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Phase transition on the fluctuation of the structure of random walk ranges

Arka Adhikari, Izumi Okada

TL;DR

This work analyzes fluctuations of the random walk range taken as a random medium by studying graph distance, cut points, and effective resistance. A dimension-dependent phase transition emerges at six dimensions: for $d\ge 7$ the fluctuations satisfy a central limit theorem with standard scaling, while at $d=6$ a nonstandard $\sqrt{n\log n}$-type scaling governs the normal limit along subsequences. For $d\le 5$, including $d=5$ and $d=4$, fluctuations are degenerate under the standard CLT normalization, with precise variance growth $\mathrm{Var}(X_n) \asymp n^{3/2}$ at $d=5$ and $\mathrm{Var}(X_n) \asymp n^2/(\log n)^{2+o(1)}$ at $d=4$, driven by heavy-tailed cross-terms $E_n$ arising from a path decomposition. The paper develops a refined decomposition of the random walk path into mesoscopic blocks and cross-terms, and shows that the dominant contributions switch from i.i.d.-type terms in higher dimensions to cross-term effects in lower dimensions, yielding the observed phase transitions for both the graph distance, cut points, and the effective resistance. These results deepen understanding of transport in random media and highlight the delicate interplay between geometry and randomness in high-dimensional stochastic systems.

Abstract

We investigate fluctuation phenomena for the graph distance and the number of cut points associated with random media arising from the range of a random walk. Our results demonstrate a sequence of dimension-dependent phase transitions in the scaling behavior of these fluctuations, leading to qualitatively different regimes across dimensions lower than six, equal to six, and higher than six. In particular, in six dimensions, we show that the convergence to the normal distribution occurs under a scaling different from that of the standard central limit theorem. We also give the related results for the effective resistance of the random walk range.

Phase transition on the fluctuation of the structure of random walk ranges

TL;DR

This work analyzes fluctuations of the random walk range taken as a random medium by studying graph distance, cut points, and effective resistance. A dimension-dependent phase transition emerges at six dimensions: for the fluctuations satisfy a central limit theorem with standard scaling, while at a nonstandard -type scaling governs the normal limit along subsequences. For , including and , fluctuations are degenerate under the standard CLT normalization, with precise variance growth at and at , driven by heavy-tailed cross-terms arising from a path decomposition. The paper develops a refined decomposition of the random walk path into mesoscopic blocks and cross-terms, and shows that the dominant contributions switch from i.i.d.-type terms in higher dimensions to cross-term effects in lower dimensions, yielding the observed phase transitions for both the graph distance, cut points, and the effective resistance. These results deepen understanding of transport in random media and highlight the delicate interplay between geometry and randomness in high-dimensional stochastic systems.

Abstract

We investigate fluctuation phenomena for the graph distance and the number of cut points associated with random media arising from the range of a random walk. Our results demonstrate a sequence of dimension-dependent phase transitions in the scaling behavior of these fluctuations, leading to qualitatively different regimes across dimensions lower than six, equal to six, and higher than six. In particular, in six dimensions, we show that the convergence to the normal distribution occurs under a scaling different from that of the standard central limit theorem. We also give the related results for the effective resistance of the random walk range.
Paper Structure (25 sections, 31 theorems, 260 equations)

This paper contains 25 sections, 31 theorems, 260 equations.

Table of Contents

  1. Random walk on random walk range
  2. Preparations in $d\ge 5$
  3. Proof of CLT for $d\ge 7$
  4. Proof of variance
  5. Proof of CLT
  6. 6 Dimensions
  7. Lower and Upper Bounds on the Variance
  8. Graph distance and cut points
  9. Effective resistance
  10. The Limit in Distribution of $\frac{X_n -\mathbb{E}[X_n]}{\sqrt{\mathrm{Var}(X_n)}}$
  11. Graph distance and cut points
  12. Effective resistance
  13. The fluctuations converge to 0 in dimension $d=5$
  14. The fluctuations converge to 0 in dimension $d=4$
  15. Lower bounds for the Cross Term
  16. ...and 10 more sections

Key Result

Proposition 1.1

It holds that, as $n\to \infty$, for some constant $\alpha^i_d >0$ for $i=1,2$.

Theorems & Definitions (69)

  • Proposition 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Remark 1.6
  • Lemma 2.1
  • Remark 2.2
  • proof : Proof of \ref{['upper+']}
  • proof : Proof of \ref{['lower+']}
  • Remark 2.3
  • ...and 59 more