Random sparse sampling in a Gibbs weighted tree

TL;DR

This paper investigates how a very sparse random sampling of a Gibbs measure on [0,1] affects its multifractal structure, revealing a phase transition at a critical sampling density and uncovering new multifractal properties of the sampled measure.

Contribution

It demonstrates the phase transition in reconstructing the original measure from sparse samples and analyzes the multifractal properties of the resulting sampled measure.

Findings

Reconstruction possible when sampling density exceeds a critical threshold

Sampled measure exhibits two phase transitions in its $L^q$-spectrum

New multifractal and thermodynamic properties of the sampled measure

Abstract

Let $\mu$ be the geometric realization on $[0,1]$ of a Gibbs measure on $\Sigma=\{0,1\}^{\mathbb{N}}$ associated with a H\"older potential. The thermodynamic and multifractal properties of $\mu$ are well known to be linked via the multifractal formalism. In this article, the impact of a random sampling procedure on this structure is studied. More precisely, let $\{I_w\}_{w\in \Sigma^*}$ stand for the collection of dyadic subintervals of $[0,1]$ naturally indexed by the set of finite dyadic words $\Sigma^*$. Fix $\eta\in(0,1)$, and a sequence $(p_w)_{w\in \Sigma^*}$ of independent Bernoulli variables of parameters $2^{-|w|(1-\eta)}$ ($|w|$ is the length of $w$). We consider the (very sparse) remaining values $\widetilde\mu=\{\mu(I_w): w\in \Sigma^*, p_w=1\}$. We prove that when $\eta<1/2$, it is possible to entirely reconstruct $\mu$ from the sole knowledge of $\widetilde\mu$, while it is not possible when $\eta>1/2$, hence a first phase transition phenomenon. We show that, for all $\eta \in (0,1)$, it is possible to reconstruct a large part of the initial multifractal structure of $\mu$, via the fine study of $\widetilde\mu$. After reorganization, these coefficients give rise to a random capacity with new remarkable scaling and multifractal properties: its $L^q$-spectrum exhibits two phase transitions, and has a rich thermodynamic and geometric structure.