The seminar is jointly sponsored by Temple and Penn. The organizers are Brian Rider and Atilla Yilmaz (Temple), and Jiaoyang Huang, Jiaqi Liu, 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.
Yuri Bakhtin, Courant Institute
We study white noise perturbations of planar dynamical systems with heteroclinic networks in the limit of vanishing noise. We show that the probabilities of transitions between various cells that the network tessellates the plane into decay as powers of the noise magnitude, and we describe the underlying mechanism. A metastability picture emerges, with a hierarchy of time scales and clusters of accessibility, similar to the classical Freidlin-Wentzell picture but with shorter transition times. We discuss applications of our results to homogenization problems and to the invariant distribution asymptotics. At the core of our results are local limit theorems for exit distributions obtained via methods of Malliavin calculus. Joint work with Hong-Bin Chen and Zsolt Pajor-Gyulai.
Dor Elboim, Princeton University
In the interchange process on a graph $G=(V, E)$, distinguished particles are placed on the vertices of $G$ with independent Poisson clocks on the edges. When the clock of an edge rings, the two particles on the two sides of the edge interchange. In this way, a random permutation $\pi _\beta: V\to V$ is formed for any time $\beta >0$. One of the main objects of study is the cycle structure of the random permutation and the emergence of long cycles.
We prove the existence of infinite cycles in the interchange process on $\mathbb Z ^d$ for all dimensions $d\ge 5$ and all large $\beta$, establishing a conjecture of Bálint Tóth from 1993 in these dimensions.
In our proof, we study a self-interacting random walk called the cyclic time random walk. Using a multiscale induction we prove that it is diffusive and can be coupled with Brownian motion. One of the key ideas in the proof is establishing a local escape property which shows that the walk will quickly escape when it is entangled in its history in complicated ways.
This is a joint work with Allan Sly.
Kavita Ramanan, Brown University
A mean-field game is a game with a continuum of players, describing the limit as n tends to infinity of Nash equilibria of certain n-player games, in which agents interact symmetrically through the empirical measure of their state processes. We first study the asymptotic behavior of Nash equilibria in static games with a large number of agents. In particular, we establish law of large number limits and large deviation principles for the set of Nash equilibria and discuss applications to congestion games and the price of anarchy. Then we discuss stochastic differential games, which are often understood via the so-called "master equation", which is an infinite-dimensional PDE for the value function. We will show how analysis of sufficiently smooth solutions to the master equation play a role in analyzing large deviation principles for mean-field games. This is based on joint works with Francois Delarue and Daniel Lacker.
Yu Gu, University of Maryland
I will present the recent work with Tomasz Komorowski and Alex Dunlap in which we derived optimal variance bounds on the solution to the KPZ equation on a large torus, in certain regimes where the size of the torus increases with time. We only use stochastic calculus and I will try to give a heuristic explanation of the 2/3 and 1/3 exponents in the 1+1 KPZ universality class.
Alex Dunlap, Courant Institute
I will discuss a two-dimensional stochastic heat equation with a nonlinear noise strength, and consider a limit in which the correlation length of the noise is taken to 0 but the noise is attenuated by a logarithmic factor. The limiting pointwise statistics can be related to a stochastic differential equation in which the diffusivity solves a PDE somewhat reminiscent of the porous medium equation. This relationship is established through the theory of forward-backward SDEs. I will also explain several cases in which the PDE can be solved explicitly, some of which correspond to known probabilistic models. This talk will be based on current joint work with Cole Graham and earlier joint work with Yu Gu.
Konstantin Tikhomirov, Carnegie Mellon University
We consider the space of d-regular directed simple graphs, where two graphs are connected whenever there is a simple switching operation transforming one graph to the other. For constant d, we prove optimal bounds on the modified Log-Sobolev constant of the associated Markov chain on the space of graphs. This implies that the total variation mixing time of the chain is of order n log(n), which settles an old open problem. Based on joint work with Pierre Youssef.
Morris Ang, Columbia University
Sheffield showed that conformally welding a \gamma-Liouville quantum gravity (LQG) surface to itself gives a Schramm-Loewner evolution (SLE) curve with parameter \kappa = \gamma^2 as the interface, and Duplantier-Miller-Sheffield proved similar stories for \kappa = 16/\gamma^2 for \gamma-LQG surfaces with boundaries decorated by looptrees of disks or by continuum random trees. We study these dynamics for LQG surfaces coming from Liouville conformal field theory (LCFT). At stopping times depending only on the curve, we give an explicit description of the surface and curve in terms of LCFT and SLE. This has applications to both LCFT and SLE. We prove the boundary BPZ equation for LCFT, which is crucial to solving boundary LCFT. With Yu we prove the reversibility of whole-plane SLE for \kappa ≥ 8 via a novel radial mating-of-trees.
Eren C. Kızıldağ, Columbia University
Many computational problems involving randomness exhibit a statistical-to-computational gap (SCG): the best known polynomial-time algorithm performs strictly worse than the existential guarantee. In this talk, we focus on the SCG of the symmetric binary perceptron (SBP), a random constraint satisfaction problem as well as a toy model of a single-layer neural network. We establish that the solution space of the SBP exhibits intricate geometrical features, known as the multi Overlap Gap Property (m-OGP). By leveraging the m-OGP, we obtain nearly sharp hardness guarantees against the class of stable and online algorithms, which capture the best known algorithms for the SBP. Our results mark the first instance of intricate geometry yielding tight algorithmic hardness against classes beyond stable algorithms.
Time permitting, I will discuss how the same program extends also to other models, including (a) discrepancy minimization, and (b) random number partitioning problem.
Based on joint works with David Gamarnik, Will Perkins, and Changji Xu.
Wenpin Tang, Columbia University
In this talk, I will discuss two mean-field models in which a certain phase transition occurs. I first describe McKean-Vlasov equations involving hitting times which arise as the mean-field limit of particle systems with annihilation. One such example is the super-cool Stefan problem. It is well known that such a system may have blow-ups. We provide some sufficient conditions on the model data to assure either blow-ups or no blow-ups. In the second part, I will discuss the convergence rate of second-order mean-field games to first-order ones, motivated from numerical challenges in first-order mean-field PDEs and the weak noise theory in KPZ universality. When the Hamiltonian and the coupling function have a certain growth, the rate is independent of the dimension; on the other hand, the rate decays in dimension (curse of dimensionality) when the Hamiltonian and the coupling function have small growth. These are based on joint work with Yuming Paul Zhang.
Yeor Hafouta, University of Maryland
We obtain optimal rates in the central limit theorem (CLT) for additive functionals of uniformly elliptic inhomogeneous Markov chains without any assumptions on the growth rates of the variance of the underlying partial sums. (The CLT itself is due to Dobrushin (1956) and it holds in greater generality.)
We will also discuss Edgeworth expansions (i.e., the correction terms in the CLT) of order one for general classes of functionals, which provide a structural characterization of having better than optimal CLT rates.
Finally, for several classes of additive functionals (e.g., Holder continuous), we will provide optimal conditions for Edgeworth expansions of an arbitrary order.
The talk is based on a joint work with Dmitry Dolgopyat.
Qian Yu, Princeton University
In the study of Ising models on large locally tree-like graphs, in both rigorous and non-rigorous methods one is often led to understanding the so-called belief propagation distributional recursions and its fixed points. We prove that there is at most one non-trivial fixed point for Ising models with zero or certain random external fields. Previously this was only known for sufficiently ``low-temperature'' models.
Our result simultaneously closes the following 6 conjectures in the literature: 1) independence of robust reconstruction accuracy to leaf noise in broadcasting on trees (Mossel-Neeman-Sly'16); 2) uselessness of global information for a labeled 2-community stochastic block model, or 2-SBM (Kanade-Mossel-Schramm'16); 3) optimality of local algorithms for 2-SBM under noisy side information (Mossel-Xu'16); 4) uniqueness of BP fixed point in broadcasting on trees in the Gaussian (large degree) limit (ibid); 5) boundary irrelevance in broadcasting on trees (Abbe-Cornacchia-Gu-Polyanskiy'21); 6) characterization of entropy (and mutual information) of community labels given the graph in 2-SBM (ibid).
This is a joint work with Yury Polyanskiy.
Lingfu Zhang, UC Berkeley
In this talk, I will discuss the colored Asymmetric Simple Exclusion Process (ASEP) in a finite interval. This Markov chain is also known as the biased card shuffling or random Metropolis scan, and its study dates back to Diaconis-Ram (2000). A total-variation cutoff was proved for this chain a few years ago using hydrodynamic techniques (Labbé-Lacoin, 2016). In this talk, I will explain how to obtain more precise information on its cutoff, specifically to establish the conjectured GOE Tracy-Widom cutoff profile. The proof relies on coupling arguments, as well as symmetries obtained from the Hecke algebra. I will also discuss some related open problems.
Ron Peled, Tel Aviv University, IAS and Princeton University
A minimal surface in a random environment (MSRE) is a surface which minimizes the sum of its elastic energy and its environment potential energy, subject to prescribed boundary values. Apart from their intrinsic interest, such surfaces are further motivated by connections with disordered spin systems and first-passage percolation models. We wish to study the geometry of d-dimensional minimal surfaces in a (d+n)-dimensional random environment. Specializing to a model that we term harmonic MSRE, we rigorously establish bounds on the geometric and energetic fluctuations of the minimal surface, as well as a scaling relation that ties together these two types of fluctuations.
Joint work with Barbara Dembin, Dor Elboim and Daniel Hadas.
Louis Fan, Indiana University
Stochastic reaction-diffusion equations are important models in mathematics and in applied sciences such as spatial population genetics and ecology. These equations describe a quantity (density/concentration of an entity) that evolves over space and time, taking into account random fluctuations. However, for many reaction terms and noises, the solution notion of these equations is still missing in dimension two or above, hindering the study of the spatial effect on stochastic dynamics through these equations.
In this talk, I will discuss a new approach, namely, to study these equations on general metric graphs that flexibly parametrize the underlying space. This enables us to not only bypass the ill-posedness issue of these equations in higher dimensions, but also assess the impact of space and stochasticity on the coexistence and the genealogies of interacting populations. We will focus on the computation of the probability of extinction, the quasi-stationary distribution, the asymptotic speed and other long-time behaviors for stochastic reaction-diffusion equations of Fisher-KPP type.
Hao Shen, University of Wisconsin-Madison
Quantum Yang-Mills model is a type of quantum field theory with gauge symmetry. The rigorous construction of quantum Yang-Mills is a central problem in mathematical physics. Stochastic quantization formulates the problem as stochastic dynamics, which can be studied using tools from analysis, PDE and stochastic PDE. We will discuss stochastic quantization of Yang-Mills on the 2 and 3 dimensional tori. To this end we need to address a number of questions, such as the construction of a singular orbit space, together with class gauge invariant observables (singular holonomies or Wilson loops), solving a stochastic PDE using regularity structures, and projecting the solution to the orbit space. Mostly based on joint work with Chandra, Chevyrev and Hairer.
Luke Peilen, Temple University
We study the statistical mechanics of the log gas, an interacting particle system with applications in random matrix theory and statistical physics, for general potential and inverse temperature. By means of a bootstrap procedure, we prove local laws on a novel next order energy quantity that are valid down to microscopic length scales. Simultaneously, we exhibit a control on fluctuations of linear statistics that is also valid down to microscopic scales. Using these local laws, we exhibit for the first time a CLT at arbitrary mesoscales, improving upon previous results of Bekerman and Lodhia.
The methods we use are suitable for generalization to higher dimensional Riesz interactions; we will discuss some generalizations of the above approach and partial results for the Riesz gas in higher dimensions.
Daniel Slonim, University of Virginia
We introduce the model of random walks in random environments (RWRE), which are random Markov chains on the integer lattice. These random walks are well understood in the nearest-neighbor, one-dimensional case due to reversibility of almost every Markov chain. For example, directional transience and limiting speed can be characterized in terms of simple expectations involving the transition probabilities at a single site. The reversibility is lost, however, if we go up to higher dimensions or relax the nearest-neighbor assumption by allowing jumps, and therefore much less is known in these models. Despite this non-reversibility, certain special cases have proven to be more tractable. Random walks in Dirichlet environments (RWDE), where the transition probability vectors are drawn according to a Dirichlet distribution, have been fruitfully studied in the nearest-neighbor, higher dimensional setting. We look at RWDE in one dimension with jumps and characterize when the walk is ballistic: that is, when it has non-zero limiting velocity. It turns out that in this model, there are two factors which can cause a directionally transient walk to have zero limiting speed: finite trapping and large-scale backtracking. Finite trapping involves finite subsets of the graph where the walk is liable to get trapped for a long time. It is a highly local phenomenon that depends heavily on the structure of the underlying graph. Large-scale backtracking is a more global and one-dimensional phenomenon. The two operate "independently" in the sense that either can occur with or without the other. Moreover, if neither factor on its own is enough to cause zero speed, then the walk is ballistic, so the two factors cannot conspire together to slow a walk down to zero speed if neither is sufficient to do so on its own. This appearance of two independent factors affecting ballisticity is a new feature not seen in any previously studied RWRE models.
Yier Lin, University of Chicago
The KPZ equation is a stochastic PDE that plays a central role in a class of random growth phenomena. In this talk, we will explore the Freidlin-Wentzell LDP for the KPZ equation through the lens of the variational principle. Additionally, we will explain how to extract various limits of the most probable shape of the KPZ equation using the variational formula. We will also discuss an alternative approach for studying these quantities using the method of moments.
This talk is based in part on joint works with Pierre Yves Gaudreau Lamarre and Li-Cheng Tsai.
Tomas Berggren, MIT
Random dimer models (or equivalently tiling models) have been a subject of extensive research in mathematics and physics for several decades. In this talk, we will discuss the doubly-periodic Aztec diamond dimer model of growing size, with arbitrary periodicity and only mild conditions on the edge weights. In this limit, we see three types of macroscopic regions — known as rough, smooth and frozen regions. We will discuss how the geometry of the arctic curves, the boundary of these regions, can be described in terms of an associated amoeba and an action function. In particular, we determine the number of frozen and smooth regions and the number of cusps on the arctic curves. We will also discuss the convergence of local fluctuations to the appropriate translation-invariant Gibbs measures. Joint work with Alexei Borodin.
Joshua McGinnis, UPenn
We review recent results regarding the rigorous approximation of 1D and 2D disordered (random, independent masses and/or springs) harmonic lattices by effective wave equations in the long wave limit. In this linear setting, we show the homogenization argument and highlight the tools used from probability theory to control the stochastic error terms such as the Law of the Iterated Logarithm and Hoeffding’s inequality. With our discussion of the linear problem serving as a springboard, we then present a new result regarding the approximation of an FPUT lattice with random masses by a KdV equation. Specifically, we are able to bound the approximation error in terms of the small parameter from the long wave scaling in an almost sure sense. In our theorem, we require a technical condition on the random masses, which we call transparency. Our proof relies on the incorporation of an auto-regressive process into an approximating ansatz, which itself is approximated by solutions to the KdV equation. We discuss the role of the auto-regressive process as well as the condition of transparency in the proof and give numerical evidence supporting the result. We conclude by discussing open questions such as the apparent lack of KdV dynamics in an FPUT lattice with independent, random masses.
Adrián González-Casanova, UC Berkeley
Heuristically, two processes are dual if one can find a function to study one process by using the other. Sampling duality is a duality which uses a duality function S(n,x) of the form "what is the probability that all the members of a sample of size n are of a certain type, given that the number (or frequency) of that type of individuals is x". Implicitly, this technique can be traced back to the work of Blaise Pascal. Explicitly, it was studied in a paper of Martin Möhle in 1999 in the context of population genetics. We will discuss examples for which this technique is useful, including an application to the Simple Exclusion Process with reservoirs. The last part of the lecture is based in recent joint work with Simone Floriani.
Cooper Boniece, Drexel University
The quadratic variation of a semimartingale plays an important role in a variety of applications, particularly so in financial econometrics, where it is closely linked to volatility. It contains information pertaining to both continuous and discontinuous path behavior of the underlying process, and separating its continuous and discontinuous parts based on high-frequency observations is a problem that has been tackled through a variety of approaches to-date.
However, despite the favorable asymptotic statistical properties of many of these approaches, their use in practice requires heuristic selection of tuning parameters that can greatly impact their estimation performance.
In this talk, I will discuss some recent work concerning an iterative approach that circumvents the "tuning problem."
This is based on joint work with J. E. Figueroa-López and Y. Han.
Yuchen Wu, Penn
Many statistical estimation problems can be reduced to the reconstruction of a low-rank n×d matrix when observed through a noisy channel. While tremendous positive results have been established, relatively few works focus on understanding the fundamental limitations of the proposed models and algorithms. Understanding such limitations not only provides practitioners with guidance on algorithm selection, but also spurs the development of cutting-edge methodologies. In this talk, I will present some recent progress in this direction from two perspectives in the context of low-rank matrix estimation. From an information-theoretic perspective, I will give an exact characterization of the limiting minimum estimation error. Our results apply to the high-dimensional regime n,d→∞ and d/n→∞ (or d/n→0) and generalize earlier works that focus on the proportional asymptotics n,d→∞, d/n→δ∈(0,∞). From an algorithmic perspective, large-dimensional matrices are often processed by iterative algorithms like power iteration and gradient descent, thus encouraging the pursuit of understanding the fundamental limits of these approaches. We introduce a class of general first order methods (GFOM), which is broad enough to include the aforementioned algorithms and many others. I will describe the asymptotic behavior of any GFOM, and provide a sharp characterization of the optimal error achieved by the GFOM class.
Pax Kivimae, Courant Institute, NYU
A classical picture in the theory of complex high-dimensional random functions is that an exponentially large number of critical points causes the gradient dynamics of the function to become slow and "glassy", becoming trapped in local minima. In non-gradient dynamics however, another case is possible. Here, one may have an exponentially large number of equilibria, but have none that are stable, leading to an endless cycle of wandering around saddles. This is believed to occur when the strength of the non-gradient terms is brought past a certain point, a phenomenon coined by Ben Arous, Fyodorov, and Khoruzhenko as the relative-absolute instability transition, and since predicted to occur in a variety of models.
We confirm such a transition occurs in the case of the asymmetric p-spin model, the first such rigorous confirmation of the existence of this transition in any model. To do so, we demonstrate concentration of the quenched complexity of stable and general equilibria around their annealed values. Our methods rely on generalizing the recent framework of Ben Arous, Bourgade, and McKenna on the Kac-Rice formula to the non-relaxational case, as well as a computation of moments of the characteristic polynomial of the elliptic ensemble.
Arka Adhikari, Stanford University
We consider general mixed $p$-spin mean-field spin glass models and provide a method to prove that the spectral gap of the Dirichlet form associated with the Gibbs measure is of order one at sufficiently high temperature. Our proof is based on an iteration scheme relating the spectral gap of the $N$-spin system to that of suitably conditioned subsystems.
Based on joint work w/ C. Brennecke, C. Xu, and H.-T. Yau.
Alejandro Ramírez, NYU Shanghai
We consider Wick-ordered solutions to the planar stochastic heat equation, corresponding to a Skorokhod interpretation in the Duhamel integral representation of the equation. We prove that the fluctuations far from the center are given by the stochastic heat equation. This talk is based on a joint work with Jeremy Quastel and Balint Virag.
Zongrui Yang, Columbia University
We prove that the stationary measures for the geometric last passage percolation and log-gamma polymer models on a diagonal strip are given by the marginals of objects we call two-layer Gibbs measures. Taking an intermediate disorder limit of the log-gamma polymer stationary measure, we recover the conjectural description of the open KPZ equation stationary measure for all choices of boundary parameters. This is a joint work with Guillaume Barraquand and Ivan Corwin.