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Timetable (SRQW03)

Scaling limits & SPDEs: recent developments and future directions

Monday 10th December 2018 to Friday 14th December 2018

Monday 10th December 2018
09:00 to 09:50 Registration
09:50 to 10:00 Welcome from David Abrahams (Isaac Newton Institute) INI 1
10:00 to 11:00 Martin Hairer (Imperial College London)
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INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Benjamin Schlein (Universität Zürich)
Bogoliubov Excitations of dilute Bose-Einstein Condensates
We consider systems of N bosons confined in a box with volume one and interacting through a potential with
short scattering length of the order 1/N (Gross-Pitaevskii regime). We determine the low-energy spectrum, i.e. the
ground state energy and low-lying excitations, up to errors that vanishes in the limit of large N, confirming
the validity of Bogoliubov’s predictions. This talk is based on joint work with C. Boccato, C. Brennecke, S. Cenatiempo.
INI 1
12:30 to 13:30 Lunch at Churchill College
13:30 to 14:30 Panagiotis Souganidis (University of Chicago)
New regularity results and long time behavior of pathwise (stochastic) Hamilton-Jacobi equations
I will discuss two new regularity results (regularizing effect and propagation of regularity) for viscosity solutions of uniformly convex Hamilton-Jacobi equations. In turn, the new estimates yield new intermittent stochastic regularization for pathwise (stochastic) viscosity solutions of Hamilton-Jacobi equations with uniformly convex Hamiltonians and rough multiplicative time dependence. The intermittent regularity estimates are then used to study the long time behavior of the pathwise (stochastic) viscosity solutions of convex Hamilton-Jacobi equations. This is joint work with P. L. Lions.

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INI 1
14:30 to 15:30 Christophe Garban (Université Claude Bernard Lyon 1)
Quantum field theory and SPDEs under the light of near-criticality and noise sensitivity
Based on a joint work with Martin Hairer and Antti Kupiainen.
INI 1
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Ivan Corwin (Columbia University)
Dynamic ASEP
The long-time behavior of the standard ASEP height function relates to the KPZ fixed point and the KPZ equation. In this talk we consider "dynamic" variants of ASEP in which the rates of increase / decrease for the height function depend on the height in such a way as to have height reversion to a preferred level. Very little is known about the long-time behavior of this type of process. We prove an SPDE limit of the model under very weakly asymmetric scaling. Along the way, we also touch on dynamic ASEP's Markov duality and relation to orthogonal polynomials.

This talk relates to a joint work with Alexei Borodin from last year (on the duality) and a new joint work with Konstantin Matetski and Promit Ghosal (on the SPDE limit).
INI 1
17:00 to 18:00 Welcome Wine Reception at INI
Tuesday 11th December 2018
09:00 to 10:00 Jeremy Quastel (University of Toronto)
The KPZ fixed point - 1
A determinantal formula for the transition probabilities of TASEP with right finite initial data allows us to pass to the limit and obtain the invariant Markov process at the centre of  the KPZ universality class.
INI 1
10:00 to 11:00 Pierre Le Doussal (CNRS - Ecole Normale Superieure Paris)
Large devations for the KPZ equation
INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Yan Fyodorov (King's College London)
Freezing Transition in Decaying Burgers Turbulence and Random Matrix Dualities
We study one-dimensional decaying Burgers turbulence with the covariance of initial profile
of Gaussian-distributed velocities decaying as inverse square of the distance. Combining the
heuristic replica trick of statistical mechanics with insights from the random matrix theory we
reveal a freezing phase transition with decreasing viscosity. In the low-viscosity phase the
one-point probability density of velocities becomes non-Gaussian reflecting a spontaneous
one step replica symmetry breaking in the associated statistical mechanics problem. We
obtain the low orders velocity cumulants analytically which favourably agree with numerical
simulation. The presentation will be based on a joint work with P. Le Doussal and A. Rosso.
INI 1
12:30 to 13:30 Lunch at Churchill College
13:30 to 14:30 Eveliina Peltola (Université de Genève)
Crossing Probabilities of Multiple Ising Interfaces
In this talk, I discuss crossing probabilities of multiple interfaces in the critical Ising model with alternating boundary conditions. In the scaling limit, they are conformally invariant expressions given by so-called pure partition functions of multiple SLE(kappa) with kappa=3. I also describe analogous results for critical percolation and the Gaussian free field. Joint work with Hao Wu (Yau Mathematical Sciences Center, Tsinghua University)
INI 1
14:30 to 15:30 Remi Rhodes (Aix-Marseille University)
Sine-Gordon model revisited (again)
In this talk, I will discuss the Sine-Gordon model and its relation with the 2D neutral Coulomb gas. This is a model of electric charges with positive/negative signs interacting through the Coulomb potential. In the case when particles are confined to the boundary of a 2D domain (called boundary Sine-Gordon model) I will present a short proof that allows us to completely solve the ultraviolet renormalization of this model, including correlation functions. The method is purely probabilistic and relies on concentration methods for martingales.
INI 1
15:30 to 16:00 Afternoon Tea
Wednesday 12th December 2018
09:00 to 10:00 Hugh Osborn (University of Cambridge)
Seeking fixed points
Fixed points are crucial in understanding the RG flow of quantum field theories. The conformal bootstrap has proved a wonderful tool in determining the properties of CFTs at fixed points but tends to require guidance in terms of what symmetries to impose and what is the spectrum of relevant operators. Here I review what can be said in general by using the time honoured epsilon expansion. Although qualitatively this is not nowadays the most efficient method it provides qualitative information about possible fixed points. Finding fixed points which cannot be linked to the epsilon expansion could provide a clue to non Lagrangian theories.
INI 1
10:00 to 11:00 Alessandro Giuliani (Università degli Studi Roma Tre)
Scaling limit and universal finite size corrections in 2D interacting Ising models
In the last few years, the methods of constructive fermionic Renormalization Group have successfully been applied to the study of the scaling limit of several 2D statistical mechanics models at the critical point, including the 2D Ising with finite range interactions. Different instances of universality have been proved in these context, including the facts that the scaling limit of the bulk energy-energy correlations and the central charge (computed from the leading sub-leading contribution to the critical free energy) are independent of the interaction. More recently, we extended our constructive RG method to the study of more subtle finite size observables: I will report ongoing progress on the scaling limit of the energy correlations in finite domains, and on the computation of the modular-invariant function associated with the sub-leading contribution to the critical free energy. Our methods may be relevant in a more general context for the study of the flow of the effective boundary conditions in critical models in a finite domain.
Based on joint work with V. Mastropietro, R. Greenblatt and G. Antinucci.
INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Jeremy Quastel (University of Toronto)
The KPZ fixed point - 2
A determinantal formula for the transition probabilities of TASEP with right finite initial data allows us to pass to the limit and obtain the invariant Markov process at the centre of the KPZ universality class.
INI 1
12:30 to 13:30 Lunch buffet at the INI
13:30 to 17:00 Free afternoon
19:30 to 22:00 Formal Dinner at Christs College (Hall)
Thursday 13th December 2018
09:00 to 10:00 Hendrik Weber (University of Bath)
Space-time localisation for the dynamic $\Phi^4_3$ model
We prove an a priori bound for solutions of the dynamic $\Phi^4_3$ equation. This bound provides a control on solutions on a compact space-time set only in terms of the realisation of the noise on an enlargement of this set, and it does not depend on any choice of space-time boundary conditions.  
We treat the  large and small scale behaviour of solutions with completely different arguments.
For small scales we use bounds akin to those presented in Hairer's theory of regularity structures. For large scales we use a PDE argument based on the maximum principle. Both regimes are connected by a solution-dependent regularisation procedure.  
The fact that our bounds do not depend on space-time boundary conditions makes them useful for the analysis of large scale properties of solutions. They can for example be used  in a compactness argument to construct solutions on the full space and their invariant measures.  

Joint work with A. Moinat.
INI 1
10:00 to 11:00 Cedric Bernardin (Université de Nice Sophia Antipolis); (NICE)
Hydrodynamic limit for a disordered harmonic chain
We consider a one-dimensional unpinned chain of harmonic oscillators with random masses. We prove that after hyperbolic scaling of space and time the distributions of the elongation, momentum and energy converge to the solution of the Euler equations. Anderson localization decouples the mechanical modes from the thermal modes, allowing the closure of the energy conservation equation even out of thermal equilibrium. This example shows that the derivation of Euler equations rests primarily on scales separation and not on ergodicity.

Joint with F. Huveneers and S. Olla
INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Weijun Xu (University of Warwick)
On mass-critical stochastic nonlinear Schrodinger equation
We report some recent joint works with Chenjie Fan on construction of global solutions to defocusing mass-critical stochastic nonlinear Schrodinger equation.
INI 1
12:30 to 13:30 Lunch buffet at the INI
13:30 to 14:30 Franco Flandoli (Scuola Normale Superiore di Pisa)
Remarks on 2D inverse cascade turbulence
Recently the interest in certain invariant measures of 2D Euler equations was renewed, motivated for instance by questions of existence for almost every initial condition, similarly to the case of dispersive equations where probability on initial conditions allowed very succesful progresses.
Obviusly invariant measures of 2D Euler equations are primarily of interest for turbulence but those known are not realistic from several viewpoints, beside some element of great interest.
We discuss this issue and show modifications, unfortunately mostly heuristic, that would give much better results for turbulence.
INI 1
14:30 to 15:30 Hao Shen (University of Wisconsin-Madison)
SPDE limits of six-vertex model
The theme of the talk is deriving stochastic PDE limits as description of large-scale fluctuations of the six-vertex (6V) model in various regimes.
We will consider two types of 6V model: stochastic 6V and symmetric 6V.
For stochastic 6V in a weakly asymmetric regime, under parabolic scaling the height function fluctuation converges to solution of KPZ equation after suitable re-centering and tilting. For symmetric 6V, in a regime where parameters are tuned into the ferroelectric/disordered phase critical point, under parabolic scaling the line density fluctuations in a one-parameter family of Gibbs states converge to solution of stationary stochastic Burgers.
Again for stochastic 6V, in a regime where the corner-shape vertex weights are tuned to zero, under hyperbolic scaling, the height fluctuation converges to the solution of stochastic telegraph equation.
We will discuss challenges and new techniques in the proofs.
Based on a joint work with Ivan Corwin, Promit Ghosal and Li-Cheng Tsai, and a joint work with Li-Cheng Tsai.
INI 1
15:30 to 16:00 Afternoon Tea
Friday 14th December 2018
09:00 to 10:00 Herbert Spohn (Technische Universität München)
Generalized hydrodynamics and the classical Toda chain
In the context of integrable quantum many-body systems, much progress has been achieved in
deriving and analysing the infinite set of coupled local conservation laws constituting
"generalized hydrodynamics". In my presentation I will outline the scheme for the
classical Toda chain exploring unexpected connections to random matrix theory.

INI 1
10:00 to 11:00 Claudio Landim (IMPA - Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro)
Potential theory for nonreversible dynamics and corrections to the hydrodynamical limit
We present in this talk two results. First, variational formulas for the capacity of non-reversible Markovian
dynamics and some applications. Then we derive the viscous Burger's equation from an interacting particle system
as a correction to the hydrodynamic limit.
INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Makiko Sasada (University of Tokyo)
Some generalization of Pitman’s transform and invariant measures for discrete integrable systems
In this talk, I will introduce some generalization of Pitman’s transform and show that the dynamics of several discrete integrable systems, such as the discrete Korteweg-de Vries equation, the ultra-discrete Korteweg-de Vries equation, the ultra-discrete Toda equation, the box-ball system, are given by them. We apply this observation to define infinite space versions of these models and study their invariant measures using the generalized Pitman’s transform. This talk is based on a joint work with David Croydon, Tsuyoshi Kato and Satoshi Tsujimoto.
INI 1
12:30 to 13:30 Lunch at Churchill College
13:30 to 14:30 Ian Melbourne (University of Warwick)
Deterministic homogenization with Levy process limits
We consider homogenization of deterministic fast-slow systems in the situation where the limiting SDE is driven by a stable Levy process. This is joint work with Chevyrev, Friz and Korepanov.
INI 1
14:30 to 15:30 Gianbattista Giacomin (Université Denis Diderot)
Two dimensional Ising model with columnar disorder and continuum limit of random matrix products
I will present results taken from a recent work with R. L. Greenblatt and F. Comets on the continuum limit of random matrix products. The focus will be on one of the applications: two dimensional Ising model with columnar disorder. 50 years ago McCoy and Wu pointed out that the free energy density of the two dimensional Ising model (on the square lattice, with nearest neighbor interactions) can be written in terms of the Lyapunov exponent of products of suitable random two by two matrices. Moreover they extracted from this remarkable formula a number of (even more remarkable) conclusions. I will present their approach and explain how some of the steps can be made mathematically rigorous. I will also explain what is missing to get to the McCoy and Wu conclusions.
INI 1
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons