Scaling limits, rough paths, quantum field theory

Programme theme

The goal of statistical mechanics is to calculate the properties, at macroscopic length scales, of a system composed of a large number of interacting microscopic subsystems. To formalise having a large ratio between largest and the smallest length scales, limits such as infinite volume limits, hydrodynamic limits and scaling limits are studied. These limits are random fields or, in cases where there is dynamics, solutions of nonlinear partial differential equations driven by white noise. Such limits can have symmetries that are not present before taking the limit; for example infinite volume limits may be translation invariant and scaling limits by construction are scale invariant. Increased symmetry leads to very special, beautiful, objects such as euclidean quantum field theories and specific partial differential equations driven by white noise. Then statistical mechanical models can be classified into universality classes characterised by these limits. We think of this as a search for far reaching extensions of the central limit theorem and the theory of large deviations. The possible limits are characterised by very few parameters. A new feature of these extensions is that limits have to be expressed in the correct variables because divergences are inherent in limits that have enhanced symmetries. This is the famous problem of renormalisation in quantum field theory. Divergences arise from the volume of non-compact symmetry groups of translations and dilations. Likewise for partial differential equations driven by white noise divergences appear in naive attempts to define the nonlinear terms in the equations. The solutions are too rough to permit ordinary pointwise multiplication. In the last few years, the theory of rough paths, existence, uniqueness and large deviations for singular partial differential equations has been making very rapid progress. Our four month program has been designed to foster a natural alliance with mathematical quantum field theory, specifically the theory of the renormalisation group, continuation in dimension, operator product expansions and conformally invariant quantum field theory. We aim for progress in global existence of solutions of stochastic pde, dynamical critical exponents, equilibrium critical exponents, bosonisation in two dimensions, better and more complete constructions of euclidean quantum fields.

The Institute kindly requests that any papers published as a result of this programme’s activities are credited as such. Please acknowledge the support of the Institute in your paper using the following text:

The author(s) would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme Scaling limits, rough paths, quantum field theory, where work on this paper was undertaken. This work was supported by EPSRC grant EP/R014604/1.

SRQ programme: interview with organisers Massimiliano Gubinelli and David Brydges

Preprints

Title	Year	Programme	Publication Date
Spatial moments for high-dimensional critical contact process, oriented percolation and lattice trees Authors: Gordon Slade, Akira Sakai Download PDF	2018	SRQ	21 October 2022
A PDE construction of the Euclidean $\Phi^4_3$ quantum field theory Authors: Massimiliano Gubinelli, Martina Hofmanová Download PDF	2018	SRQ	21 October 2022
The infinitesimal generator of the stochastic Burgers equation Authors: Nicolas Perkowski, Massimiliano Gubinelli Download PDF	2018	SRQ	21 October 2022
Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity Authors: Herbert Koch, Massimiliano Gubinelli, Tadahiro Oh Download PDF	2018	SRQ	20 October 2022
Elliptic stochastic quantization Authors: Massimiliano Gubinelli, Sergio Albeverio, Francesco C. De Vecchi Download PDF	2018	SRQ	20 October 2022

Workshops

SRQW01

Renormalisation in quantum field theory and in stochastic partial differential equations: a gentle introduction and some recent developments

3 September 2018 to 7 September 2018

SRQW02

Quantum field theory, renormalisation and stochastic partial differential equations

22 October 2018 to 26 October 2018

SRQW03

Scaling limits & SPDEs: recent developments and future directions

10 December 2018 to 14 December 2018

Seminars

Monday 10th September 2018
11:00 to 12:30	David Brydges University of British Columbia Reflection Positivity which plays an important role in QFT and statistical mechanics.	Room 2

Wednesday 12th September 2018
15:00 to 16:30	Margherita Disertori Rheinische Friedrich-Wilhelms-Universität Bonn Why fermions?	Room 2

Friday 14th September 2018
11:00 to 12:30	David Brydges University of British Columbia ; Hendrik Weber University of Bath Chessboard estimates and their role in the 1976	Room 2

Monday 17th September 2018
11:00 to 12:30	Hendrik Weber University of Bath Renormalisation in QFT, statistical mechanics and stochastic PDE's	Room 2

Wednesday 19th September 2018
15:30 to 16:30	Margherita Disertori Rheinische Friedrich-Wilhelms-Universität Bonn Why Fermions, part II	Room 2

Friday 21st September 2018
11:00 to 12:30	Hendrik Weber University of Bath Renormalisation in QFT, statistical mechanics and stochastic PDE's	Room 2

Thursday 27th September 2018
11:00 to 12:00	Peter Friz Technische Universität Berlin Rough paths	Room 2

Friday 28th September 2018
11:00 to 12:30	Takashi Hara Kyushu University Why Yang-Mills: a review of no-go results for phi^4_4	Room 2
16:00 to 17:00	Martin Hairer Imperial College London Rothschild Lecture: Random rubber bands	Room 1

Monday 1st October 2018
11:00 to 12:30	Thierry Levy Sorbonne Université 2-dimensional Yang-Mills theory and the Makeenko-Migdal equations (I)	Room 2

Wednesday 3rd October 2018
15:30 to 16:30	Thierry Levy Sorbonne Université 2-dimensional Yang-Mills theory and the Makeenko-Migdal equations (II)	Room 2

Friday 5th October 2018
11:00 to 12:30	Hendrik Weber University of Bath Local estimates for reaction diffusion equations with applications to $\Phi^4_3$	Room 1

Monday 8th October 2018
11:00 to 12:30	Fabio Toninelli Lecture 1 - The interacting dimer model	Room 2

Wednesday 10th October 2018
15:30 to 17:00	Fabio Toninelli Lecture 2 - The interacting dimer model	Room 2

Friday 12th October 2018
11:00 to 12:30	Fabio Toninelli Lecture 3 - The interacting dimer model (copy)	Room 2

Monday 15th October 2018
11:00 to 12:30	Alessandro Giuliani Università degli Studi Roma Tre Lecture 4 - The interacting dimer model	Room 2

Wednesday 17th October 2018
15:00 to 16:30	Alessandro Giuliani Università degli Studi Roma Tre Lecture 5 - The interacting dimer model	Room 2

Friday 19th October 2018
11:00 to 12:30	Alessandro Giuliani Università degli Studi Roma Tre Lecture 6 - The interacting dimer model	Room 2

Monday 29th October 2018
11:00 to 12:30	Milton Jara IMPA - Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro Non-equilibrium fluctuations of interacting particle systems, lecture 1	Room 2

Wednesday 31st October 2018
15:00 to 16:30	Milton Jara IMPA - Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro Non-equilibrium fluctuations of interacting particle systems, lecture 1	Room 2

Thursday 1st November 2018
11:00 to 12:30	Milton Jara IMPA - Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro Non-equilibrium fluctuations of interacting particle systems, lecture 3	Room 2

Friday 2nd November 2018
11:00 to 12:30	Giovanni Jona-Lasinio Università degli Studi di Roma La Sapienza Towards a non-equilibrium thermodynamics: the role of large fluctuations	Room 2

Monday 5th November 2018
10:00 to 11:30	Giovanni Jona-Lasinio Università degli Studi di Roma La Sapienza Towards a non-equilibrium thermodynamics: the role of large fluctuations	Room 2

Wednesday 7th November 2018
15:00 to 15:30	Augustin Moinat University of Warwick Local bounds for the dynamic $\Phi^4_3$ model	Room 2
15:30 to 16:00	Susanne Hilger Rheinische Friedrich-Wilhelms-Universität Bonn Renormalisation Group Analysis of a gradient model with nonconvex interaction	Room 2
16:00 to 16:30	Henri Elad altman Laboratoire de Probabilités et Modèles Aléatoires Bessel S(P)DEs : a story of renormalization	Room 2

Friday 9th November 2018
11:00 to 12:30	David Brydges University of British Columbia The Higgs Mechanism	Room 2

Monday 12th November 2018
11:00 to 12:30	Pronob Mitter CNRS (Centre national de la recherche scientifique); Université de Montpellier Introduction to the Renormalisation Group	Room 2

Wednesday 14th November 2018
15:00 to 16:30	Pronob Mitter CNRS (Centre national de la recherche scientifique); Université de Montpellier Introduction to the Renormalisation Group	Room 2

Friday 16th November 2018
11:00 to 12:30	Pronob Mitter CNRS (Centre national de la recherche scientifique); Université de Montpellier Introduction to the Renormalisation Group	Room 2

Monday 19th November 2018
11:00 to 12:30	Massimiliano Gubinelli Universität Bonn A variational approach to Phi^4_3	Room 2

Wednesday 21st November 2018
15:00 to 16:30	Thierry Levy Sorbonne Université The partition function of the 2-dimensional Yang—Mills model	Room 2

Friday 23rd November 2018
11:00 to 12:30	Ilya Chevyrev University of Oxford Yang-Mills measure on the two-dimensional torus as a random distribution	Room 2

Monday 26th November 2018
11:00 to 12:30	Mircea Petrache Pontificia Universidad Católica de Chile Large-N asymptotics of energy-minimizing measures on N-point configurations	Room 2

Wednesday 28th November 2018
15:00 to 16:00	Benjamin Gess Technische Universität Berlin Generation of random dynamical systems for SPDE with nonlinear noise.	Room 2
16:00 to 16:30	Trishen Gunaratnam University of Warwick Quasi-invariant Gaussian Measures for the 3D Nonlinear Wave Equation	Room 2

Friday 30th November 2018
11:00 to 12:30	Sandra Cerrai University of Maryland, College Park Large time behavior of infinite dimensional systems under the Smoluchowski-Kramers approximation	Room 2

Monday 3rd December 2018
11:00 to 12:30	Tyler Helmuth University of Bristol Isomorphism theorems, random walks, and spin systems.	Room 2

Wednesday 5th December 2018
15:00 to 16:30	Ajay Chandra University of Warwick Stochastic quantization of Yang Mills	Room 2

Friday 7th December 2018
11:00 to 12:30	Roman Kotecky University of Warwick An abstract framework for a non-perturbative renormalisation	Room 2

Monday 17th December 2018
11:00 to 12:30	Massimiliano Gubinelli Universität Bonn Elliptic stochastic quantisation	Room 2

Scaling limits, rough paths, quantum field theory

Organisers

David Brydges University of British Columbia
Alessandro Giuliani Università degli Studi Roma Tre
Massimiliano Gubinelli Universität Bonn
Antti Kupiainen University of Helsinki
Hendrik Weber University of Bath
Lorenzo Zambotti Université Pierre et Marie Curie Paris

Speakers

David Brydges University of British Columbia
Sandra Cerrai University of Maryland, College Park
Ajay Chandra University of Warwick
Ilya Chevyrev University of Oxford
Margherita Disertori Rheinische Friedrich-Wilhelms-Universität Bonn
Henri Elad Altman Laboratoire de Probabilités et Modèles Aléatoires
Peter Friz Technische Universität Berlin
Benjamin Gess Max-Planck-Institut für Mathematik, Leipzig
Alessandro Giuliani Università degli Studi Roma Tre
Massimiliano Gubinelli Universität Bonn
Trishen Gunaratnam University of Warwick
Martin Hairer Imperial College London
Takashi Hara Kyushu University
Tyler Helmuth University of Bristol
Susanne Hilger Rheinische Friedrich-Wilhelms-Universität Bonn
Milton Jara IMPA - Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro
Giovanni Jona-Lasinio Università degli Studi di Roma La Sapienza
Roman Kotecky University of Warwick; Charles University, Prague
Thierry Levy Sorbonne Université
Pronob Mitter CNRS (Centre national de la recherche scientifique); Université de Montpellier
Augustin Moinat University of Warwick
Mircea Petrache Pontificia Universidad Católica de Chile
Fabio Toninelli
Hendrik Weber University of Bath

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Participants

Abdelmalek Abdesselam University of Virginia
Stefan Adams University of Warwick
Giovanni Antinucci Universität Zürich
Roland Bauerschmidt University of Cambridge
Nils Berglund Université d'Orléans
Thierry Bodineau École Polytechnique
Yvain Bruned Imperial College London
Stefano Bruno University of Bath
David Brydges University of British Columbia
Serena Cenatiempo Gran Sasso Science Institute, L'Aquila
Sandra Cerrai University of Maryland, College Park
Ajay Chandra University of Warwick
Ilya Chevyrev University of Oxford
Codina Cotar University College London
Giuseppe Da Prato Scuola Normale Superiore di Pisa
Margherita Disertori Rheinische Friedrich-Wilhelms-Universität Bonn
Bertrand Duplantier CEA/Saclay; IHES
Henri Elad Altman Laboratoire de Probabilités et Modèles Aléatoires
Luca Fresta Universität Zürich
Peter Friz Technische Universität Berlin
Juerg Froehlich ETH Zürich
Tadahisa Funaki University of Tokyo
Christophe Garban Université Claude Bernard Lyon 1
Krzysztof Gawedzki ENS - Lyon
Benjamin Gess Max-Planck-Institut für Mathematik, Leipzig
Alessandro Giuliani Università degli Studi Roma Tre
Patricia Gonçalves Instituto Superior Técnico, Lisboa
Paolo Grazieschi University of Bath
Massimiliano Gubinelli Universität Bonn
Trishen Gunaratnam University of Warwick
Martin Hairer Imperial College London
Takashi Hara Kyushu University
Tyler Helmuth University of Bristol
Susanne Hilger Rheinische Friedrich-Wilhelms-Universität Bonn
Martina Hofmanova Bielefeld University
Yichao Huang University of Helsinki
John Imbrie University of Virginia
Milton Jara IMPA - Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro
Giovanni Jona-Lasinio Università degli Studi di Roma La Sapienza
Horst Knoerrer ETH Zürich
Roman Kotecky University of Warwick; Charles University, Prague
Antti Kupiainen University of Helsinki
Cyril Labbe Université Paris-Dauphine
Thierry Levy Sorbonne Université
Martin Lohmann University of British Columbia
Vieri Mastropietro Università degli Studi di Milano
Pronob Mitter CNRS (Centre national de la recherche scientifique); Université de Montpellier
Augustin Moinat University of Warwick
Joona Oikarinen University of Helsinki
Etienne Pardoux Aix Marseille Université
Sabrina Pasterski Harvard University
Eveliina Peltola Université de Genève
Nicolas Perkowski Humboldt-Universität zu Berlin
Mircea Petrache Pontificia Universidad Católica de Chile
Marcello Porta Eberhard Karls Universität Tübingen
Jeremy Quastel University of Toronto
Remi Rhodes Aix-Marseille University
Michael Rockner Universität Bielefeld
Slava Rychkov IHES; École Normale Supérieure
Manfred Salmhofer Universität Heidelberg
Hao Shen
Gordon Slade University of British Columbia
Thomas Spencer Institute for Advanced Study, Princeton
Alexandra Symeonides Universidade de Lisboa
Fabio Toninelli
Balint Toth University of Bristol; Renyi Institute for Mathematics, Hungarian Academy of Sciences
Vincent Vargas CNRS (Centre national de la recherche scientifique)
Christian Webb Aalto University
Hendrik Weber University of Bath
Lorenzo Zambotti Université Pierre et Marie Curie Paris

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Scaling limits, rough paths, quantum field theory

Programme theme

SRQ programme: interview with organisers Massimiliano Gubinelli and David Brydges

Preprints

Spatial moments for high-dimensional critical contact process, oriented percolation and lattice trees

A PDE construction of the Euclidean $\Phi^4_3$ quantum field theory

The infinitesimal generator of the stochastic Burgers equation

Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity

Elliptic stochastic quantization