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Expected First Return Times for Random Walks on Bounded Grids

Nan An

TL;DR

This paper develops a general formula for the expected first return time to a fixed vertex in a finite Markov chain by augmenting the chain with a waiting-room structure, yielding a block form $M=QR0T$ with a sub-stochastic $Q$ and a finite fundamental matrix $N=(I-Q)^{-1}$. The key result expresses the first-return time as $E = U_{o,o} + s N^2 (s')^T$, and the approach enables explicit, closed-form calculations on bounded rectangular grids under different boundary conditions. The authors derive concrete formulas for grid return times under periodic, absorbing-stay, and reflecting boundaries, illustrating how boundary effects alter $E$ and enabling efficient computation.

Abstract

We derive a general formula for computing the expected first return time of a random walk on a finite graph. Using this framework, we calculate the expected first return time in various settings over bounded rectangular grids with different boundary conditions.

Expected First Return Times for Random Walks on Bounded Grids

TL;DR

This paper develops a general formula for the expected first return time to a fixed vertex in a finite Markov chain by augmenting the chain with a waiting-room structure, yielding a block form with a sub-stochastic and a finite fundamental matrix . The key result expresses the first-return time as , and the approach enables explicit, closed-form calculations on bounded rectangular grids under different boundary conditions. The authors derive concrete formulas for grid return times under periodic, absorbing-stay, and reflecting boundaries, illustrating how boundary effects alter and enabling efficient computation.

Abstract

We derive a general formula for computing the expected first return time of a random walk on a finite graph. Using this framework, we calculate the expected first return time in various settings over bounded rectangular grids with different boundary conditions.
Paper Structure (3 sections, 6 theorems, 25 equations)

This paper contains 3 sections, 6 theorems, 25 equations.

Table of Contents

  1. Problem Setup
  2. The Waiting Room Construction
  3. Over Bounded Grids

Key Result

Proposition 2.2

The matrices $Q$, $R$, and $T$ arising from the "waiting room" construction satisfy:

Theorems & Definitions (14)

  • Definition 1.1
  • Definition 2.1: Waiting Room Construction
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof : Sketch of Proof
  • Proposition 2.5
  • proof
  • ...and 4 more