Duality and shock measures in stochastic interacting particle systems

• Grant number: 15/15258-9
• Support Opportunities: Research Grants - Visiting Researcher Grant - International
• Duration: January 01, 2016 - March 27, 2016
• Field of knowledge: Physical Sciences and Mathematics - Probability and Statistics - Probability
• Principal Investigator: Vladimir Belitsky
• Grantee: Vladimir Belitsky
• Visiting researcher: Gunter Markus Schutz
• Visiting researcher institution: Forschungszentrum Jülich, Germany
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil

Abstract

The aim of the project is to obtain duality functions for stochastic interacting particle systems whose transition matrix is build by the generators of quotients of the Hecke-algebra and which is therefore symmetric under the action of an associated quantum algebra. The main emphasis will be on the asymmetric simple exclusion process withn>2 classes of particles which is symmetric under the action of the quantum algebra Uq[gl/(n+1)]. We shall first derive the representation matrices that commute with the transition matrix of the exclusion process from the co-product structure of the quantum algebra. Then (i) probabilistic and combinatorial techniques will be employed to construct invariant measures of the process, both of blocking type and translation invariant, which will allow for a deeper probabilistic understanding of the so-called matrix product construction of invariant measures. (ii) Duality functions will be derived for the finite system with reflecting boundaries, for the infinite volume limit, and for periodic boundary conditions. (iii) The latter allows for a generalization of the second-class particle technique to mark the microscopic position of shocks in the process conditioned on an atypical current.From this we shall obtain for arbitrary hopping asymmetry a microscopic description of the so-called dynamical phase transition which so far has been proved only for the single-species weakly asymmetric simple exclusion process using macroscopic fluctuation theory. (iv) The generalized second-class technique will also be used to trace the motion of microscopic fluctuations to obtain better bounds on the boundary terms that appear in generalized Onsager relations and that are important for the study of fluctuations in the framework of non-linear fluctuating hydrodynamics. (AU)

Scientific publications

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

BELITSKY, V.; SCHUETZ, G. M.. Self-duality and shock dynamics in the n-species priority ASEP. Stochastic Processes and their Applications, v. 128, n. 4, p. 1165-1207, APR 2018. (15/15258-9)