Blocking measures for interacting particle systems

Student thesis: Doctoral ThesisDoctor of Philosophy (PhD)


In this thesis we consider questions for interacting particle systems which are members of the blocking family described by Balázs and Bowen [4]. Informally, these are systems in which a typical state would have a block like structure to the particle configuration. For these systems we have product reversible stationary measures (known as blocking measures). This reversibility lets us ask and answer some interesting questions for these systems. The two main themes for this thesis are the question of correspondences for blocking particle systems and the relation to combinatorial objects as well as studying a multi-species asymmetric simple exclusion process under the blocking measure framework.
We discuss the work of Balázs and Bowen [4], where they prove the Jacobi triple product identity by considering the ASEP-AZRP correspondence. It is natural to ask if we can prove other identities of combinatorial significance; to answer this we review recent work of the author, Balázs and Fretwell [6]. Here we fully parameterise a subfamily of the blocking family with 0, 1 or 2 particles at each site, in terms of the jump rates. We then construct a corresponding family of restricted particle systems. By comparing the stationary measures for each family we prove (believed to be new) three variable Jacobi style identities. We show that these correspond to generating functions for generalised Frobenius partitions with a 2-repetition condition on the rows. By specialising to known members of this 0-1-2 family we recover known two variable identities of combinatorial significance (for example studied by Andrews [1]). The family of k-exclusion processes for arbitrary k will also be considered and we will prove similar Jacobi style identities relating to counting generalised Frobenius partitions with a k-repetition condition. Naturally we would like to consider the most general family of blocking processes with up to k particles allowed at a site. We will discuss the problems we face when considering this generalisation and consider what we might expect to be able prove combinatorially in this case.
For the 0-1-2 family we also consider another correspondence with certain evolving lattice paths; these are (mostly) down-right paths but where we can take a single up step at certain sites. We look to find the stationary measure for this stochastic lattice path model, and find it as a limiting solution of some second order recurrence relation. We therefore have a way of linking interacting particle systems, generalised Frobenius partitions and these types of lattice paths.
Classical particle systems assume that all the particles in space are all of a single species, i.e. somewhat similar. However, there are many interesting questions for multi-species particle systems. For example, one may want to study how a second class particle behaves within a well-known classical system. In this thesis we will consider the asymmetric simple exclusion process under its natural blocking measure and describe the behaviour of d second class particles in this system. The second class particles are defined via a basic coupling argument and the distribution for the position of the second class particles is given. Along the way we again see and use the relationship between particle systems with blocking measure and combinatorial objects, in particular integer partitions.
Date of Award24 Jan 2023
Original languageEnglish
Awarding Institution
  • The University of Bristol
SupervisorMarton Balazs (Supervisor)

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