Scaling limits & SPDEs: recent developments and future directions
Monday 10th December 2018 to Friday 14th 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) tba 
INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Benjamin Schlein (Universität Zürich) Bogoliubov Excitations of dilute BoseEinstein 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 (GrossPitaevskii regime). We determine the lowenergy spectrum, i.e. the ground state energy and lowlying 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) HamiltonJacobi equations
I will discuss two new regularity results (regularizing effect and propagation of regularity) for viscosity solutions of uniformly convex HamiltonJacobi equations. In turn, the new estimates yield new intermittent stochastic regularization for pathwise (stochastic) viscosity solutions of HamiltonJacobi 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 HamiltonJacobi equations. This is joint work with P. L. Lions.  
INI 1  
14:30 to 15:30 
Christophe Garban (Université Claude Bernard Lyon 1) Quantum field theory and SPDEs under the light of nearcriticality 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 longtime 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 longtime 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 
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 onedimensional decaying Burgers turbulence with the covariance of initial profile of Gaussiandistributed 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 lowviscosity phase the onepoint probability density of velocities becomes nonGaussian 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 socalled 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 (AixMarseille University) SineGordon model revisited (again)
In this talk, I will discuss the SineGordon 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 SineGordon 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 
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 energyenergy correlations and the central charge (computed from the leading subleading 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 modularinvariant function associated with the subleading 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) 
09:00 to 10:00 
Hendrik Weber (University of Bath) Spacetime 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 spacetime 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 spacetime 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 solutiondependent regularisation procedure. The fact that our bounds do not depend on spacetime 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 onedimensional 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 masscritical stochastic nonlinear Schrodinger equation
We report some recent joint works with Chenjie Fan on construction of global solutions to defocusing masscritical 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 WisconsinMadison) SPDE limits of sixvertex model
The theme of the talk is deriving stochastic PDE limits as description of largescale fluctuations of the sixvertex (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 recentering 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 oneparameter family of Gibbs states converge to solution of stationary stochastic Burgers. Again for stochastic 6V, in a regime where the cornershape 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 LiCheng Tsai, and a joint work with LiCheng Tsai. 
INI 1  
15:30 to 16:00  Afternoon Tea 
09:00 to 10:00 
Herbert Spohn (Technische Universität München) Generalized hydrodynamics and the classical Toda chain
In the context of integrable quantum manybody 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 nonreversible 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 Kortewegde Vries equation, the ultradiscrete Kortewegde Vries equation, the ultradiscrete Toda equation, the boxball 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 fastslow
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 