UC San Diego Probability Seminar (2020-2021)

The seminar meets on Thursdays via Zoom from 11 AM to 12 PM PT unless noted otherwise. Please send me an email if you would like to be added to the mailing list or if you have a suggestion for a speaker. Invite links to talks will be sent to the mailing list.

October 8
Jorge Garza-Vargas (UC Berkeley)
Title: Spectral stability under random perturbations
Abstract: Consider an \(n\times n\) deterministic matrix \(A\) and a random matrix \(M\) with independent standard Gaussian entries. In this talk I will discuss recent results that state that, if \(||A||\leq 1\), for any \(\delta>0\), with high probability \(A+\delta M\) has eigenvector condition number of order poly\((n/\delta)\) and eigenvalue gaps of order poly\((\delta/n)\), which implies that the randomly perturbed matrix has a stable spectrum. This has useful applications to numerical analysis and was used to obtain the fastest known provable algorithm for diagonalizing an arbitrary matrix.
This is joint work with Jess Banks, Archit Kulkarni and Nikhil Srivastava.

October 29 (Special time: 10 AM to 11 AM PST)
Guillaume Dubach (IST Austria)
Title: Overlaps between eigenvectors of non-Hermitian random matrices
Abstract: Right and left eigenvectors of non-Hermitian matrices form a bi-orthogonal system to which one can associate homogeneous quantities known as overlaps. The matrix of overlaps quantifies the stability of the spectrum and characterizes the joint eigenvalues increments under Dyson-type dynamics. Overlaps first appeared in the physics literature: Chalker and Mehlig calculated their conditional expectation for complex Ginibre matrices (1998). For the same model, we extend their results by deriving the distribution of the overlaps and their correlations (joint work with P. Bourgade). Similar results have been obtained in other integrable settings, namely quaternionic Gaussian matrices, as well as matrices from the spherical and truncated unitary ensembles.

November 19
Ben Landon (MIT)
Title: Fluctuations of the 2-spin spherical Sherrington Kirkpatrick model
Abstract: The 2-spin spherical Sherrington Kirkpatrick (SSK) model was introduced by Kosterlitz, Thouless and Jones as a simplification of the usual Sherrington-Kirkpatrick model with Ising spins. The SSK model admits tractable formulas for many of its observables, allowing for a detailed analysis of its fluctuations using techniques from random matrix theory. We discuss recent results on the fluctuations of the SSK model with a magnetic field as well as at critical temperature. Based on joint work with P. Sosoe.

December 3
Amber Puha (CSU San Marcos and UC San Diego)
Title: Scaling Limits for Shortest Remaining Processing Time Queues
Abstract: In an SRPT queue, the job with the shortest remaining processing time is served first, with preemption. The SRPT scheduling rule is of interest due to its optimality properties; it minimizes queue length (number of jobs in system). However, even with Markovian distributional assumptions on the processing times, an exact analysis is not possible. Hence, approximations in the form of a fluid (functional law of large numbers) limit or a diffusion (functional central limit theorem) limit can provide insights into system performance. It was shown by Gromoll, Kruk and Puha (2011) that, if the processing time distribution has unbounded support, then, under standard heavy traffic conditions, the diffusion limit of the queue length process is identically equal to zero. This exhibits the queue length minimization property of SRPT in sharpest relief. It also demonstrates that the SRPT queue length process is orders of magnitude smaller than the workload process in the diffusion limit. (The workload process tracks the time it will take the server to process the work associated with each job in system.) In this talk, we report on progress in characterizing this order of magnitude difference. We find that distribution dependent scaling must be used to obtain a nontrivial limit for the queue length and the associated measure valued state descriptor. The scaling captures the order of magnitude difference, and the nature of the limit is dependent on the tail decay of the processing time distribution.

This work is joint with Sayan Banerjee (UNC) and Amarjit Budhiraja (UNC).

January 7
Yilin Wang (MIT)
Title: SLE, energy duality, and foliations by Weil-Petersson quasicircles
Abstract: The Loewner energy for Jordan curves first arises from the small-parameter large deviations of Schramm-Loewner evolution (SLE). It is finite if and only if the curve is a Weil-Petersson quasicircle, an interesting class of Jordan curves appearing in Teichmuller theory, geometric function theory, and string theory with currently more than 20 equivalent definitions. In this talk, I will show that the large-parameter large deviations of SLE gives rise to a new Loewner-Kufarev energy, which is dual to the Loewner energy via foliations by Weil-Petersson quasicircles and exhibits remarkable features and symmetries. Based on joint works with Morris Ang and Minjae Park (MIT) and with Fredrik Viklund (KTH).

January 14
Dan Daniel Erdmann-Pham (UC Berkeley)
Title: Hydrodynamics of the inhomogeneous l-TASEP and its Application to Protein Synthesis
Abstract: The inhomogeneous l-TASEP is an interacting particle process wherein particles stochastically enter, unidirectionally traverse, and finally exit a one-dimensional lattice segment at rates that may depend on a particle's location within the lattice. Its homogeneous version is known to exhibit various phase transitions in macroscopic observables like particle density and current, with fluctuations governed by what is known as the KPZ equation. In this talk, we begin to extend such results to the inhomogeneous setting by developing the so-called hydrodynamic limit, which governs the system dynamics on an LLN-type scale. If time permits, we apply our results to elucidate the key determinants of protein synthesis, which motivated the introduction of TASEP fifty years ago. This is based on joint work with Khanh Dao Duc and Yun S. Song.

January 21
Gwen McKinley (UC San Diego)
Title: Counting integer partitions with the method of maximum entropy
Abstract: We give an asymptotic formula for the number of partitions of an integer n where the sums of the kth powers of the parts are also fixed, for some collection of values k. To obtain this result, we reframe the counting problem as an optimization problem, and find the probability distribution on the set of all integer partitions with maximum entropy among those that satisfy our restrictions in expectation (in essence, this is an application of Jaynes' principle of maximum entropy). This approach leads to an approximate version of our formula as the solution to a relatively straightforward optimization problem over real-valued functions. To establish more precise asymptotics, we prove a local central limit theorem using an equidistribution result of Green and Tao.
A large portion of the talk will be devoted to outlining how our method can be used to re-derive a classical result of Hardy and Ramanujan, with an emphasis on the intuitions behind the method, and limited technical detail. This is joint work with Marcus Michelen and Will Perkins.

February 25
Hao Shen (UW Madison)
Title: Stochastic quantization and Yang-Mills
Abstract: We briefly overview the current developments of rigorous constructions in stochastic quantization - an active field linking quantum field theory with stochastic PDE. We then focus on stochastic quantization of the Yang-Mills model in 2 and 3 space dimensions. This includes constructing the Langevin dynamic for the formal Yang-Mills measure, defining the state space of gauge orbits, proving gauge equivariance of the dynamic, and making sense of Wilson loop observables in this context. We will also discuss some future directions.
The talk is based on several works mostly joint with A.Chandra, I.Chevyrev, and M.Hairer.

April 15
Subhabrata Sen (Harvard)
Title: Large deviations for dense random graphs: beyond mean-field
Abstract: In a seminal paper, Chatterjee and Varadhan derived an LDP for the dense Erdős-Rényi random graph, viewed as a random graphon. This directly provides LDPs for continuous functionals such as subgraph counts, spectral norms, etc. In contrast, very little is understood about this problem if the underlying random graph is inhomogeneous or constrained.
In this talk, we will explore large deviations for dense random graphs, beyond the mean-field setting. In particular, we will study large deviations for uniform random graphs with given degrees, and a family of dense block model random graphs. We will establish the LDP in each case, and identify the rate function. In the block model setting, we will use this LDP to study the upper tail problem for homomorphism densities of regular sub-graphs. Our results establish that this problem exhibits a symmetry/symmetry-breaking transition, similar to one observed for Erdős-Rényi random graphs.
Based on joint works with Christian Borgs, Jennifer Chayes, Souvik Dhara, Julia Gaudio and Samantha Petti.
Papers: arxiv.org/abs/1904.07666 and arxiv.org/abs/2007.14508

April 29
March Boedihardjo (UCLA)
Title: Spectral norms of Gaussian matrices with correlated entries
Abstract: We give a non-asymptotic bound on the spectral norm of a \(d\times d\) matrix \(X\) with centered jointly Gaussian entries in terms of the covariance matrix of the entries. In some cases, this estimate is sharp and removes the \(\sqrt{\log d}\) factor in the noncommutative Khintchine inequality. Joint work with Afonso Bandeira.
Paper: arxiv.org/abs/2104.02662

May 13
Sourav Chatterjee (Stanford)
Title: New results for surface growth
Abstract: The growth of random surfaces has attracted a lot of attention in probability theory in the last ten years, especially in the context of the Kardar-Parisi-Zhang (KPZ) equation. Most of the available results are for exactly solvable one-dimensional models. In this talk I will present some recent results for models that are not exactly solvable. In particular, I will talk about the universality of deterministic KPZ growth in arbitrary dimensions, and if time permits, a necessary and sufficient condition for superconcentration in a class of growing random surfaces.
Papers: arxiv.org/abs/2102.13131 and arxiv.org/abs/2103.09199

May 27
Amol Aggarwal (Columbia)
Title: Spectral Statistics of Lévy Matrices
Abstract: Lévy matrices are symmetric matrices whose entries are random variables with infinite variance; they are governed by a parameter \(\alpha \in (0, 2)\) dictating the power law decay of their entries. For \(\alpha < 1\), they are believed to serve as one of the few examples of a matrix model exhibiting a mobility edge, also called an Anderson transition, that separates chaotic (GOE) eigenvalue spacing statistics from ordered (Poisson) ones. In this talk we describe results concerning the statistics for the eigenvalue spacings and eigenvector entries of Lévy matrices. In particular, for \(\alpha \in (1, 2)\) their eigenvalue statistics asymptotically follow those of the GOE throughout the spectrum, and for \(\alpha < 1\) the same statement holds around small eigenvalues. These describe joint works with Patrick Lopatto, Jake Marcinek, and Horng-Tzer Yau.
Papers: arxiv.org/abs/1806.07363 and arxiv.org/abs/2002.09355

June 3
Duncan Dauvergne (Princeton)
Title: The directed landscape
Abstract: The directed landscape is a random `directed metric' on the spacetime plane that arises as the scaling limit of integrable models of last passage percolation. It is expected to be the universal scaling limit for all models in the KPZ universality class for random growth. In this talk, I will describe its construction in terms of the Airy line ensemble via an isometric property of the Robinson-Schensted-Knuth correspondence, and discuss some surprising Brownian structures that arise from this construction. Based on joint work with M. Nica, J. Ortmann, B. Virag, and L. Zhang.

June 10
Alisa Knizel (UChicago)
Title: Stationary measure for the open KPZ equation
Abstract: The Kardar-Parisi-Zhang (KPZ) equation is the stochastic partial differential equation that models interface growth. In the talk I will present the construction of a stationary measure for the KPZ equation on a bounded interval with general inhomogeneous Neumann boundary conditions. Along the way, we will encounter classical orthogonal polynomials, the asymmetric simple exclusion process, and precise asymptotics of q-Gamma functions. This construction is a joint work with Ivan Corwin.