Correlations in multispecies asymmetric exclusion processes

Abstract

The main aim of this thesis is to study the correlations in multispecies exclusion processes inspired by the research of Ayyer and Linusson (Trans. AMS., 2017) where they studied correlations in the multispecies TASEP on a ring with one particle of each species. The focus is on studying various models, such as multispecies TASEP on a continuous ring, multispecies PASEP on a ring, multispecies B-TASEP and multispecies TASEP on a ring with multiple copies of each particle. The primary goal is to investigate the two-point correlations of adjacent particles in these models. The details of these models are given below:

We study the multispecies TASEP on a continuous ring and prove a conjecture by Aas and Linusson (AIHPD, 2018) regarding the two-point correlations of adjacent particles. We use the theory of multiline queues developed by Ferrari and Martin (Ann. Probab., 2007) to interpret the conjecture in terms of the placements of numbers in triangular arrays. Additionally, we use projections to calculate correlations in the continuous multispecies TASEP using a distribution on these placements.

Next, we study the correlations of adjacent particles on the first two sites in the multispecies PASEP on a finite ring. To prove the results, we use the multiline process defined by Martin (Electron. J. Probab., 2020), which is a generalisation of the multiline process defined earlier by Ferrari and Martin.

We then study the multispecies B-TASEP with open boundaries. Aas, Ayyer, Linusson and Potka (J. Physics A, 2019) conjectured a formula for the correlations between adjacent particles on the last two sites in the multispecies B-TASEP. To approach this problem, we use a Markov chain that is a 3-species TASEP defined on the Weyl group of type B. This allows us to make conjectures and prove some results towards the above conjecture.

Finally, we study a more general multispecies TASEP with multiple particles for each species. We extend the results of Ayyer and Linusson to this case and prove formulas for two-point correlations and the TASEP speed process.