Department of Mathematics and Systems Analysis


Workshop on

Probabilistic Field Theories

June 17-19, 2024

Aalto University, Finland

HomeLocal InformationProgramParticipantsRegistration

All the lectures will be held in the D-hall on Otaniemi Campus (Otakaari 1, Espoo, Finland), see local information for directions.
The registration desk will be close to the D-hall.

There is a Reception and Poster Session on Monday, June 17, 17:00 in the department coffee room with snacks and beverages.

The Social Dinner on Tuesday, June 18, 17:00 takes place in Restaurant Elm, Puistokatu 4, Helsinki.

Schedule (tentative)

  Monday, June 17
Tuesday, June 18 Wednesday, June 19
9:00 – 9:20 Registration
9:20 – 9:30 Opening    
9:30 – 10:30 Webb Garban Berestycki
  Coffee Break Coffee Break  Coffee Break
11:00 – 12:00 Carrance Wolfram Cerclé
12:00 – 13:00 Park Armstrong Barrera
  Lunch Break Lunch Break Lunch Break
14:30 – 15:30 Bou-Rabee Kupiainen Chandra
  Coffee Break Coffee Break Coffee Break
16:00 – 17:00 Singh   Gordina
17:00 – Reception / Poster Session
(Department Coffee Room) 
Social Dinner
(Restaurant Elm)


Scott Armstrong

(Courant Institute at NYU)

Renormalization group and homogenization in high contrast

Many problems in mathematical physics involve disorder across various length scales, and a central question is to predict (or estimate the size of) critical length scales beyond which disorder can be integrated out. This is often addressed heuristically using renormalization group arguments. In this lecture, we will present a rigorous approach to this problem in the context of homogenization of diffusion equations. We consider equations which exhibit a very large ellipticity contrast ratio, and are interested in the length scale at which we see homogenization (with high probability). We introduce a coarse-graining scheme, allowing us to integrate out the disorder scale-by-scale, which can be seen as a rigorous renormalization group argument. Our techniques may be iterated across many scales, allowing us to treat random fields with multifractal structure (leading in some cases to anomalous diffusion). One such application will be presented in the talk of Ahmed Bou-Rabee. This is joint work with Tuomo Kuusi.

Gerardo Barrera

(University of Helsinki)

Coupling estimates between heat equation and the Goldstein–Kac telegraph equation and its relation with the velocity flip model

In this presentation we will obtain a non-asymptotic process level control between the telegraph process (a.k.a. Goldstein–Kac equation/process) and a Brownian motion with explicit diffusivity constant via a transportation Wasserstein path-distance with quadratic average cost.

We stress that the marginals of the telegraph process solves a partial linear differential equation of the hyperbolic type for which explicit computations can be carried by in terms of Bessel functions. In the present talk, I will discuss a coupling approach, which is a robust technique that in principle can be used for more general PDEs.

The proof is done via the interplay of the following probabilistic couplings: coin-flip coupling, synchronous coupling and the celebrated Komlós–Major–Tusnády coupling.

Using the previous result, we derive a probabilistic coupling between a multivariate (non-commutative) geometric Brownian motion and the celebrated velocity flip model with quadratic interaction.

The talk is based on joint work with Jani Lukkarinen, University of Helsinki, Finland.

Nathanaël Berestycki

(University of Vienna)

Thick points of 4D critical branching Brownian motion

We show that branching Brownian motion in dimension four is governed by a nontrivial multifractal geometry and compute the associated exponents. As a part of this, we establish very precise estimates on the probability that a ball is hit by an unusually large number of particles, sharpening earlier works by Angel, Hutchcroft, and Jarai (2020) and Asselah and Schapira (2022) and allowing us to compute the Hausdorff dimension of the set of “a-thick” points for each a > 0. Surprisingly, we find that the exponent for the probability of a unit ball to be “a-thick” has a double phase transition, where it is differentiable but not twice differentiable at a = 2, whereas the dimension of the set of thick points is positive until a = 4. If time permits, we will also discuss a new strong coupling theorem for branching random walk that allows us to prove analogues of some of our results in the discrete case.

Ahmed Bou-Rabee

(Courant Institute at NYU)

Superdiffusion for Brownian motion with random drift

A Brownian particle subject to a random, divergence-free drift will have enhanced diffusion. The correlation structure of the drift determines the strength of the diffusion and there is a critical threshold, bordering the diffusive and superdiffusive regimes. Physicists have long expected logarithmic-type superdiffusivity at this threshold, and recently some progress in this direction has been made by mathematicians. I will discuss joint work with Scott Armstrong and Tuomo Kuusi in which we identify and obtain the sharp rate of superdiffusivity. We also establish a quenched invariance principle under this scaling. Our proof is a quantitative renormalization group argument made rigorous by ideas from
stochastic homogenization.

Ariane Carrance

(École Polytechnique, Paris)

A new family of random planar maps with a stable flavour

The success of the study of the Brownian sphere relies crucially on its natural construction from the Brownian snake, i.e. a Brownian tree endowed with Brownian spatial displacements. In a recent joint work with Eleanor Archer and Laurent Ménard, we introduce and study its generalisation to the case of alpha-stable trees, with 1<alpha<2. The resulting "stable spheres" are expected to be the scaling limits of specific random models of planar quadrangulations, with heavy-tailed vertex degrees. In this talk, I will present the construction of these continuum and discrete models, and the main properties that we have established so far, including the identification of their Hausdorff dimension. I will also highlight the main differences between these models and uniform or Boltzmann random maps, that give rise to interesting challenges to further study this new family.

Baptiste Cerclé


A probabilistic approach to Toda Conformal Field Theories

Conformal invariance is a feature that (most of the time conjecturally) arises for a large class of models of statistical physics at criticality. To address the issue of understanding the conformal field theory (CFT) thus defined, Belavin-Polyakov-Zamolodchikov designed in 1984 a general method for solving such a theory, dubbed conformal bootstrap. However there is a large class of models, such as the critical three-states Potts model, that enjoy in addition to conformal invariance an enhanced level of symmetry called higher-spin symmetry. Capturing this additional feature led to the introduction by Zamolodchikov of the notion of W-algebras, which are Vertex Algebras containing the Virasoro algebra. In this talk we will explain how this higher-spin symmetry manifests itself for Toda CFTs, generalizations of Liouville CFT that enjoy this higher level of symmetry. To do so we will rely on a probabilistic definition of these theories based on (vectorial) Gaussian Free Fields and Gaussian Multiplicative Chaos.

Ajay Chandra

(Imperial College London)

Stochastic quantization for a non-local field theory

I will discuss quartic melonic tensor field theories, a class of field theories built using a non-local quartic interaction term. These resemble the more well-known $\Phi^4_d$ models studied in constructive quantum field theory, but they behave differently with regards to power-counting and the structure of their divergences. In particular, these models are conjectured to be non-trivial in their critical dimension, in contrast with $\Phi^4_4$. I will then describe joint work with Léonard Ferdinand where we use stochastic analysis methods to construct the $\Phi^4_2$ and $\Phi^4_3$ analogs of these models.

Christophe Garban

(Université Lyon 1)

Fourier analysis of Gaussian multiplicative chaos

In this talk, I will start by introducing a natural class of random measures on $R^d$ (called "Gaussian multiplicative chaos") whose support is "multi-fractal". These measures play a key role in several areas of physics and probability. The goal of this talk will be to show that Gaussian multiplicative chaos also provides interesting measures from the point of view of harmonic analysis: in particular, we will focus on the asymptotic behavior of its Fourier coefficients (for multiplicative chaos defined on the torus $T^d$). Our main theorem is that a Gaussian multiplicative chaos is almost certainly a Rajchman measure (i.e. a measure whose Fourier coefficients tend to zero).

Based on a joint work with Vincent Vargas (University of Geneva).

Maria Gordina

(University of Connecticut)

Fermionic stochastic analysis

I will talk about stochastic analysis of Grassmann random variables suitable for the stochastic quantization of Euclidean fermionic quantum field theories. The idea of using Grassmann random variables for fermionic systems goes back to Osterwalder and Schrader, and to implement it we used tools from non-commutative analysis. I will describe some of the stochastic analytical tools such as Grassmann Brownian motion and non-​commutative $L^{p}$ spaces, and mention some of the applications.

Based on the joint work with F. De Vecchi, L. Fresta and M. Gubinelli.

Antti Kupiainen

(University of Helsinki)

Wess-Zumino-Witten models and path integrals

The Wess-Zumino-Witten (WZW) model is a 2 dimensional conformal quantum field theory where the field takes values in a Lie group G or its coset space. For a compact G  this CFT is rational and its cosets G/H include for instance all unitary rational CFTs (e.g. the Ising model). WZW model has a formal path integral representation but it has proved to be very hard to make mathematical sense of it and in fact most of its conjectured properties have been discussed using the representation theory of affine Lie algebras. In this talk I will review the basic facts about the path integral formulation of WZW models and then discuss the coset theory SL(2,C)/SU(2). This theory can be formulated  in terms of field taking values in the 3-dimensional hyperbolic space and by the work of Ribault, Teschner and Schomerus it has been argued to have a mapping  to the Liouville CFT. It has interesting connections to the 3d Chern-Simons topological QFT as well as a “quantum” deformation of the geometric and analytic Langlands correspondence. I will explain briefly how this theory can be formulated probabilistically using the Gaussian Multiplicative Chaos and how on a general Riemann surface the correlation functions of its primary fields can be mapped to those of the Liouville CFT.

Minjae Park

(University of Chicago)

Yang-Mills theory and random surfaces

​I will discuss some recent work on Yang-Mills theory and its connection to the theory of random surfaces. In particular, Wilson loop expectations (important observables in Yang-Mills) can be represented as surface sums in two settings: ​- 2D continuum Euclidean Yang-Mills for classical Lie groups of any matrix size N (based on joint work with Joshua Pfeffer, Scott Sheffield, and Pu Yu) - Any dimensional lattice Yang-Mills for classical Lie groups of any matrix size N and any inverse temperature β (based on joint work with Sky Cao and Scott Sheffield) In addition, our probabilistic approaches provide alternative derivations and interpretations of several recent theorems, including Brownian motion limits (Dahlqvist), lattice string trajectories (Chatterjee and Jafarov), and surface sums (Magee and Puder).

Harprit Singh

(Imperial College London)

Singular SPDEs on Geometric Spaces

We shall discuss the solution theory to a large class of singular SPDEs in various non- translation invariant settings. In particular, I shall present results on Riemannian manifolds and homogeneous Lie groups as well as related results for non-translation invariant parabolic and hypo-elliptic equations on Euclidean space. Based on joint work with M. Hairer, respectively A. Mayorcas.

Christian Webb

(University of Helsinki)

Bosonization in critical and near critical models of statistical field theory

Bosonization is a curious duality occurring in certain two-dimensional field theories. It allows relating bosonic (probabilistic) models with fermionic models. I will review some recent results about bosonization in the setting of the Gaussian free field, the sine-Gordon model, the Ising model, and free fermions.

This is based on joint works with R. Bauerschmidt, B. Bayraktaroglu, K. Izyurov, S. Mason, S.C. Park, and T. Virtanen.

Catherine Wolfram


The dimer model in 3D

A dimer tiling of Z^d is a collection of edges such that every vertex is covered exactly once. In 2000, Cohn, Kenyon, and Propp showed that 2D dimer tilings satisfy a large deviations principle. In joint work with Nishant Chandgotia and Scott Sheffield, we prove an analogous large deviations principle for dimers in 3D. A lot of the results for dimers in two dimensions use tools and exact formulas (e.g. the height function representation of a tiling or the Kasteleyn determinant formula) that are specific to dimension 2. I will explain how to formulate the large deviations principle in 3D, show simulations, try to give some intuition for why three dimensions is different from two. Time permitting, I will explain some of the ways that we use a smaller set of tools (e.g. Hall’s matching theorem or a double dimer swapping operation) in our arguments.

Poster session

Petri Laarne

(University of Helsinki)

Metastability of hyperbolic $\Phi^4$ model

We consider a wave equation with a double-well cubic nonlinearity and possibly a stochastic forcing term. We show that the mean transition time between the potential wells can be computed from the invariant measure of the equation. This result holds on two- and three-dimensional small tori, and extends the previously known 1D and parabolic 2D results.

Aapo Pajala

(Aalto University)

SLE curves and rational functions

In their 2020 paper on large deviations of multichordal SLE, Peltola and Wang showed that the preimage of the extended real line under a real rational function with real critical points corresponds to an SLE(0) multichord. SLE(0) curves are obtained as deterministic limits of their stochastic counterparts, SLE(k) curves. The surprising link between rational functions and SLE gives rise to new connections. Examples include a new proof of the rational function case of the well-known Shapiro conjecture by Peltola and Wang, and dynamics of the poles of rational functions by Alberts, Byun, Kang and Makarov in 2022.

Emanuele Pasqui

(University of Padua)

Extremes and entropic repulsion for the Gaussian Free Field with bond disorder

Randomly fluctuating interfaces naturally arise in the context of coexistence of two homogeneous phases and are studied in a variety of statistical mechanics models. The Gaussian process on the integer lattice with zero mean and covariances given by the Green function of the simple random walk on the lattice is known as the (Lattice) Gaussian Free Field. It has been determined by [1] that in dimension $d\ge 3$ the probability of the event that all the spins of the field are positive in a box of volume $N^d$ decays exponentially at speed $N^(d-2) \log N$. In our work we focus on the Gaussian Free Field with bond disorder, in which the underlying graph is the integer lattice with weights sampled according to a stationary and ergodic distribution. We aim at revisiting the above results in this random environment, computing explicitly the corresponding constants. We also discuss the phenomenon of entropic repulsion, that is the equality between the asymptotic behaviour of the field under the conditioning on the event of positive spins on a box of volume $N^d$, and the law of a stationary Gaussian field with mean shifted by a height of order $\sqrt(\log N)$.

References: [1] Erwin Bolthausen, Jean-Dominique Deuschel, and Ofer Zeitouni, Entropic repulsion of the lattice free field, Communications in Mathematical Physics 170 (1995), no. 2, 417-443.

Yi Tian

(University of Bonn)

Relations of Permutons and Meanders to Random Geometry

We explore several advanced topics which focus on space-filling SLE, LQG, and their interplay with combinatorial structures such as meanders, meandric systems, and O(n) loop model. We present conjectures and theorems that describe their scaling limits and connections to LQG and SLE. We also study random permutons constructed from LQG surfaces decorated by a pair of space-filling SLEs, which can be used to described the scaling limit of various types of random permutations such as meandric permutations and Baxter permutations.

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