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| * '''When''': Thursdays at 2:30 pm | | * '''When''': Thursdays at 2:30 pm |
| * '''Where''': 901 Van Vleck Hall | | * '''Where''': 901 Van Vleck Hall |
| * '''Organizers''': Hanbaek Lyu, Tatyana Shcherbyna, David Clancy | | * '''Organizers''': Hongchang Ji, Ander Aguirre, Hai-Xiao Wang |
| * '''To join the probability seminar mailing list:''' email probsem+subscribe@g-groups.wisc.edu. | | * '''To join the probability seminar mailing list:''' email probsem+subscribe@g-groups.wisc.edu. |
| * '''To subscribe seminar lunch announcements:''' email lunchwithprobsemspeaker+subscribe@g-groups.wisc.edu | | * '''To subscribe seminar lunch announcements:''' email lunchwithprobsemspeaker+subscribe@g-groups.wisc.edu |
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| [[Past Seminars]] | | [[Past Seminars]] |
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| | == Fall 2025 == |
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| = Fall 2024 =
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| <b>Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b> | | <b>Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b> |
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| We usually end for questions at 3:20 PM. | | We usually end for questions at 3:20 PM. |
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| == September 5, 2024: == | | == September 4, 2025: No seminar == |
| No seminar
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| == September 12, 2024: Hongchang Ji (UW-Madison) == | | == September 11, 2025: David Renfrew (Binghamton U.) == |
| '''Spectral edge of non-Hermitian random matrices'''
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| We report recent progress on spectra of so-called deformed i.i.d. matrices. They are square non-Hermitian random matrices of the form $A+X$ where $X$ has centered i.i.d. entries and $A$ is a deterministic bias, and $A$ and $X$ are on the same scale so that their contributions to the spectrum of $A+X$ are comparable. Under this setting, we present two recent results concerning universal patterns arising in eigenvalue statistics of $A+X$ around its boundary, on macroscopic and microscopic scales. The first result shows that the macroscopic eigenvalue density of $A+X$ typically has a jump discontinuity around the boundary of its support, which is a distinctive feature of $X$ by the \emph{circular law}. The second result is edge universality for deformed non-Hermitian matrices; it shows that the local eigenvalue statistics of $A+X$ around a typical (jump) boundary point is universal, i.e., matches with those of a Ginibre matrix $X$ with i.i.d. standard Gaussian entries.
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| Based on joint works with A. Campbell, G. Cipolloni, and L. Erd\H{o}s.
| | '''Singularities in the spectrum of random block matrices''' |
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| | We consider the density of states of structured Hermitian and non-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. |
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| == September 19, 2024: Miklos Racz (Northwestern) == | | == September 18, 2025: JE Paguyo (McMaster U.) == |
| '''The largest common subtree of uniform attachment trees''' | | '''Asymptotic behavior of the hierarchical Pitman-Yor and Dirichlet processes''' |
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| Consider two independent uniform attachment trees with n nodes each -- how large is their largest common subtree? Our main result gives a lower bound of n^{0.83}. We also give some upper bounds and bounds for general random tree growth models. This is based on joint work with Johannes Bäumler, Bas Lodewijks, James Martin, Emil Powierski, and Anirudh Sridhar.
| | The Pitman-Yor process is a discrete random measure specified by a concentration parameter, discount parameter, and base distribution, and is used as a fundamental prior in Bayesian nonparametrics. The hierarchical Pitman-Yor process (HPYP) is a generalization obtained by randomizing the base distribution through a draw from another Pitman-Yor process. It is motivated by the study of groups of clustered data, where the group specific Pitman-Yor processes are linked through an intergroup Pitman-Yor process. Setting both discount parameters to zero recovers the celebrated hierarchical Dirichlet process (HDP), first introduced by Teh et al. |
| | In this talk, we discuss our recent work on the asymptotic behavior of the HPYP and HDP. First, we establish limit theorems associated with the power sum symmetric polynomials for the vector of weights of the HDP as the concentration parameters tend to infinity. These objects are related to the homozygosity in population genetics, the Simpson diversity index in ecology, and the Herfindahl-Hirschman index in economics. Second, we consider a random sample of size $N$ from a population whose type distribution is given by the vector of weights of the HPYP and study the large $N$ asymptotic behavior of the number of clusters in the sample. Our approach relies on a random sample size representation of the number of clusters through the corresponding non-hierarchical process. This talk is based on joint work with Stefano Favaro and Shui Feng. |
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| == September 26, 2024: Dmitry Krachun (Princeton) == | | == September 25, 2025: Chris Janjigian (Purdue U.) == |
| '''A glimpse of universality in critical planar lattice models'''
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| Abstract: Many models of statistical mechanics are defined on a lattice, yet they describe behaviour of objects in our seemingly isotropic world. It is then natural to ask why, in the small mesh size limit, the directions of the lattice disappear. Physicists' answer to this question is partially given by the Universality hypothesis, which roughly speaking states that critical properties of a physical system do not depend on the lattice or fine properties of short-range interactions but only depend on the spatial dimension and the symmetry of the possible spins. Justifying the reasoning behind the universality hypothesis mathematically seems virtually impossible and so other ideas are needed for a rigorous derivation of universality even in the simplest of setups.
| | == October 2, 2025: Elliot Paquette (McGill U.) == |
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| In this talk I will explain some ideas behind the recent result which proves rotational invariance of the FK-percolation model. In doing so, we will see how rotational invariance is related to universality among a certain one-dimensional family of planar lattices and how the latter can be proved using exact integrability of the six-vertex model using Bethe ansatz.
| | == October 9, 2025: No seminar (Midwest Probability Colloquium) == |
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| Based on joint works with Hugo Duminil-Copin, Karol Kozlowski, Ioan Manolescu, Mendes Oulamara, and Tatiana Tikhonovskaia.
| | == October 16, 2025: Zachary Selk (Florida State U.) == |
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| == October 3, 2024: Joshua Cape (UW-Madison) ==
| | '''<br />On the Onsager-Machlup Function for the \Phi^4 Measure''' |
| '''A new random matrix: motivation, properties, and applications''' | |
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| In this talk, we introduce and study a new random matrix whose entries are dependent and discrete valued. This random matrix is motivated by problems in multivariate analysis and nonparametric statistics. We establish its asymptotic properties and provide comparisons to existing results for independent entry random matrix models. We then apply our results to two problems: (i) community detection, and (ii) principal submatrix localization. Based on joint work with Jonquil Z. Liao.
| | The \Phi^4 measure is a measure arising in effective quantum field theory as arguably the simplest example of a nontrivial QFT, modelling the self-interaction of a single scalar quantum field. This measure can be constructed through a procedure known as stochastic quantization. Stochastic quantization seeks to construct a measure on an infinite dimensional space with a given Gibbs-type ``density function" as the invariant measure of a stochastic PDE, in analogy with Langevin dynamics of stochastic ODEs. Both the \Phi^4 measure and its associated stochastic quantization PDE involve nonlinearities of distributions, necessitating renormalization procedures via tools like Wick calculus, regularity structures or paracontrolled calculus. Although the \Phi^4 measure has been constructed in dimensions 1,2 and 3, the question of whether these measures have the desired ``density function" remains open. Although in infinite dimensions, density functions are typically thought to not exist as there is no reference Lebesgue measure, there is a notion of a probability density function that extends to infinite dimensions called the Onsager-Machlup (OM) functional. One pathology of OM theory is that different metrics can lead to different OM functionals, or OM functionals can fail to exist under reasonable metrics. In a joint work with Ioannis Gasteratos (TU Berlin), we study the OM functional for the \Phi^4 measure. In dimension 1, the OM functional is what is desired under naive choices of metrics. In dimension 2, the OM functional is what is desired if we choose a metric analogous to the rough paths metric. In dimension 3, naive approaches don't work and the situation is complicated. |
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| == October 10, 2024: Midwest Probability Colloquium ==
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| == October 17, 2024: Kihoon Seong (Cornell) == | | ==October 23, 2025: Alex Dunlap (Duke U.)== |
| '''Gaussian fluctuations of focusing Φ^4 measure around the soliton manifold'''
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| I will explain the central limit theorem for the focusing Φ^4 measure in the infinite volume limit. The focusing Φ^4 measure, an invariant Gibbs measure for the nonlinear Schrödinger equation, was first studied by Lebowitz, Rose, and Speer (1988), and later extended by Bourgain (1994), Brydges and Slade (1996), and Carlen, Fröhlich, and Lebowitz (2016).
| | ==October 30, 2025: Ander Aguirre (UW-Madison)== |
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| Rider previously showed that this measure is strongly concentrated around a family of minimizers of the associated Hamiltonian, known as the soliton manifold. In this talk, I will discuss the fluctuations around this soliton manifold. Specifically, we show that the scaled field under the focusing Φ^4 measure converges to white noise in the infinite volume limit, thus identifying the next-order fluctuations, as predicted by Rider.
| | '''Edgeworth expansion and random polynomials''' |
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| This talk is based on joint work with Philippe Sosoe (Cornell).
| | ==November 6, 2025: Sudeshna Bhattacharjee (Indian Institute of Science)== |
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| == October 24, 2024: Jacob Richey (Alfred Renyi Institute) ==
| | == November 13, 2025: Jiaoyang Huang (U. Penn) == |
| TBD
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| == October 31, 2024: David Clancy (UW-Madison) ==
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| TBD
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| == November 7, 2024: Zoe Huang (UNC Chapel Hill) == | |
| TBD
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| == November 14, 2024: Deb Nabarun (University of Chicago) ==
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| TBD
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| == November 21, 2024: Reza Gheissari (Northwestern) ==
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| TBD
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| == November 28, 2024: Thanksgiving ==
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| No seminar
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| == December 5, 2024: Erik Bates (NC State) ==
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| TBD
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| = Spring 2024 =
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| <b>Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b>
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| We usually end for questions at 3:20 PM.
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| == January 25, 2024: Tatyana Shcherbina (UW-Madison) ==
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| '''Characteristic polynomials of sparse non-Hermitian random matrices'''
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| We consider the asymptotic local behavior of the second correlation functions of the characteristic polynomials of sparse non-Hermitian random matrices $X_n$ whose entries have the form $x_{jk}=d_{jk}w_{jk}$ with iid complex standard Gaussian $w_{jk}$ and normalized iid Bernoulli$(p)$ $d_{jk}$. If $p\to\infty$, the local asymptotic behavior of the second correlation function of characteristic polynomials near $z_0\in \mathbb{C}$ coincides with those for Ginibre ensemble of non-Hermitian matrices with iid Gaussian entries: it converges to a determinant of the Ginibre kernel in the bulk $|z_0|<1$, and it is factorized if $|z_0|>1$. It appears, however, that for the finite $p>0$, the behavior is different and it exhibits the transition between three different regimes depending on values $p$ and $|z_0|^2$. This is the joint work with Ie. Afanasiev.
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| == February 1, 2024: [https://lopat.to/index.html Patrick Lopatto (Brown)] ==
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| '''Optimal rigidity and maximum of the characteristic polynomial of Wigner matrices'''
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| We consider two related questions about the extremal statistics of Wigner matrices (random symmetric matrices with independent entries). First, how much can their eigenvalues fluctuate? It is known that the eigenvalues of such matrices display repulsive interactions, which confine them near deterministic locations. We provide optimal estimates for this “rigidity” phenomenon. Second, what is the behavior of the maximum of the characteristic polynomial? This is motivated by a conjecture of Fyodorov–Hiary–Keating on the maxima of logarithmically correlated fields, and we will present the first results on this question for Wigner matrices. This talk is based on joint work with Paul Bourgade and Ofer Zeitouni.
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| == February 8, 2024: Benoit Dagallier (NYU), online talk: https://uwmadison.zoom.us/j/95724628357 ==
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| '''Stochastic dynamics and the Polchinski equation'''
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| I will discuss a general framework to obtain large scale information in statistical mechanics and field theory models. The basic, well known idea is to build a dynamics that samples from the model and control its long time behaviour. There are many ways to build such a dynamics, the Langevin dynamics being a typical example. In this talk I will introduce another, the Polchinski dynamics, based on renormalisation group ideas. The dynamics is parametrised by a parameter representing a certain notion of scale in the model under consideration. The Polchinski dynamics has a number of interesting properties that make it well suited to study large-dimensional models. It is also known under the name stochastic localisation. I will mention a number of recent applications of this dynamics, in particular to prove functional inequalities via a generalisation of Bakry and Emery's convexity-based argument. The talk is based on joint work with Roland Bauerschmidt and Thierry Bodineau and the recent review paper <nowiki>https://arxiv.org/abs/2307.07619</nowiki> .
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| == February 15, 2024: [https://math.temple.edu/~tue86896/ Brian Rider (Temple)] ==
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| '''A matrix model for conditioned Stochastic Airy'''
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| There are three basic flavors of local limit theorems in random matrix theory, connected to the spectral bulk and the so-called soft and hard edges. There also abound a collection of more exotic limits which arise in models that posses degenerate (or “non-regular”) points in their equilibrium measure. What is more, there is typically a natural double scaling about these non-regular points, producing limit laws that transition between the more familiar basic flavors. Here I will describe a general beta matrix model for which the appropriate double scaling limit is the Stochastic Airy Operator, conditioned on having no eigenvalues below a fixed level. I know of no other random matrix double scaling fully characterized outside of beta = 2. This is work in progress with J. Ramirez (University of Costa Rica).
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| == February 22, 2024: No talk this week ==
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| '''TBA'''
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| == February 29, 2024: Zongrui Yang (Columbia) ==
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| '''Stationary measures for integrable models with two open boundaries'''
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| We present two methods to study the stationary measures of integrable systems with two open boundaries. The first method is based on Askey-Wilson signed measures, which is illustrated for the open asymmetric simple exclusion process and the six-vertex model on a strip. The second method is based on two-layer Gibbs measures and is illustrated for the geometric last-passage percolation and log-gamma polymer on a strip. This talk is based on joint works with Yizao Wang, Jacek Wesolowski, Guillaume Barraquand and Ivan Corwin.
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| == March 7, 2024: Atilla Yilmaz (Temple) ==
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| '''Stochastic homogenization of nonconvex Hamilton-Jacobi equations'''
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| After giving a self-contained introduction to the qualitative homogenization of Hamilton-Jacobi (HJ) equations in stationary ergodic media in spatial dimension ''d ≥ 1'', I will focus on the case where the Hamiltonian is nonconvex, and highlight some interesting differences between: (i) periodic vs. truly random media; (ii) ''d = 1'' vs. ''d ≥ 2''; and (iii) inviscid vs. viscous HJ equations.
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| == March 14, 2024: Eric Foxall (UBC Okanagan) ==
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| '''Some uses of ordered representations in finite-population exchangeable ancestry models''' (ArXiv: https://arxiv.org/abs/2104.00193)
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| For a population model that encodes parent-child relations, an ordered representation is a partial or complete labelling of individuals, in order of their descendants’ long-term success in some sense, with respect to which the ancestral structure is more tractable. The two most common types are the lookdown and the spinal decomposition(s), used respectively to study exchangeable models and Markov branching processes. We study the lookdown for an exchangeable model with a fixed, arbitrary sequence of natural numbers, describing population size over time. We give a simple and intuitive construction of the lookdown via the complementary notions of forward and backward neutrality. We discuss its connection to the spinal decomposition in the setting of Galton-Watson trees. We then use the lookdown to give sufficient conditions on the population sequence for the existence of a unique infinite line of descent. For a related but slightly weaker property, takeover, the necessary and sufficient conditions are more easily expressed: infinite time passes on the coalescent time scale. The latter property is also related to the following question of identifiability: under what conditions can some or all of the lookdown labelling be determined by the unlabelled lineages? A reasonably good answer can be obtained by comparing extinction times and relative sizes of lineages.
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| == March 21, 2024: Semon Rezchikov (Princeton) ==
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| '''Renormalization, Diffusion Models, and Optimal Transport'''
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| To this end, we will explain how Polchinski’s formulation of the renormalization group of a statistical field theory can be seen as a gradient flow equation for a relative entropy functional. We will review some related work applying this idea to problems in mathematical physics; subsequently, we will explain how this idea can be used to design adaptive bridge sampling schemes for lattice field theories based on diffusion models which learn the RG flow of the theory. Based on joint work with Jordan Cotler.
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| == March 28, 2024: Spring Break ==
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| '''TBA'''
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| == April 4, 2024: Zijie Zhuang (Upenn) via zoom https://uwmadison.zoom.us/j/99288619661 ==
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| '''Percolation Exponent, Conformal Radius for SLE, and Liouville Structure Constant'''
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| In recent years, a technique has been developed to compute the conformal radii of random domains defined by SLE curves, which is based on the coupling between SLE and Liouville quantum gravity (LQG). Compared to prior methods that compute SLE related quantities via its coupling with LQG, the crucial new input is the exact solvability of structure constants in Liouville conformal field theory. It appears that various percolation exponents can be expressed in terms of conformal radii that can be computed this way. This includes known exponents such as the one-arm and polychromatic
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| two-arm exponents, as well as the backbone exponents, which is unknown previously. In this talk we will review this method using the derivation of the backbone exponent as an example, based on a joint work with Nolin, Qian, and Sun.
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| == April 11, 2024: Bjoern Bringman (Princeton) ==
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| '''Global well-posedness of the stochastic Abelian-Higgs equations in two dimensions.'''
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| There has been much recent progress on the local solution theory for geometric singular SPDEs. However, the global theory is still largely open. In this talk, we discuss the global well-posedness of the stochastic Abelian-Higgs model in two dimension, which is a geometric singular SPDE arising from gauge theory. The proof is based on a new covariant approach, which consists of two parts: First, we introduce covariant stochastic objects, which are controlled using covariant heat kernel estimates. Second, we control nonlinear remainders using a covariant monotonicity formula, which is inspired by earlier work of Hamilton.
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| == April 18, 2024: Christopher Janjigian (Purdue) ==
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| '''Infinite geodesics and Busemann functions in inhomogeneous exponential last passage percolation'''
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| This talk will discuss some recent progress on understanding the structure of semi-infinite geodesics and their associated Busemann functions in the inhomogeneous exactly solvable exponential last-passage percolation model. In contrast to the homogeneous model, this generalization admits linear segments of the limit shape and an associated richer structure of semi-infinite geodesic behaviors. Depending on certain choices of the inhomogeneity parameters, we show that the model exhibits new behaviors of semi-infinite geodesics, which include wandering semi-infinite geodesics with no asymptotic direction, isolated asymptotic directions of semi-infinite geodesics, and non-trivial intervals of directions with no semi-infinite geodesics.
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| Based on joint work-in-progress with Elnur Emrah (Bristol) and Timo Seppäläinen (Madison)
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| == April 25, 2024: Colin McSwiggen (NYU) ==
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| '''Large deviations and multivariable special functions'''
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| This talk introduces techniques for using the large deviations of interacting particle systems to study the large-N asymptotics of generalized Bessel functions. These functions arise from a versatile approach to special functions known as Dunkl theory, and they include as special cases most of the spherical integrals that have captured the attention of random matrix theorists for more than two decades. I will give a brief introduction to Dunkl theory and then present a result on the large-N limits of generalized Bessel functions, which unifies several results on spherical integrals in the random matrix theory literature. These limits follow from a large deviations principle for radial Dunkl processes, which are generalizations of Dyson Brownian motion. If time allows, I will discuss some further results on large deviations of radial Heckman-Opdam processes and/or applications to asymptotic representation theory. Joint work with Jiaoyang Huang.
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| == May 2, 2024: Anya Katsevich (MIT) ==
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| '''The Laplace approximation in high-dimensional Bayesian inference'''
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| Computing integrals against a high-dimensional posterior is the major computational bottleneck in Bayesian inference. A popular technique to reduce this computational burden is to use the Laplace approximation, a Gaussian distribution, in place of the true posterior. Despite its widespread use, the Laplace approximation's accuracy in high dimensions is not well understood. The body of existing results does not form a cohesive theory, leaving open important questions e.g. on the dimension dependence of the approximation rate. We address many of these questions through the unified framework of a new, leading order asymptotic decomposition of high-dimensional Laplace integrals. In particular, we (1) determine the tight dimension dependence of the approximation error, leading to the tightest known Bernstein von Mises result on the asymptotic normality of the posterior, and (2) derive a simple correction to this Gaussian distribution to obtain a higher-order accurate approximation to the posterior.
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Back to Probability Group
- When: Thursdays at 2:30 pm
- Where: 901 Van Vleck Hall
- Organizers: Hongchang Ji, Ander Aguirre, Hai-Xiao Wang
- To join the probability seminar mailing list: email probsem+subscribe@g-groups.wisc.edu.
- To subscribe seminar lunch announcements: email lunchwithprobsemspeaker+subscribe@g-groups.wisc.edu
Past Seminars
Fall 2025
Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom
We usually end for questions at 3:20 PM.
September 4, 2025: No seminar
September 11, 2025: David Renfrew (Binghamton U.)
Singularities in the spectrum of random block matrices
We consider the density of states of structured Hermitian and non-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.
September 18, 2025: JE Paguyo (McMaster U.)
Asymptotic behavior of the hierarchical Pitman-Yor and Dirichlet processes
The Pitman-Yor process is a discrete random measure specified by a concentration parameter, discount parameter, and base distribution, and is used as a fundamental prior in Bayesian nonparametrics. The hierarchical Pitman-Yor process (HPYP) is a generalization obtained by randomizing the base distribution through a draw from another Pitman-Yor process. It is motivated by the study of groups of clustered data, where the group specific Pitman-Yor processes are linked through an intergroup Pitman-Yor process. Setting both discount parameters to zero recovers the celebrated hierarchical Dirichlet process (HDP), first introduced by Teh et al.
In this talk, we discuss our recent work on the asymptotic behavior of the HPYP and HDP. First, we establish limit theorems associated with the power sum symmetric polynomials for the vector of weights of the HDP as the concentration parameters tend to infinity. These objects are related to the homozygosity in population genetics, the Simpson diversity index in ecology, and the Herfindahl-Hirschman index in economics. Second, we consider a random sample of size $N$ from a population whose type distribution is given by the vector of weights of the HPYP and study the large $N$ asymptotic behavior of the number of clusters in the sample. Our approach relies on a random sample size representation of the number of clusters through the corresponding non-hierarchical process. This talk is based on joint work with Stefano Favaro and Shui Feng.
September 25, 2025: Chris Janjigian (Purdue U.)
October 2, 2025: Elliot Paquette (McGill U.)
October 9, 2025: No seminar (Midwest Probability Colloquium)
October 16, 2025: Zachary Selk (Florida State U.)
On the Onsager-Machlup Function for the \Phi^4 Measure
The \Phi^4 measure is a measure arising in effective quantum field theory as arguably the simplest example of a nontrivial QFT, modelling the self-interaction of a single scalar quantum field. This measure can be constructed through a procedure known as stochastic quantization. Stochastic quantization seeks to construct a measure on an infinite dimensional space with a given Gibbs-type ``density function" as the invariant measure of a stochastic PDE, in analogy with Langevin dynamics of stochastic ODEs. Both the \Phi^4 measure and its associated stochastic quantization PDE involve nonlinearities of distributions, necessitating renormalization procedures via tools like Wick calculus, regularity structures or paracontrolled calculus. Although the \Phi^4 measure has been constructed in dimensions 1,2 and 3, the question of whether these measures have the desired ``density function" remains open. Although in infinite dimensions, density functions are typically thought to not exist as there is no reference Lebesgue measure, there is a notion of a probability density function that extends to infinite dimensions called the Onsager-Machlup (OM) functional. One pathology of OM theory is that different metrics can lead to different OM functionals, or OM functionals can fail to exist under reasonable metrics. In a joint work with Ioannis Gasteratos (TU Berlin), we study the OM functional for the \Phi^4 measure. In dimension 1, the OM functional is what is desired under naive choices of metrics. In dimension 2, the OM functional is what is desired if we choose a metric analogous to the rough paths metric. In dimension 3, naive approaches don't work and the situation is complicated.
October 23, 2025: Alex Dunlap (Duke U.)
October 30, 2025: Ander Aguirre (UW-Madison)
Edgeworth expansion and random polynomials
November 6, 2025: Sudeshna Bhattacharjee (Indian Institute of Science)
November 13, 2025: Jiaoyang Huang (U. Penn)