Probability and Mathematical Physics Seminar

Synopsis:

I will explain some recent results which show that the law of the stochastic burgers equation at a fixed time t is absolutely continuous with respect to the natural Gaussian measure on the spatial domain. The results will apply to forcing just up to the point where the roughness of the forcing corresponds to the classical KPZ equation in the Burgers setting. As one approaches  this level of roughness, the equations must be understood as a Singular SPDEs (in the sense of Hairer ).  As such the construction helps illuminate the structure of the equation and makes clear in what sense we might call these equations “truly” elliptic in this infinite dimensional setting. I will also make comments connecting back to previous results on the 2D Naiver Stokes equation. This work is joint with Marco Romito and Langxuan Su and builds on work with Andrea Watkins Hairston.

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We present recent progress on the massless sine-Gordon model at the free fermion point. In particular, we prove that the fractional charge (vertex operator) correlation functions exist, and they can be described by certain renormalised determinants of massive free fermions with branching. The latter were studied by Palmer, whose results combined with ours should yield explicit Hilbert-Schmidt determinant formulas for the correlation functions, and also imply predictions of Bernard-Leclair that the fractional two-point function satisfies a certain PDE.

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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 single-resolvent and multi-resolvent 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.

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We study the dynamics of two local optimization algorithms, online stochastic gradient descent (SGD) and gradient flow, within the framework of the multi-spiked covariance and tensor model in the high-dimensional regime. This multi-index model arises from the well-known 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 signal-to-noise 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 \(N^{p-2}\), aligning with the computational threshold identified in the rank-one 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 \(N^{p-2}\). However, recovering the subsequent directions requires the number of samples to scale as \(N^{p-1}\). Our results are obtained through a detailed analysis of a low-dimensional 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.

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Synopsis:

We propose a new concept of lifts of reversible diffusion processes and 
show that various well-known non-reversible 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 non-asymptotic 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 space-time Poincaré
inequalities can be rephrased in the language of lifts and how it can be 
applied to find optimal lifts.

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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.

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We will discuss the typical behavior of M-Lipschitz functions on d-regular expander graphs, where an M-Lipschitz 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 M-Lipschitz function on an n-vertex 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 M-Lipschitz function has height at most C'*M*loglog n with high probability, showing that the typical height of an M-Lipschitz function is much smaller than the extreme case. Peled-Samotij-Yehudayoff'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.

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Fixing integers q,d>2, denote by Q(n,T,B) the ferromagnetic q-Potts measures on graphs G(n), at temperature T>0 and non-negative external field strength B, where as n grows the uniformly sparse G(n) of n vertices converge locally to the infinite d-regular 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 edge-expander G(n), we also establish a pure-state 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 q-Potts measure on the infinite tree, with a boundary wired at color k.

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