Quantum Field Theory with Gaussian Multiplicative Chaos

Project Information

QuantGMC

Grant agreement ID: 837793

DOI

10.3030/837793

Project closed

EC signature date 11 April 2019

Start date 9 March 2020

End date 8 May 2021

Funded under

• EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions

Total cost

€ 114 746,24

EU contribution

€ 114 746,24

Coordinated by

UNIVERSITE D'AIX MARSEILLE
France

Periodic Reporting for period 1 - QuantGMC (Quantum Field Theory with Gaussian Multiplicative Chaos)

Reporting period: 2020-03-09 to 2021-05-08

Summary of the context and overall objectives of the project

The scientific activities of the researcher during the project have given raise to a variety of results in three main directions,
-Complex Gaussian Multiplicative Chaos,
-Scaling limit for directed polymers and the Stochastic Heat Equation,
-Mixing time for effective interface models and particle systems.

Complex Gaussian Multiplicative Chaos.

The development of Complex Gaussian Multiplicative Chaos theory has been presented as research direction 2 in the DoA. Complex Gaussian Multiplicative Chaos is a random distribution obtained by taking the complex exponential of a Gaussien field whose covariance diverges logarithmically. It is a very natural occurence of a random distribution with fractal properties and has connection with several problems in Theoretical Physics in particular Quantum Field Theory.

Scaling limit for directed polymers and the Stochastic Heat Equation

In a collaborative effort with Quentin Berger (Université de Paris), we investigated the problem of scaling limit of directed polymer models with an heavy-tailed environment. The directed polymer in a random environment is one of the most studied model for diffusion in a random medium.



Mixing time for effective interface models and particle systems.

Together with Cyril Labbé (Université Paris Dauphine), Pietro Caputo (Universitá degli Studi Roma Tre) and Shangjie Yang (IMPA - Rio de Janeiro), the researcher also brought new developments on the research concerning the relaxation to equilibrium of physical systems described by a Markov chain. One part of this effort concerns so-called effective interfaces, which are simplified models to account for the time evolution of phase separation in a system. The other is about particle systems in a random environment.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

Our work has been exploited and disseminated in the following publications and seminars.

[P1] H. Lacoin, A universality result for subcritical Complex Gaussian Multiplicative Chaos, Annals of
Applied Probability (in press) arXiv:2003.14024
[P2] Q. Berger, H. Lacoin, The scaling limit of the directed polymer with power-law tail disorder, Com-
munication in Mathematical Physics (in press) arXiv:2010.09592
[P3] P. Caputo, C. Labbé, H. Lacoin, Spectral gap and cutoff phenomenon for the Gibbs sampler of ∇φ
interfaces with convex potential, Annales de l’Institut Henri Poincaré (in press) arXiv:2007.10108 .
[P4] H. Lacoin, S. Yang, Metastability for expanding bubbles on a sticky substrate, preprint arXiv:2007.07832
[P5] Q. Berger, H. Lacoin, The continuum directed polymer in Lévy Noise, preprint arXiv:2007.06484
[P6] H. Lacoin, Convergence in law for Complex Gaussian Multiplicative Chaos in phase III, preprint arXiv:2011.08033
[P7] H. Lacoin, S. Yang, Mixing time for the asymmetric simple exclusion process in a random environment, preprint arXiv:2102.02606.

Conferences and seminars : The researcher has presented his research results in the following venues
(mostly online)
[C1] Seminaire de probabilité de d’analyse du CEREMADE, Université Paris Dauphine, April 21th 2020.
[C2] One World Probability Seminar (www.owprobability.org) April 30th 2020.
[C3] Stochastic PDE Seminar, Columbia University, October 16th 2020.
[C4] Workshop "Random Polymers and Networks", Porquerolles Island, September 7-11 2020.
[C5] Séminaire de probabilités, Aix-Marseille Université, September 25th 2020. Groupe de Travail modélisation stochastique, Université de Paris, November 26th 2020.
[C6] Probability and Statistical Physics Seminar, University of Chicago, January 15th 2021.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

The scientific impact of our research can in each research direction can be summarized as follows:

Complex Gaussian Multiplicative Chaos.

Our work on Complex Gaussian Multiplicative Chaos during the period of the project allowed for a rigorous proof of the universality of the object when the complex parameter in the exponential varies in a range called the subcritical phase. Also in another region of its phase diagramm (called phase III) we proved the convergence of the complex GMC towards a Gaussian white noise.


Scaling limit for directed polymers and the Stochastic Heat Equation

We manage to prove that when the size and the temperature of the system are properly rescaled, there exists a continuum limit of the model in which the limiting environment is given by a Lévy white noise. This continuum limit is intimately related with the Stochastic Heat Equation with multiplicative noise and has been constructed and studied in a separate work.


Mixing time for effective interface models and particle systems.

Our work brought valuable estimates concerning the time that these systems need to relax to their equilibrium state when starting from an out-of-equilibrium configuration.

A scientific conference in 2020