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Abstracts

Abdelmalek Abdesselam (Virginia): A renormalization group construction of a Bosonic com­posite field with anomalous scaling dimension

In this talk I will explain the recent rigorous nonperturbative construction of a composite field with dynamically generated anomalous dimension. The model is a hierarchical version of the three-dimensional phi-four model with fractional Laplacian considered by Brydges, Mitter and Scoppola. The two new ideas on which this result is based are the development of a renormalization group framework for spatially inhomogeneous couplings and a partial linearization theorem in the analytic category which generalizes Koenigs’ Theorem from classical one-dimensional complex dynamics to an infinite-dimensional setting. This is joint work with Ajay Chandra and Gianluca Guadagni.

Roland Bauerschmidt (IAS Princeton): Renormalisation group analysis of |$\varphi$|4 models in 4D

I will discuss recent results on renormalisation group analysis of |$\varphi$|4 models in the critical dimension 4. I will outline the scope and general aspects of the method, including a sketch of the involved perturbative calculations, and application to the analysis of the susceptibility. We prove that the susceptibility has a logarithmic correction to mean field (free field) behavior with power (n+2)/(n+8) for the logarithm. My talk sets the stage and provides motivation for the talk of D. Brydges who discusses a general framework for non-perturbative aspects of the analysis, which are an essential ingredient in this work.
This is joint work with D. Brydges and G. Slade.

Marek Biskup (UCLA): Extrema of two-dimensional Gaussian Free Field

Recent years have witnessed a lot of progress in the understanding of the two-dimensional Discrete Gaussian Free Field (DGFF). In my talk I will discuss the asymptotic law of the extreme point process for the DGFF on lattice approximations of bounded open sets in the complex plane with zero boundary conditions outside. For points arising from nearly-maximal local maxima, the limit process is Poisson with intensity that is a product of a random measure in the spatial coordinate and the Gumbel intensity in the field coordinate. The random measure obeys a canonical transformation rule under conformal maps of the domain and can be linked to the measure representing the volume form of the critical 2D Liouville Quantum Gravity. Based on joint work with Oren Louidor.

Erwin Bolthausen (Zürich): Scaling limits for a weakly pinned Gaussian random field at a critical parameter

We consider the standard harmonic crystal in a finite but large region with positive boundary conditions, and a weak pinning at the base wall. It is well known that with boundary condition zero, the field is localized near the wall with exponentially decaying correlations. If one increases the boundary conditions, then at a certain critical value, the surface detaches completely from the wall. In a special case, we prove that at the critical point, the surface still is partially attached to the wall and completely detaches only for boundary values strictly above the critical one. We conjecture that this is a general phenomenon in more than one dimension. (joint work with Taizo Chiyonobu and Tadahisa Funaki)

David Brydges (UBC Vancouver): Control of irrelevant terms in the Wilson renormalisation group

This is a continuation of the lecture of Roland Bauerschmidt. In his lecture he will have asserted without proof that for a model such as |$\varphi$|4 there exists a sequence of equivalent models labeled by increasing length scales, where equivalence means that they have the same scaling limit. My lecture will validate his assertion by defining a space of models on which the renormalisation group acts as a dynamical system to generate the sequence of equivalent models, and by explaining why the renormalisation group trajectory in this space is well approximated by perturbation theory. The details of perturbation theory will not be needed to follow my lecture.

Ajay Chandra (Virginia): Proving Scale Invariance of a Massless QFT over the p-adics

We will discuss how one can use various techniques from statistical mechanics to prove full scale invariance of a φ4 model corresponding to a Renormalization Group fixed point. The ingredients include basic correlation inequalities, a result of Aizenman, Barsky, and Fernandez on the sharpness of certain phase transitions, and the work of Lebowitz and Presutti on unbounded spin systems with superstable interactions. This is joint work with Abdelmalek Abdesselam and Gianluca Guadagni.

Codina Cotar (UCL London): Gradient interfaces with disorder

We consider - in uniformly strictly convex potential regime - two versions of random gradient models with disorder. In model (A) the interface feels a bulk term of random fields while in model (B) the disorder enters though the potential acting on the gradients. We assume a general distribution on the disorder with uniformly-bounded finite second moments.
It is well known that for gradient models without disorder there are no Gibbs measures in infinite-volume in dimension d=2, while there are shift-invariant gradient Gibbs measures describing an infinite-volume distribution for the gradients of the field, as was shown by Funaki and Spohn. Van Enter and Kuelske proved that adding a disorder term as in model (A) prohibits the existence of such gradient Gibbs measures for general interaction potentials in d=2. Cotar and Kuelske proved the existence of shift-covariant random gradient Gibbs measures for model (A) when d≥3, the disorder is i.i.d and has mean zero, and for model (B) when d≥1 and the disorder has stationary distribution. In the present work, we prove existence and uniqueness of shift-covariant random gradient Gibbs measures with a given expected tilt u and with the corresponding annealed measure being ergodic: for model (A) when d≥3 and the disordered random fields are i.i.d. and symmetrically-distributed, and for model (B) when d≥1 and for any stationary disorder dependence structure. We also compute for both models for the corresponding annealed gradient Gibbs measure, when the disorder is i.i.d. and its distribution satisfies a Poincare inequality assumption, the optimal decay of covariances with respect to the averaged-over-the-disorder gradient Gibbs measure. This is joint work with Christof Kuelske.

Jian Ding (Chicago): On multiple peaks for general Gaussian processes

In his two papers of 2008 and 2009, Chatterjee established for the first time a rigorous connection among superconcentration, chaos, and multiple peaks for general Gaussian processes. Informally, these three properties can be understood as the following: (1) Superconcentration: the standard deviation of the maximum is substantially smaller than what is guaranteed by Borell-Sudakov-Tsirelson inequality; (2) Chaos: with a slight perturbation to the Gaussian process, the location of the new maximizer is uncorrelated with the previous one; (3) Multiple peaks: there are a large number of Gaussian variables (where the underlying pairwise Gaussian correlations are near 0) which achieve values close to the global maximum. In this talk, I will present a recent result improving the connection between superconcentration and multiple peaks. If time permits, I will also discuss some connections between the expectation of the maximum and its concentration property. The talk is based on joint work with Ronen Eldan and Alex Zhai.

Margherita Disertori (Bonn): Some results on history dependent stochastic processes

Edge reinforced random walk (ERRW) and vertex reinforced jump processes are history dependent stochastic process, where the particle tends to come back more often on sites it has already visited in the past. For a particular scheme of reinforcement these processes are mixtures of reversible Markov chains whose mixing measure can be related to a non-linear sigma model introduced in the context of random matrices. I will give an overview on these models and explain some recent results in joint work with F. Merkl and S. Rolles.

Martin Hairer (Warwick): Weak universality of the KPZ equation

The KPZ equation is a popular model of one-dimensional interface propagation. From heuristic consideration, it is expected to be ”universal” in the sense that any ”weakly asymmetric” or ”weakly noisy” microscopic model of interface propagation should converge to it if one sends the asymmetry (resp. noise) to zero and simultaneously looks at the interface at a suitable large scale. The only microscopic models for which this has been proven so far all exhibit very particular that allow to perform a microscopic equivalent to the Cole-Hopf transform. The main bottleneck for generalisations to larger classes of models was that until recently it was not even clear what it actually means to solve the equation, other than via the Cole-Hopf transform. In this talk, we will see that there exists a rather large class of continuous models of interface propagation for which convergence to KPZ can be proven rigorously. The main tool for both the proof of convergence an d the identification of the limit is the recently developed theory of regularity structures, but with an interesting twist.

SPECIAL TALK - AWARD OF REGIUS PROFESSORSHIP

Martin Hairer (Warwick): Taming infinities

Some physical and mathematical theories have the unfortunate feature that if one takes them at face value, many quantities of interest appear to be infinite! Various techniques, usually going under the common name of renormalisation have been developed over the years to address this, allowing mathematicians and physicists to tame these infinities. We will dip our toes into some of the mathematical aspects of these techniques and we will see how they have recently been used to make precise analytical statements about the solutions of some equations whose meaning was not even clear until now.

Tyler Helmuth (UBC Vancouver): Loop Weighted Walk

Loop weighted walk with weight λ (λ-LWW) is a non-Markovian model of random walks that are discouraged (or encouraged, depending on λ) from completing loops: a walk receives a weight λn if it contains n loops. An important and challenging feature of this model is that it is not purely repulsive: the weight of the future of a walk may either increase or decrease if the past is forgotten. I will describe a representation of λ-LWW that enables a lace expansion analysis in high dimensions.

Takashi Kumagai (Kyoto): Simple random walk on the two-dimensional uniform spanning tree and the scaling limits

In this talk, we will first summarize known results about anomalous asymptotic behavior of simple random walk on the two-dimensional uniform spanning tree. We then show the existence of subsequential scaling limits for the random walk, and describe the limits as diffusions on the limiting random real trees embedded into Euclidean space. Anomalous heat transfer on the random real trees will be observed by estimating heat kernels of the diffusions. This is an on-going joint project with M.T. Barlow (UBC) and D. Croydon (Warwick)

J.-C. Mourrat (Paris) Fluctuations in Homogenization

Rémi Rhodes (Paris): Liouville Field Theory

Liouville Field Theory (LFT) is a continuum model of two dimensional random surfaces, which is for instance involved in 2d-string theory or in the description of the fluctuations of metrics in 2d-Liouville quantum gravity. This is a probabilistic model that consists in weighting the classical Free Field action with an interaction term given by the exponential of a Gaussian multiplicative chaos. The purpose of this talk is the study of the semiclassical limit of the theory, which is a prescribed asymptotic regime of LFT of interest in physics literature, by deriving exact formulas for the Laplace transform of the Liouville field. This shows that the Liouville field concentrates on the solution of the classical Liouville equation with negative scalar curvature. Time permitting, I will also discuss the leading fluctuations and a large deviation principle.

Hao Shen (Princeton): Dynamical Sine-Gordon model at high temperature

We discuss a dynamical version of the Sine-Gordon model at high temperature. The model, being a time-dependent Gaussian free field perturbed by a Sine-type interaction, shows up in the context of Coulomb gas system, dynamic of liquid-vapour interfaces, and crystal surface fluctuations. We renormalise the model in a suitable way in order to pass into the continuum limit. The theory of regularity structure, recently developed by Martin Hairer, is applied. The main technical challenge is the estimates for certain observables constructed from the time-dependent Gaussian free field, and in these estimates we carry out multi-scale analysis. This is joint work with Prof. Martin Hairer.

Thomas Spencer (IAS Princeton): Statistical Mechanics and Random Matrices

Alex Tomberg (UBC Vancouver): Critical correlation functions for the 4-dimensional n-component |$\varphi$|4 model

This is a continuation of the lectures of Roland Bauerschmidt and David Brydges on the renormal-isation group analysis of the n-component |$\varphi$|4 spin model in 4 dimensions.
In his talk, D. Brydges will have discussed the non-perturbative aspects of the analysis. My talk will focus on the perturbative calculations needed to study the asymptotic decay of the critical correlation functions, including the logarithmic corrections to mean-field scaling. The details of the non-perturbative analysis will not be needed to follow my lecture.
This is joint work with R Bauerschmidt and G. Slade.

Vincent Vargas (Paris): Complex Gaussian multiplicative Chaos

The mathematical theory of Gaussian multiplicative chaos was founded by J. P. Kahane in 1985. This theory has numerous applications in mathematical physics: 2d Liouville quantum gravity in the conformal gauge (boundary and non boundary Liouville measure, KPZ equation), the Kolmogorov-Obukhov model of energy dissipation in 3d turbulence, the maximum of log-correlated fields, etc... In this talk, we will review the theory and discuss numerous extensions that have been developed the past few years, in particular to the complex case. Special emphasis will be made on applications which motivate these extensions. This is based on joint works with B. Duplantier, H. Lacoin, T. Madaule, R. Rhodes, S. Sheffield.

Hendrik Weber (Warwick): Large deviations for white-noise driven, nonlinear stochastic PDEs in two and three dimensions

We study the stochastic Allen-Cahn equation driven by a noise term with noise intensity √c and correlation length 8. We study diagonal limits 8, c -+ 0 and describe fully the large deviation be­haviour depending on the relationship between 8 and c. The recently developed theory of regularity structures allows to fully analyse the behaviour of solutions for vanishing correlation length 8 and fixed noise intensity c in two and three space dimensions. One key fact is that in order to get non­trivial limits as 8 -+ 0, it is in general necessary to introduce diverging counter terms. The theory of regularity structures allows to rigorously analyse this renormalisation procedure for a number of interesting equations. Our main result is a large deviation principle for these renormalised solutions. One interesting feature of this result is that the diverging renormalisation constants disappear at the level of the large deviations rate function. We apply this result to derive a sharp condition on 8, c that guarantees a large deviation principle for diagonal schemes c, 8 -+ 0 for the equation without renormalisation. Joint work with Martin Hairer.

Ofer Zeitouni (Rehovot): Extremal processes and freezing

Recently, the structure of the extremal point process has been described in several models of interest; in all these cases, the resulting process is a shifted decorated Poisson point process. On the other hand, the phonomenon of freezing (whereby certain quantities become independent of temperature) has been described, mostly in the physics literature. We introduce an generalized notion of freezing and study its relation with the structure of the extremal point process. Joint work with Eliran Subag.