Seminar Schedule:
The seminar meets on Tuesdays from 4:15 to 5:15 pm EST, beginning February 6th.
Abstract: Recently, there has been much progress in understanding stationary measures for coloured (also called multi-species or multi-type) interacting particle systems motivated by asymptotic phenomena and rich underlying algebraic and combinatorial structures (such as nonsymmetric Macdonald polynomials). I will describe a unifying approach to constructing stationary measures for most known such systems (including the classical multispecies Asymmetric Simple Exclusion Process) based on integrable stochastic vertex models and the Yang-Baxter equation. Joint work with Amol Aggarwal and Matthew Nicoletti.
Abstract: Measures of tree balance play an important role in different research areas ranging from evolutionary biology to theoretical computer science. The balance of a tree is usually quantified in a single number, called a balance or imbalance index, and several such indices exist in the literature. Some of them are well-understood, while for others there are still open questions regarding their mathematical properties.
In this talk, I will give a comprehensive introduction to tree balance and introduce different tree balance indices. I will then focus on presenting some curious results related to tree balance indices, describing recent advances and novel contributions in the field, and highlighting some open questions and directions for future research.
Abstract: We show new lower bounds for sphere packings in high dimensions and for independent sets in graphs with not too large co-degrees. For dimension d, this achieves a sphere packing of density (1 + o(1)) d log d / 2^(d+1). In general dimension, this provides the first asymptotically growing improvement for sphere packing lower bounds since Rogers’ bound of c*d/2^d in 1947. The proof amounts to a random (very dense) discretization together with a new theorem on constructing independent sets on graphs with not too large co-degree. Both steps will be discussed, and no knowledge of sphere packings will be assumed or required. Central to the analysis is the study of a random process on a graph. This is based on joint work with Marcelo Campos, Matthew Jenssen and Julian Sahasrabudhe.
Abstract: Directed polymers are a statistical mechanics model for random growth. Their partition functions are solutions to a discrete stochastic heat equation. This talk will discuss the logarithmic derivatives of the partition functions, which are solutions to a discrete stochastic Burgers equation. Of interest is the success or failure of the “one force-one solution principle” for this equation. I will reframe this question in the language of polymers, and share some surprising results that follow. Based on joint work with Louis Fan and Timo Seppäläinen.
Abstract: Physicists starting in the 1980s proposed that some systems display “self-organized criticality” as an explanation for how real-life systems with no obvious phase transition might exhibit self-similarity and power-law tails typical of a statistical mechanics model at criticality. The activated random walk model (ARW) was eventually proposed as a universal model for self-organized criticality. The crux of its proposed behavior is the density conjecture: ARW has a traditional phase transition on an infinite lattice at some critical density, while on a finite box with removal of particles at the boundary, its density converges over time to the same critical value. We prove the density conjecture in one dimension, giving the first confirmation that the model demonstrates self-organized criticality in any setting. We prove that critical values for ARW in several other settings are also identical, providing further evidence for the universality of ARW. Joint work with Chris Hoffman and Matt Junge.
Abstract: I shall recall generalized Ray-Knight theorems in the context of several models of self-interacting random walks (SIRWs) introduced by Balint Toth in the late 1990s. These theorems are very useful in studying the scaling limits of SIRWs. Nevertheless, just by themselves, they do not always correctly identify the scaling limit or imply that such a limit exists. I shall describe some open problems, conjectures, and work in progress with Thomas Mountford (EPFL) and Jonathon Peterson (Purdue University).
Abstract: This talk will present recent work on the Minkowski content of the scaling limit of the three-dimensional loop-erased random walk (LERW). Since essential tools in the continuum are lacking, key parts of the analysis occur in discrete space. In particular, we will overview some sharp estimates on the one-point function and ball-hitting probabilities for the LERW on Z^3. Based on joint work with Xinyi Li and Daisuke Shiraishi.
Abstract: We focus on the Ising p-spin glass model for large p. Confirming a prediction of Kirkpatrick and Thirumalai from 1987, we show the existence of a shattering phase: with high probability, the configuration space breaks down into exponentially many well-separated clusters such that (a) each cluster has an exponentially small Gibbs mass, and (b) the clusters collectively contain all but a vanishing fraction of Gibbs mass. I will then discuss shattering in random computational problems (in particular, the random k-SAT) and algorithmic implications of shattering.
In the second part of the talk, we focus on the algorithmic problem of optimizing the Hamiltonian of this model. We establish the presence of the multi Overlap Gap Property (m-OGP), an intricate geometrical property known to be a rigorous barrier for broad classes of algorithms. In particular, we show that (a) as p grows, the onset of the m-OGP asymptotically matches the algorithmic threshold of the Random Energy Model—the formal p->∞ limit of the p-spin model, and (b) the m-OGP exhibits a sharp phase transition.
To best of our knowledge, these are the first shattering result for Ising spin glasses and the first sharp phase transition result for m-OGP.
Based on several papers, some of which are joint work with David Gamarnik (MIT) and Aukosh Jagannath (Waterloo).
Abstract: We will present a matching upper and lower bound for the right tail probability of the maximum of a random model of the zeta function over intervals of varying length. In particular, we show that the distribution of the maximum interpolates between that of log-correlated and IID random variables as the interval varies in length. This result follows the recent work of Arguin-Dubach-Hartung and is inspired by a conjecture by Fyodorov-Hiary-Keating on the maximum over short intervals.
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