Probability and Mathematical Physics Seminar
This seminar covers a wide range of topics in pure and applied probability and in mathematical physics. Unless otherwise noted, the talks take place on Fridays, 11:10am–12pm, in room WWH 1302. See directions to the Courant Institute.
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Twice every semester, we will have a joint Columbia–Courant meeting (hosted once at each institution) as the Probability and the City seminar.
Seminar Organizer(s): Eyal Lubetzky, Paul Bourgade, Klara Courteaut, and the probability group
Upcoming Events

Friday, October 11, 202411:10AM, Warren Weaver Hall 1302
Nonreversible lifts of reversible diffusion processes and relaxation times
Andreas Eberle, Universität BonnSynopsis:
We propose a new concept of lifts of reversible diffusion processes and
show that various wellknown nonreversible Markov processes arising in
applications are lifts in this sense of simple reversible diffusions.
For example, (kinetic) Langevin dynamics and randomised Hamiltonian
Monte Carlo are lifts of overdamped Langevin dynamics. Furthermore, we
introduce a concept of nonasymptotic relaxation times and show that these can at most be reduced by a square root through lifting, generalising a related result in discrete time. Finally, we demonstrate how the recently developed
approach to quantitative hypocoercivity based on spacetime Poincaré
inequalities can be rephrased in the language of lifts and how it can be
applied to find optimal lifts. 
Friday, October 18, 202411:10AM, Warren Weaver Hall 1302
Highdimensional optimization for multispiked tensor PCA
Cédric Gerbelot, Courant Institute of Mathematical SciencesSynopsis:
We study the dynamics of two local optimization algorithms, online stochastic gradient descent (SGD) and gradient flow, within the framework of the multispiked covariance and tensor model in the highdimensional regime. This multiindex model arises from the wellknown tensor principal component analysis (PCA) problem, where the goal is to infer \(r\) unknown, orthogonal signal vectors within the \(N\)dimensional unit sphere through maximum likelihood estimation (MLE) from noisy observations of an order\(p\) tensor. We determine the number of samples and the conditions on the signaltonoise ratios (SNRs) required to efficiently recover the unknown spikes from natural initializations. Specifically, we distinguish between three types of recovery: exact recovery of each spike, recovery of a permutation of all spikes, and recovery of the correct subspace spanned by the unknown signal vectors. We show that with online SGD, it is possible to recover all spikes provided a number of sample scaling as \(N^{p2}\), aligning with the computational threshold identified in the rankone tensor PCA problem~\cite{arous2020algorithmic,arous2021online}. For gradient flow, we show that the algorithmic threshold to efficiently recover the first spike is also of order \(N^{p2}\). However, recovering the subsequent directions requires the number of samples to scale as \(N^{p1}\). Our results are obtained through a detailed analysis of a lowdimensional system that describes the evolution of the correlations between the estimators and the unknown vectors, while controlling the noisy part of the dynamics. In particular, the hidden vectors are recovered one by one according to a sequential elimination phenomenon: as one correlation exceeds a critical threshold that depends on the order \(p\) of the tensor and on the SNRs, all correlations that share a row or column index become sufficiently small to be negligible, allowing the subsequent correlation to grow and become macroscopic. The sequence in which correlations become macroscopic depends on their initial values and the associated SNRs.

Friday, October 25, 202411:10AM, Warren Weaver Hall 1302
Zigzag strategy for random matrices
Joscha Henheik, IST AustriaSynopsis:
It is a remarkable property of random matrices, that their resolvents tend to concentrate around a deterministic matrix as the dimension of the matrix tends to infinity, even for a small imaginary part of the involved spectral parameter.
These estimates are called local laws and they are the cornerstone in most of the recent results in random matrix theory.
In this talk, I will present a novel method of proving singleresolvent and multiresolvent local laws for random matrices, the Zigzag strategy, which is a recursive tandem of the characteristic flow method and a Green function comparison argument. Novel results, which we obtained via the Zigzag strategy, include the optimal Eigenstate Thermalization Hypothesis (ETH) for Wigner matrices, uniformly in the spectrum, and universality of eigenvalue statistics at cusp singularities for correlated random matrices.
Based on joint works with G. Cipolloni, L. Erdös, O. Kolupaiev, and V. Riabov.
Past Events

Friday, October 4, 202411:10AM, Warren Weaver Hall 1302
Generative modeling with flows and diffusions
Eric VandenEijnden, Courant InstituteSynopsis:
Generative models based on dynamical transport have recently led to significant advances in unsupervised learning. At mathematical level, these models are primarily designed around the construction of a map between two probability distributions that transform samples from the first into samples from the second. While these methods were first introduced in the context of image generation, they have found a wide range of applications, including in scientific computing where they offer interesting ways to reconsider complex problems once thought intractable because of the curse of dimensionality. In this talk, I will discuss the mathematical underpinning of generative models based on flows and diffusions, and show how a better understanding of their inner workings can help improve their design. These results indicate how to structure the transport to best reach complex target distributions while maintaining computational efficiency, both at learning and sampling stages. I will also discuss applications of generative AI in scientific computing, in particular in the context of Monte Carlo sampling, with applications to the statistical mechanics and Bayesian inference, as well as probabilistic forecasting, with application to fluid dynamics and atmosphere/ocean science.

Friday, September 27, 202411:10AM, Warren Weaver Hall 1302
Lipschitz functions on expanders
Jinyoung Park, Courant InstituteSynopsis:
We will discuss the typical behavior of MLipschitz functions on dregular expander graphs, where an MLipschitz function means any two adjacent vertices admit integer values differ by at most M. While it is easy to see that the maximum possible height of an MLipschitz function on an nvertex expander graph is about C*M*log n, where C depends (only) on d and the expansion of the given graph, it was shown by Peled, Samotij, and Yehudayoff (2012) that a uniformly chosen random MLipschitz function has height at most C'*M*loglog n with high probability, showing that the typical height of an MLipschitz function is much smaller than the extreme case. PeledSamotijYehudayoff's result holds under the condition that, roughly, subsets of the expander graph expand by the rate of about M*log(dM). We will show that the same result holds under a much weaker condition assuming that d is large enough. This is joint work with Robert Krueger and Lina Li.

Friday, September 13, 202411:10AM, Warren Weaver Hall 1302
Ferromagnetic Potts measures on large locally treelike graphs
Amir Dembo, Stanford UniversitySynopsis:
Fixing integers q,d>2, denote by Q(n,T,B) the ferromagnetic qPotts measures on graphs G(n), at temperature T>0 and nonnegative external field strength B, where as n grows the uniformly sparse G(n) of n vertices converge locally to the infinite dregular tree. I will review a joint work with Anirban Basak and Allan Sly, showing that the convergence of the Potts free energy density to its Bethe replica symmetric prediction (which was proved for even d, or when B=0), yields the local weak convergence of Q(n,T,B) to the corresponding free or wired Potts measure on the infinite tree. One gets the free versus wired limit, according to which has the larger Potts Bethe functional value, with mixtures of these two appearing as limit points at the critical temperature T_c(q,B), where these two values of the Bethe functional coincide. For edgeexpander G(n), we also establish a purestate decomposition by showing that below the critical temperature, conditionally on having a dominant color k, the measures Q(n,T,0) converge locally to the qPotts measure on the infinite tree, with a boundary wired at color k.