[Paper Review] Invariant Measures, Hausdorff Dimension and Dimension Drop of some Harmonic Measures on Galton-Watson Trees

Pierre Rousselin|arXiv (Cornell University)|Aug 23, 2017

Stochastic processes and statistical mechanics16 references4 citations

TL;DR

This paper introduces algebraic conditions for constructing explicit invariant measures on marked Galton-Watson trees, enabling the computation of Hausdorff dimensions of harmonic measures for $λ$-biased and random-length random walks. It establishes the dimension drop phenomenon in both cases, providing explicit formulas for the harmonic measure's dimension where previous results were non-constructive.

ABSTRACT

We consider infinite Galton-Watson trees without leaves together with i.i.d.~random variables called marks on each of their vertices. We define a class of flow rules on marked Galton-Watson trees for which we are able, under some algebraic assumptions, to build explicit invariant measures. We apply this result, together with the ergodic theory on Galton-Watson trees developed in \\cite{LPP95}, to the computation of Hausdorff dimensions of harmonic measures in two cases. The first one is the harmonic measure of the (transient) $\\lambda$-biased random walk on Galton-Watson trees, for which the invariant measure and the dimension were not explicitly known. The second case is a model of random walk on a Galton-Watson trees with random lengths for which we compute the dimensions of the harmonic measure and show dimension drop phenomenon for the natural metric on the boundary and another metric that depends on the random lengths.

Motivation & Objective

• To construct explicit invariant measures for a class of flow rules on marked Galton-Watson trees under algebraic conditions.
• To extend ergodic theory on Galton-Watson trees to compute Hausdorff dimensions of harmonic measures.
• To resolve the lack of explicit formulas for the invariant measure and dimension in $λ$-biased random walks on supercritical Galton-Watson trees.
• To analyze the dimension drop phenomenon in a model with random edge lengths on Galton-Watson trees.

Proposed method

• Define a flow rule on marked Galton-Watson trees using a positive function $\phi$ on subtrees to determine transition probabilities for a random ray.
• Establish sufficient algebraic conditions on $\phi$ to ensure the existence of an explicit invariant measure absolutely continuous with respect to the Galton-Watson law.
• Apply ergodic theory from [12] to relate the invariant measure to the harmonic measure of the random walk.
• Use the Markov chain on marked trees induced by the flow rule to analyze long-term behavior and dimension properties.
• Leverage symmetry and stochastic identities involving uniform random variables and conductance functions to compute expectations.
• Verify consistency of dimension formulas using Tonelli’s theorem and functional identities involving the function $h(x,y,z) = \frac{xyz}{xy+yz+xz - x - y - z + 1}$.

Experimental results

Research questions

• RQ1 Can explicit invariant measures be constructed for flow rules on marked Galton-Watson trees under algebraic conditions on the function $\phi$?
• RQ2 What is the Hausdorff dimension of the harmonic measure for the $\lambda$-biased random walk on supercritical Galton-Watson trees, and does dimension drop occur?
• RQ3 How does the introduction of random edge lengths affect the Hausdorff dimension of the harmonic measure and the dimension drop phenomenon?
• RQ4 Is the invariant measure for the $\lambda$-biased random walk on Galton-Watson trees explicitly computable, and how does it relate to the harmonic measure?
• RQ5 Can the dimension of the harmonic measure be computed in a model with random edge lengths using the same framework?

Key findings

• The paper constructs an explicit invariant measure for a class of flow rules on marked Galton-Watson trees under specified algebraic conditions on $\phi$.
• For the $\lambda$-biased random walk on supercritical Galton-Watson trees, the Hausdorff dimension of the harmonic measure is computed explicitly, confirming the dimension drop phenomenon.
• In the model with random edge lengths, the harmonic measure exhibits dimension drop under both the natural metric and the length-dependent metric.
• The dimension drop is quantitatively characterized by showing that the harmonic measure's Hausdorff dimension is strictly less than that of the full boundary.
• The consistency of the derived dimension formula is verified through stochastic identities involving uniform random variables and symmetric functions of conductances.
• The method provides a general framework to compute dimensions in models where previous results were non-constructive, especially for $\lambda \neq 1$.