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Renormalisation, dynamics, and hyperbolic symmetry

Project description

Robust mathematical techniques for renormalisation group analysis of spin systems

Renormalisation group is an important mathematical tool in theoretical physics, allowing the systematic investigation of changes in a physical system at different scales. The aim of the EU-funded SPINRG project is to develop new mathematical methods for renormalisation group analysis of classical continuous spin systems. The focus will be on their stochastic dynamics and hyperbolic symmetry. Project work will build on the outcomes of previously used successful methods on renormalisation group analysis for static systems, renormalisation group approaches to Glauber dynamics, spin systems with hyperbolic symmetry and supersymmetry and random matrix theory. The renormalisation theory was key to the development of quantum electrodynamics and has now become a central technique in quantum field theory.

Objective

The objective of this proposal is to develop effective 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 and to random operators. In particular, I propose 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 random matrix theory, 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 asymptotics (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 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 which are of independent interest, and allow for some of the quantum phenomena to be reinterpreted probabilistically.

I will build on methods that I developed in previous work, including renormalisation group analysis for static systems, the development of a renormalisation group approach to Glauber dynamics, the study of spin systems with hyperbolic symmetry and supersymmetry, and my experience in random matrix theory.

Host institution

THE CHANCELLOR MASTERS AND SCHOLARS OF THE UNIVERSITY OF CAMBRIDGE
Net EU contribution
€ 1 250 043,00
Address
TRINITY LANE THE OLD SCHOOLS
CB2 1TN Cambridge
United Kingdom

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Region
East of England East Anglia Cambridgeshire CC
Activity type
Higher or Secondary Education Establishments
Links
Total cost
€ 1 250 043,00

Beneficiaries (1)