Periodic Reporting for period 3 - SPINRG (Renormalisation, dynamics, and hyperbolic symmetry)
Periodo di rendicontazione: 2023-01-01 al 2024-12-31
The objective of this project was to develop methods for the analysis of classical continuous spin systems, with focus on their stochastic dynamics and on spin systems with hyperbolic symmetry. The latter are related to reinforced random walks, random operators, and other geometric random systems. In particular, a main goal is to develop mathematical methods for renormalisation group analysis of such systems. Renormalisation is a central concept in theoretical physics, explaining a vast range of phenomena heuristically. While its rigorous implementation is difficult, when a renormalisation group approach to a problem is available, it provides very detailed control and also explains universality.
Both, in stochastic dynamics of large scale systems and in the theory of random operators and matrices, great progress has been achieved recently. In stochastic dynamics, this applies in particular to the problem of existence of solutions to SPDEs and their regularity (ultraviolet problem). This proposal focuses on the complementary regime of long time behaviour (infrared problem), where important results have been obtained via exact solutions but robust methods remain scarce. In random matrix theory, very general classes of random matrices have been understood, yet those with finite dimensional structure like the Anderson model remain mostly out of reach. Spin systems with hyperbolic symmetry and supersymmetry arise in the descriptions of such random matrices, and their simplified versions are prototypes for the understanding of the original models. They also describe linearly reinforced random walks and random forests which are of independent interest, and allow for some of the quantum phenomena to be reinterpreted probabilistically.
Both, in stochastic dynamics of large scale systems and in the theory of random operators and matrices, great progress has been achieved recently. In stochastic dynamics, this applies in particular to the problem of existence of solutions to SPDEs and their regularity (ultraviolet problem). This proposal focuses on the complementary regime of long time behaviour (infrared problem), where important results have been obtained via exact solutions but robust methods remain scarce. In random matrix theory, very general classes of random matrices have been understood, yet those with finite dimensional structure like the Anderson model remain mostly out of reach. Spin systems with hyperbolic symmetry and supersymmetry arise in the descriptions of such random matrices, and their simplified versions are prototypes for the understanding of the original models. They also describe linearly reinforced random walks and random forests which are of independent interest, and allow for some of the quantum phenomena to be reinterpreted probabilistically.
The project made significant progress to towards its goals.
In particular, in the context of the development methods for stochastic dynamics (Part 1 of the grant proposal), we resolved the central problems of the proposal, by establishing the log-Sobolev inequality for the dynamical phi^4 model (continuum), and by showing that the log-Sobolev constant for Glauber dynamics of the Ising model (discrete) diverges polynomially at the critical point in dimensions above 4. Our results opened a much more general perspective exposed in a more recent survey that outlined the connections of methods such as Stochastic Localization and the Barashkov-Gubinelli method.
In the context context of developing the renormalisation group method in the context of hyperbolic and continuous spin models (Part 2 of the grant proposal), the main goal to establish spontaneous symmetry breaking was achieved in the context of the H02 nonlinear sigma model and its related probabilistic representation of random forests. In a related direction, we were able to develop the renormalisation group method for the Discrete Gaussian model which is dual to a continuous spin model, by establishing its GFF scaling limit at high temperature. Both lines of work open the possibility to address several important related questions in the future (such as the analysis of the 2D Villain XY model). For the H22 nonlinear sigma model on the Bethe lattice, the nature of the phase transition on the Bethe lattice was rigorously understood.
Finally, a number of independent questions were studied. This includes the implementation of a renormalisation group approach for the study of extrema of non-Gaussian logarithmically correlated fields, the proof of Coleman's bosonisation conjecture at the free fermion point, and the proof of macroscopic cycles of the interchange model on regular graphs.
The above results were presented at various important conferences (including the ICMP 2021, the MSRI Workshop on Random Matrices, the IHES conferences on 100 years of the Ising, and the ICM 2022) and several longer lecture series (Cour d'IHES in March 2022, Summer school in Beijing/Zoom in July 2022).
In particular, in the context of the development methods for stochastic dynamics (Part 1 of the grant proposal), we resolved the central problems of the proposal, by establishing the log-Sobolev inequality for the dynamical phi^4 model (continuum), and by showing that the log-Sobolev constant for Glauber dynamics of the Ising model (discrete) diverges polynomially at the critical point in dimensions above 4. Our results opened a much more general perspective exposed in a more recent survey that outlined the connections of methods such as Stochastic Localization and the Barashkov-Gubinelli method.
In the context context of developing the renormalisation group method in the context of hyperbolic and continuous spin models (Part 2 of the grant proposal), the main goal to establish spontaneous symmetry breaking was achieved in the context of the H02 nonlinear sigma model and its related probabilistic representation of random forests. In a related direction, we were able to develop the renormalisation group method for the Discrete Gaussian model which is dual to a continuous spin model, by establishing its GFF scaling limit at high temperature. Both lines of work open the possibility to address several important related questions in the future (such as the analysis of the 2D Villain XY model). For the H22 nonlinear sigma model on the Bethe lattice, the nature of the phase transition on the Bethe lattice was rigorously understood.
Finally, a number of independent questions were studied. This includes the implementation of a renormalisation group approach for the study of extrema of non-Gaussian logarithmically correlated fields, the proof of Coleman's bosonisation conjecture at the free fermion point, and the proof of macroscopic cycles of the interchange model on regular graphs.
The above results were presented at various important conferences (including the ICMP 2021, the MSRI Workshop on Random Matrices, the IHES conferences on 100 years of the Ising, and the ICM 2022) and several longer lecture series (Cour d'IHES in March 2022, Summer school in Beijing/Zoom in July 2022).
Our results on large scale stochastic dynamics (Part 1) have significantly extended the state of the art of the subject by treating central problems for the first time (such as near-critical Ising models, long-time behaviour of large-scale singular SPDEs) and provide the basis for a more general theory that had several other applications. Our results on nonlinear sigma models (Part 2) have brought the important class statistical physics models of nonlinear sigma models with noncompact symmetries within reach of accessible rigorous renormalisation group methods.