2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 | 2023
The seminar is jointly sponsored by Temple and Penn. The organizers are Brian Rider and Atilla Yilmaz (Temple), and Jian Ding, Robin Pemantle and Xin Sun (Penn).
Talks are Tuesdays 3:30 - 4:30 pm and are held either in Wachman Hall (Temple) or David Rittenhouse Lab (Penn) as indicated below.
For a chronological listing of the talks, click the year above.
Fan Yang, UPenn
We consider a Hermitian random band matrix $H$ on the $d$-dimensional lattice of linear size $L$. Its entries are independent centered complex Gaussian random variables with variances $s_{xy}$, that are negligible if $|x-y|$ exceeds the band width $W$. In dimensions eight or higher, we prove that, as long as $W > L^\epsilon$ for a small constant $\epsilon>0$, with high probability, most bulk eigenvectors of $H$ are delocalized in the sense that their localization lengths are comparable to $L$. Moreover, we also prove a quantum diffusion result of this model in terms of the Green's function of $H$. Joint work with Horng-Tzer Yau and Jun Yin.
Jack Hanson, City College & The Graduate Center, CUNY
In their study of percolation, physicists have proposed "scaling hypotheses" relating the behavior of the model in the critical ($p = p_c$) and subcritical ($p < p_c$) regimes. We show a version of such a scaling hypothesis for the one-arm probability $\pi(n;p)$ — the probability that the open cluster of the origin has Euclidean diameter at least $n$.
As a consequence of our analysis, we obtain the correct scaling of the lower tail of cluster volumes and the chemical (intrinsic) distances within clusters. We also show that the number of spanning clusters of a side-length $n$ box is tight on scale $n^{d-6}$. A new tool of our analysis is a sharp asymptotic for connectivity probabilities when paths are restricted to lie in half-spaces.
Jiaming Xia, UPenn
In this talk, I will first present the high-dimensional limit of the free energy associated with the inference problem of finite-rank matrix tensor products. We compare the limit with the solution to a certain Hamilton-Jacobi equation, following the recent approach by Jean-Christophe Mourrat. The motivation comes from the averaged free energy solving an approximate Hamilton-Jacobi equation. We consider two notions of solutions which are weak solutions and viscosity solutions. The two types of solutions require different treatments and each has its own advantages. At the end of this part, I will show an example of application of our results to a model with i.i.d. entries and symmetric interactions. If time permits, I will talk about the same problem but with a different model, namely, the multi-layer generalized linear model. I will mainly explain the iteration method as an important tool used in our proof. This is based on joint works with Hong-Bin Chen and J.-C. Mourrat, NYU.
Minjae Park, MIT
Although lattice Yang-Mills theory on $\mathbb{Z}^d$ is easy to rigorously define, the construction of a satisfactory continuum theory on $\mathbb{R}^d$ is a major open problem when $d\ge 3$. Such a theory should assign a Wilson loop expectation to each suitable collection $\mathcal{L}$ of loops in $\mathbb{R}^d$. One classical approach is to try to represent this expectation as a sum over surfaces with boundary $\mathcal{L}$. There are some formal/heuristic ways to make sense of this notion, but they typically yield an ill-defined difference of infinities.
In this talk, we show how to make sense of Yang-Mills integrals as surface sums for $d=2$, where the continuum theory is already understood. We also obtain an alternative proof of the Makeenko-Migdal equation and a version of the Gross-Taylor expansion. Joint work with Joshua Pfeffer, Scott Sheffield, and Pu Yu.
David Renfrew, SUNY Binghamton
We consider the density of states of structured Hermitian random matrices with a variance profile. As the dimension tends to infinity the associated eigenvalue density can develop a singularity at the origin. The severity of this singularity depends on the relative positions of the zero submatrices. We provide a classification of all possible singularities and determine the exponent in the density blow-up.
Robert Hough, Stony Brook University
Alexopoulos proved local limit theorems for measures with a density and lattice measures in the general setting of groups of moderate growth. On the Heisenberg group, Breuillard's thesis obtained a local limit theorem for general measures subject to a condition on the characteristic function, and asked if this condition can be removed. I will discuss two new local limit theorems, one joint with Diaconis, that treat local limit theorems on nilpotent Lie groups driven by general measures. We prove Breuillard's conjecture and also solve a problem of Diaconis and Saloff-Coste on the mixing of the central coordinate in unipotent matrix walks modulo $p$.
Guillaume Remy, Columbia University
Liouville theory is a fundamental example of a conformal field theory (CFT) first introduced in physics by A. Polyakov to describe a canonical random 2d surface. In recent years it has been rigorously studied using probabilistic techniques. In this talk we will study the integrable structure of Liouville CFT on a domain with boundary by proving exact formulas for its correlation functions. Our latest result is derived using conformal welding of random surfaces, in relation with the Schramm-Loewner evolutions. We will also discuss the connection with the conformal blocks of CFT which are fundamental functions determined by conformal invariance that underlie the exact solvability of CFT. Based on joint works with Morris Ang, Promit Ghosal, Xin Sun, Yi Sun and Tunan Zhu.
Jonathon Peterson, Purdue University
Random walks in cooling random environments are a model of random walks in dynamic random environments where the random environment is re-sampled at a fixed sequence of times (called the cooling sequence) and the environment remains constant between these re-sampling times. We study the limiting distributions of the walk in the case when distribution on environments is such that a walk in a fixed environment has an $s$-stable limiting distribution for some $s \in (1,2)$. It was previously conjectured that for cooling maps whose gaps between re-sampling times grow polynomially that the model should exhibit a phase transition from Gaussian limits to $s$-stable depending on the exponent of the polynomial growth of the re-sampling gaps. We confirm this conjecture, identifying the precise exponent at which the phase transition occurs and proving that at the critical exponent the limiting distribution is a generalized tempered $s$-stable distribution. The proofs require us to prove some previously unknown facts about one-dimensional random walks in random environments which are of independent interest.
Joshua Pfeffer, Columbia University
In my talk I will discuss Loewner chains whose driving functions are complex Brownian motions with general covariance matrices. This extends the notion of Schramm-Loewner evolution (SLE) by allowing the driving function to be complex-valued and not just real-valued. We show that these Loewner chains exhibit the same phases (simple, swallowing, and space-filling) as SLE, and we explicitly characterize the values of the covariance matrix corresponding to each phase. In contrast to SLE, we show that the evolving left hulls are a.s. not generated by curves, and that they a.s. disconnect each fixed point in the plane from infinity before absorbing the point.
This talk is based on a joint work with Ewain Gwynne.
Christian Houdré, Georgia Tech
Let $(X_n)_{n \ge 0}$ be an irreducible, aperiodic, homogeneous Markov chain, with state space a totally ordered finite alphabet of size $m$. Using combinatorial constructions and weak invariance principles, we obtain the limiting shape of the associated RSK Young diagrams as a multidimensional Brownian functional. Since the length of the top row of the Young diagrams is also the length of the longest weakly increasing subsequences of $(X_k)_{1\le k \le n}$, the corresponding limiting law follows. We relate our results to a conjecture of Kuperberg by providing, under a cyclic condition, a spectral characterization of the Markov transition matrix precisely characterizing when the limiting shape is the spectrum of the $m \times m$ traceless GUE. For each $m \ge 4$, this characterization identifies a proper, non-trivial class of cyclic transition matrices producing such a limiting shape. However, for $m=3$, all cyclic Markov chains have such a limiting shape, a fact previously only known for $m=2$. For $m$ arbitrary, we also study reversible Markov chains and obtain a characterization of symmetric Markov chains for which the limiting shape is the spectrum of the traceless GUE. To finish, we explore, in this general setting, connections between various limiting laws and spectra of Gaussian random matrices, focusing in particular on the relationship between the terminal points of the Brownian motions, the diagonal terms of the random matrix, and the scaling of its off-diagonal terms, a scaling we conjecture to be a function of the spectrum of the covariance matrix governing the Brownian motion.
Joint work with Trevis Litherland.
Jacopo Borga, Stanford University
Consider a large random permutation satisfying some constraints or biased according to some statistics. What does it look like? In this seminar we make sense of this question introducing the notion of permutons. Permuton convergence has been established for several models of random permutations in various works: we give an overview of some of these results, mainly focusing on the case of pattern-avoiding permutations. The main goal of the talk is to present a new family of universal limiting permutons, called skew Brownian permutons. This family includes (as particular cases) some already studied limiting permutons, such as the biased Brownian separable permuton and the Baxter permuton. We also show that some natural families of random constrained permutations converge to some new instances of skew Brownian permutons. The construction of these new limiting objects will lead us to investigate an intriguing connection with some perturbed versions of the Tanaka SDE and the SDEs encoding skew Brownian motions. If time permits, we will present some conjectures on how it should be possible to construct these new limiting permutons directly from the Liouville quantum gravity decorated with two SLE curves.
Krishnan Mody, Courant Institute, NYU
I will discuss recent work with P. Bourgade and M. Pain in which we show that the log-characteristic polynomial for general beta ensembles converges to a log-correlated field in the large-dimension limit. The proof of this result relies on a so-called optimal local law, which I will explain and prove in the Gaussian case. I will explain how the local law is useful, and give an outline of the proof of the log-correlated field.
Louis-Pierre Arguin, Baruch College & The Graduate Center, CUNY
I will give an account of the recent progress in probability and in number theory to understand the large values of the zeta function in small intervals of the critical line. This problem has interesting connections with the extreme value statistics of IID and log-correlated random variables, as well as random matrix theory.
2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 | 2023