Brian Harry Marcus
Relevant Degree Programs
Graduate Student Supervision
Doctoral Student Supervision (Jan 2008 - May 2019)
Over the last few decades, there has been a growing interest in a measure-theoretical property of Gibbs distributions known as strong spatial mixing (SSM). SSM has connections with decay of correlations, uniqueness of equilibrium states, approximation algorithms for counting problems, and has been particularly useful for proving special representation formulas and the existence of efficient approximation algorithms for (topological) pressure. We look into conditions for the existence of Gibbs distributions satisfying SSM, with special emphasis in hard constrained models, and apply this for pressure representation and approximation techniques in Z^d lattice models. Given a locally finite countable graph G and a finite graph H, we consider Hom(G,H) the set of graph homomorphisms from G to H, and we study Gibbs measures supported on Hom(G,H). We develop some sufficient and other necessary conditions on Hom(G,H) for the existence of Gibbs specifications satisfying SSM (with exponential decay). In particular, we introduce a new combinatorial condition on the support of Gibbs distributions called topological strong spatial mixing (TSSM). We establish many useful properties of TSSM for studying SSM on systems with hard constraints, and we prove that TSSM combined with SSM is sufficient for having an efficient approximation algorithm for pressure. We also show that TSSM is, in fact, necessary for SSM to hold at high decay rate. Later, we prove a new pressure representation theorem for nearest-neighbour Gibbs interactions on Z^d shift spaces, and apply this to obtain efficient approximation algorithms for pressure in the Z² (ferromagnetic) Potts, (multi-type) Widom-Rowlinson, and hard-core lattice gas models. For Potts, the results apply to every inverse temperature except the critical. For Widom-Rowlinson and hard-core lattice gas, they apply to certain subsets of both the subcritical and supercritical regions. The main novelty of this work is in the latter, where SSM cannot hold.
No abstract available.
We study topological and measure theoretic forms of mean equicontinuity and mean sensitivityfor dynamical systems. With this we characterize well known notions like systems with discretespectrum, almost periodic functions, and subshifts with regular extensions. We also study the limitbehaviour of µ-equicontinuous cellular automata. In this thesis we prove a conjecture from  (see Corollary 2.3.18); this was indepently solved by Li-Tu-Ye in . In Chapter 3 we answer questions from .
No abstract available.
Master's Student Supervision (2010 - 2018)
This thesis will discuss the relationship between stationary Markov random fields and probability measures with a nearest neighbour Gibbs potential. While the relationship has been well explored when the measures are fully supported, we shall discuss what happens when we weaken this assumption.