands on a vertex, the sleeping frog there wakes up and begins its own simple random walk, which in turn wakes up any sleeping frogs it lands on, and so on. This process is called the frog model. I'll (mostly) answer a question posed by Serguei Popov in 2003: On an infinite d-ary tree, is the frog model recurrent or transient? That is, is each vertex visited infinitely or finitely often by frogs? The answer is that it depends on d: there's a phase transition between recurrence and transience as d grows. Furthermore, if the system starts with Poi(m) sleeping frogs on each vertex independently, for any d there's a phase transition as m grows. This is joint work with Christopher Hoffman and Matthew Junge.
Sep 08 @ Penn: Matt Junge (U. Washington)
Splitting hairs (with choice)
Sequentially place n balls into n bins. For each ball, two bins are sampled uniformly and the ball is placed in the emptier of the two. Computer scientists like that this does a much better job of evenly distributing the balls than the "choiceless" version where one places each ball uniformly. Consider the continuous version: Form a random sequence in the unit interval by having the nth term be whichever of two uniformly placed points falls in the larger gap between the previous n-1 points. We confirm the intuition that this sequence is a.s. equidistributed, resolving a conjecture from Itai Benjamini, Pascal Maillard and Elliot Paquette. The history goes back a century to Weyl and more recently to Kakutani.