[Paper Review] Lamplighter Random Walks and Entropy-Sensitivity of Languages

Ecaterina Sava‐Huss|arXiv (Cornell University)|Dec 13, 2010

Mathematical Dynamics and Fractals37 references3 citations

TL;DR

This dissertation investigates lamplighter random walks on wreath products of graphs and their connection to entropy-sensitivity in formal languages. It establishes that the Poisson boundary of lamplighter random walks on graphs with infinitely many ends or hyperbolic structures is non-trivial, and proves that languages defined on Schreier graphs—such as word problems in group theory—are entropy-sensitive under forbidden transition constraints, using Markov chains with forbidden transitions to analyze growth rates and spectral radii.

ABSTRACT

The main purpose of this thesis is to study the interplay between geometric properties of infinite graphs and analytic and probabilistic objects such as transition operators, harmonic functions and random walks on these graphs. For a transient random walk, there are several problems one is interested in: for instance to study its convergence, to describe the bounded harmonic functions for the random walk, to describe its Poisson boundary, or to study the parameter of exponential decay of the transition probabilities of the random walk. In the first part of the thesis we deal with similar problems in the context of random walks on the so-called lamplighter graphs, which are wreath products of graphs. The convergence and the Poisson boundary of lamplighter random walks is studied for different underlying graphs, and the used methods are mostly of a geometrical nature. In the second part of the thesis we consider Markov chains on directed, labelled graphs. With such graphs we associate in a natural way a class of infinite languages (sets of labels of paths in the graph) and we study the growth sensitivity (or entropy sensitivity) of these languages using Markov chains.

Motivation & Objective

• To analyze the asymptotic behavior of lamplighter random walks on infinite graphs, particularly their convergence to the boundary and Poisson boundary structure.
• To investigate the interplay between geometric properties of graphs (e.g., number of ends, hyperbolicity) and probabilistic features of lamplighter walks.
• To study the growth sensitivity of formal languages defined on labeled graphs using Markov chains with forbidden transitions.
• To prove that word problems in group theory, represented as languages on Schreier graphs, are entropy-sensitive under finite forbidden sets.
• To establish spectral radius and entropy comparisons between unrestricted and forbidden-transition Markov chains on labeled graphs.

Proposed method

• Use of wreath product constructions (Z₂ ≀ G) to model lamplighter graphs, where vertices represent lamp configurations and walker positions.
• Application of the half-space method to identify the Poisson boundary of lamplighter random walks on graphs with infinitely many ends.
• Employment of harmonic functions and transition operators to characterize convergence to the boundary and Poisson boundary structure.
• Definition of a Markov chain on labeled directed graphs with transition probabilities proportional to inverse out-degree.
• Introduction of a 'h-process' via Doob's h-transform to model random walks with forbidden transition sets F.
• Use of spectral radius comparison: ρ(PF) < ρ(P) for forbidden sets F, leading to strict entropy reduction in associated languages.

Experimental results

Research questions

• RQ1 What is the Poisson boundary of a lamplighter random walk on a graph with infinitely many ends?
• RQ2 How does the hyperbolic structure of a graph influence the Poisson boundary and convergence of lamplighter random walks?
• RQ3 Under what conditions is the language of paths in a labeled graph entropy-sensitive when certain transitions are forbidden?
• RQ4 What is the relationship between the spectral radius of a Markov chain with forbidden transitions and the original chain?
• RQ5 How does the growth rate (entropy) of word problems in group theory change when finite sets of words are forbidden?

Key findings

• The Poisson boundary of lamplighter random walks on graphs with infinitely many ends is non-trivial and can be identified via the half-space method.
• For lamplighter walks on hyperbolic graphs, the Poisson boundary is non-trivial when the graph has infinite hyperbolic boundary, even with one end.
• The spectral radius of the Markov chain with forbidden transitions satisfies sup_{x,y} ρ_{x,y}(PF) < ρ(P), implying a strict reduction in entropy.
• The entropy of the language L_F^{x,y} (paths avoiding forbidden factors) is strictly less than that of the unrestricted language L^{x,y}, proving entropy sensitivity.
• For Schreier graphs of finitely generated groups, the word problem language is entropy-sensitive under any non-empty finite forbidden set.
• The entropy of a language L^{x,y} on a uniformly connected labeled graph is given by h(L^{x,y}) = log(ρ(P) · |Σ|), where ρ(P) is the spectral radius and |Σ| the alphabet size.