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Strassen's LIL and a Phase transition for the capacity of the random walk under diameter constraints

Arka Adhikari, Izumi Okada

TL;DR

The paper analyzes the capacity of the three-dimensional simple random walk, focusing on Strassen-type laws for the capacity of the walk’s range and on how constraining the walk’s diameter induces a phase transition in the limsup behavior of the capacity. The authors introduce functional limit sets ${\mathcal S}$ and ${\mathcal K}$ via normalized capacity processes $f_n$ and $g_n$, establishing Strassen-like LIL results in $d=3$ and linking these limits to analytic objects like the Sobolev unit ball. They then show that conditioning on the diameter of the walk yields three distinct regimes, depending on a scaling parameter $k_n$ relative to a critical scale $j_3(n)$: (i) string-like, boundary-spreading configurations for $k_n\to\infty$, (ii) a sparse-surface phase for $k_n\asymp1$, and (iii) dense-sphere-wrapping when $k_n\to0$, each with corresponding limsup behavior and optimal strategies. The analysis combines Strassen-type functional limit techniques with delicate Green’s-function estimates, Brownian coupling, and Wiener sausage large-deviation principles to obtain both upper and lower bounds for multi-point capacity events, revealing deep connections between capacity geometry and path structure and suggesting extensions to more general domains beyond diameter constraints.

Abstract

We discuss the relationship between the capacity and the geometry for the range of the random walk for $d=3$. In particular, we consider how efficiently the random walk moves or what shape it forms in order to maximize its capacity. In one of our main results, we show a functional law for the capacity of the random walk. In addition, we find that there is a phase transition for the asymptotics of the capacity of the random walk when we condition the diameter of the random walk.

Strassen's LIL and a Phase transition for the capacity of the random walk under diameter constraints

TL;DR

The paper analyzes the capacity of the three-dimensional simple random walk, focusing on Strassen-type laws for the capacity of the walk’s range and on how constraining the walk’s diameter induces a phase transition in the limsup behavior of the capacity. The authors introduce functional limit sets and via normalized capacity processes and , establishing Strassen-like LIL results in and linking these limits to analytic objects like the Sobolev unit ball. They then show that conditioning on the diameter of the walk yields three distinct regimes, depending on a scaling parameter relative to a critical scale : (i) string-like, boundary-spreading configurations for , (ii) a sparse-surface phase for , and (iii) dense-sphere-wrapping when , each with corresponding limsup behavior and optimal strategies. The analysis combines Strassen-type functional limit techniques with delicate Green’s-function estimates, Brownian coupling, and Wiener sausage large-deviation principles to obtain both upper and lower bounds for multi-point capacity events, revealing deep connections between capacity geometry and path structure and suggesting extensions to more general domains beyond diameter constraints.

Abstract

We discuss the relationship between the capacity and the geometry for the range of the random walk for . In particular, we consider how efficiently the random walk moves or what shape it forms in order to maximize its capacity. In one of our main results, we show a functional law for the capacity of the random walk. In addition, we find that there is a phase transition for the asymptotics of the capacity of the random walk when we condition the diameter of the random walk.

Paper Structure

This paper contains 20 sections, 34 theorems, 317 equations.

Table of Contents

  1. Introduction and Main results
  2. Strassen's LIL for the $\limsup$ of a Random Walk Range:Notation and Estimates
  3. Key proposition
  4. Deriving Upper Bounds on the Capacity in Proposition \ref{['mainpro1']}
  5. Deriving Lower Bounds in Proposition \ref{['mainpro1']}
  6. Strassen's LIL for the $\limsup$ of a Random Walk Range: Proofs
  7. Elements not in $\mathcal{S}$ are not points of $C(\{f_n\})$
  8. Elements in $\mathcal{S}$ are points in $C(\{f_n\})$
  9. $\liminf$ estimates for $R_n$
  10. Lower bound estimates on $\mathbb{P}( \tilde{E}^{\mathcal{A},\delta}_{n,\mathcal{T}})$
  11. Upper bound estimates on $\mathbb{P}( \tilde{E}^{\mathcal{A},\delta}_{n,\mathcal{T}})$
  12. Proof of Theorem \ref{['m2']}
  13. Proof of (i) in Theorem \ref{['m2']}
  14. Preliminaries
  15. Necessary Notation
  16. ...and 5 more sections

Key Result

Theorem 1

For $d=3$, almost surely, as $n \to \infty$,

Theorems & Definitions (72)

  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Proposition 1
  • Proposition 2
  • proof : Proof of Proposition \ref{['mainpro1']} given Proposition \ref{['prop:upperboundcap']}
  • Definition 2.1
  • Remark 2.2
  • Lemma 1
  • proof
  • ...and 62 more