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Scenery Reconstruction for Random Walk on Random Scenery Systems

Tsviqa Lakrec

TL;DR

This work analyzes a random walk on a random scenery (RWRS) observed through a finite record that may be adversarially corrupted in a fraction $\\delta$ of entries. The authors develop a multiscale reconstruction framework built around a hierarchy of interval tests $\\Lambda_k$ and a carefully constructed family of sinew-like stochastic paths $\\mathcal{W}_s$, termed sinuous walks, to separate the scenery into disjoint, analyzable blocks. The core result shows that for any $\\theta<\\tfrac{1}{2}$, $p>0$, and $\\epsilon>0$, one can, with probability exceeding $1-p$, reconstruct more than $N^{\\theta}$ integers of the visited scenery from a corrupted record, and with at most $2^{\\epsilon s}$ alternative reconstructions where $s$ is the number of reconstructed sites. The approach combines a robust, multilevel testing scheme with probabilistic control of anomalous intervals and local time, yielding a constructive method that remains effective even under worst-case adversarial noise. This advances scenery reconstruction in RWRS contexts and links ergodic-theoretic insights (Kalikow-type arguments) with concrete finite-record recovery guarantees, with potential implications for isomorphism-invariant entropy questions and VWB-type analyses.

Abstract

Consider a simple random walk on $\mathbb{Z}$ with a random coloring of $\mathbb{Z}$. Look at the sequence of the first $N$ steps taken in the random walk, together with the colors of the visited locations. We call this the record. From the record one can deduce the coloring of of the interval in $\mathbb{Z}$ that was visited, which is of size approximately $\sqrt{N}$. This is called scenery reconstruction. Now suppose that an adversary may change $δN$ entries in the record that was obtained. What can be deduced from the record about the scenery now? In this paper we show that it is likely that we can still reconstruct a large part of the scenery. More precisely, we show that for any $θ<0.5,p>0,ε>0$, there are $N_{0}$ and $δ_{0}$ such that if $N>N_{0}$ and $δ<δ_{0}$ then with probability $>1-p$, the walk is such that we can reconstruct the coloring of $>N^θ$ integers, up to a number of possible reconstructions that is less than $2^{εs}$, where $s$ is the number of integers whose color we reconstruct.

Scenery Reconstruction for Random Walk on Random Scenery Systems

TL;DR

This work analyzes a random walk on a random scenery (RWRS) observed through a finite record that may be adversarially corrupted in a fraction of entries. The authors develop a multiscale reconstruction framework built around a hierarchy of interval tests and a carefully constructed family of sinew-like stochastic paths , termed sinuous walks, to separate the scenery into disjoint, analyzable blocks. The core result shows that for any , , and , one can, with probability exceeding , reconstruct more than integers of the visited scenery from a corrupted record, and with at most alternative reconstructions where is the number of reconstructed sites. The approach combines a robust, multilevel testing scheme with probabilistic control of anomalous intervals and local time, yielding a constructive method that remains effective even under worst-case adversarial noise. This advances scenery reconstruction in RWRS contexts and links ergodic-theoretic insights (Kalikow-type arguments) with concrete finite-record recovery guarantees, with potential implications for isomorphism-invariant entropy questions and VWB-type analyses.

Abstract

Consider a simple random walk on with a random coloring of . Look at the sequence of the first steps taken in the random walk, together with the colors of the visited locations. We call this the record. From the record one can deduce the coloring of of the interval in that was visited, which is of size approximately . This is called scenery reconstruction. Now suppose that an adversary may change entries in the record that was obtained. What can be deduced from the record about the scenery now? In this paper we show that it is likely that we can still reconstruct a large part of the scenery. More precisely, we show that for any , there are and such that if and then with probability , the walk is such that we can reconstruct the coloring of integers, up to a number of possible reconstructions that is less than , where is the number of integers whose color we reconstruct.

Paper Structure

This paper contains 25 sections, 23 theorems, 166 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Overview of the Proof
  4. Construction of the Set W s
  5. Abnormally Long Intervals
  6. Probability of spending too much time in $\mathop{\mathrm{BAD}}\nolimits_{m,0}^{\mathrm{upper}}\left(\omega\right)$ and $\mathop{\mathrm{BAD}}\nolimits_{m,m-1}^{\mathrm{redUpper}}\left(\omega\right)$
  7. Abnormally Short Intervals
  8. Probability of spending too much time in $\mathop{\mathrm{BAD}}\nolimits_{m,0}^{\mathrm{lower}}\left(\omega\right)$ and $\mathop{\mathrm{BAD}}\nolimits_{m,m-1}^{\mathrm{redLower}}\left(\omega\right)$
  9. Intervals of Abnormal Length
  10. Probability of spending too much time in $\mathop{\mathrm{BAD}}\nolimits_{m,0}^{\mathrm{length}}\left(\omega\right)$ and $\mathop{\mathrm{BAD}}\nolimits_{m,m-1}^{\mathrm{redLength}}\left(\omega\right)$
  11. Intervals Where the Random Walk Induces Atypical Local Time Measure
  12. Probability of spending too much time in $\mathop{\mathrm{BAD}}\nolimits_{m,m-1}^{\mathrm{local}}\left(\omega\right)$
  13. Probability of spending too much time in $\mathop{\mathrm{BAD}}\nolimits_{m,0}\left(\omega\right)$
  14. Bounding the probability that $\omega\in\mathcal{R}_{\ge}^{\mathrm{all}}$
  15. Explicit choice of the parameters
  16. ...and 10 more sections

Key Result

Theorem 1.8

Let $p,\epsilon>0$, $\theta<\frac{1}{2}$. There exist $N_{0}$ and $\delta>0$ such that for any $N>N_{0}$ there exist: so that the following holds. If $x=\left(\left(\omega\left(t\right),\sigma\left(X^{\omega}_{t}\right)\right)\right)_{t=0}^{N-1}\in\left\{\pm 1\right\}^{N}\times C^{N}$, where $\omega\in\mathcal{W}_{\mathrm{s}}$ and $\sigma\in C^{\mathbb{Z}}$, and $x'\in\left\{\pm 1\right\}^{N}\tim

Figures (5)

  • Figure 1.1: The function $\mathrm{next}$ from Definition \ref{['defn:next']}
  • Figure 2.1: The reduced random walk from Definitions \ref{['fig:red_walk']}.
  • Figure 2.2: Illustration of Definition \ref{['def:badb_asterix_at_levels']}.
  • Figure 2.3: Another illustration of Definition \ref{['def:badb_asterix_at_levels']}.
  • Figure 2.4: Idea of the proof of Lemma \ref{['lem:bound_on_badb_uppers_volume']}

Theorems & Definitions (72)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Theorem 1.8: Main Theorem
  • Definition 2.1
  • Definition 2.2
  • ...and 62 more