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Minkowski sum of fractal percolation and random sets

Tianyi Bai, Xinxin Chen, Yuval Peres

Abstract

In this paper, we prove that hitting probability of Minkowski sum of fractal percolations can be characterized by capacity. Then we extend this result to Minkowski sum of general random sets in $\mathbb Z^d$, including ranges of random walks and critical branching random walks, whose hitting probabilities are described by Newtonian capacity individually.

Minkowski sum of fractal percolation and random sets

Abstract

In this paper, we prove that hitting probability of Minkowski sum of fractal percolations can be characterized by capacity. Then we extend this result to Minkowski sum of general random sets in , including ranges of random walks and critical branching random walks, whose hitting probabilities are described by Newtonian capacity individually.

Paper Structure

This paper contains 11 sections, 13 theorems, 119 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:main']}
  3. Percolation on tree
  4. Lower bound of Theorem \ref{['thm:main']}
  5. Relation with Markov chain
  6. Upper bound of Theorem \ref{['thm:main']}
  7. Applications
  8. Continuous case, proof of Corollary \ref{['cor:main_continuous']}
  9. Changing parameter in capacity, proof of Corollary \ref{['cor:cap_a_b']}
  10. Consequences for sum of general random sets, proof of Theorem \ref{['thm:rw+rw']}
  11. Remark on the case in

Key Result

Theorem 1.1

Let $d,k\ge 1$, let $\alpha_1,\beta_1,\dots,\alpha_k,\beta_k\in(0,d)$ such that If $\mathcal{R}_1,\dots,\mathcal{R}_k$ are independent random sets in $\mathbb Z^d$ with the following property: for every finite set $A\subset\mathbb Z^d$ containing the origin and every $|x|\ge 2\mathrm{diam}(A)$, we have Then for every $|x|$ sufficiently large, Here $\asymp$ may rely on $d,k,(\alpha_{i}),(\beta_{

Figures (7)

  • Figure 1: Demonstration for construction of fractal percolation. The continuous version is limit of (union of closure of) $Z_k$, and the discrete version is collection of lattice points in the rescaled $2^kZ_k$.
  • Figure 2: An illustration of $Q_k(p)$ and its corresponding percolation on $\Gamma^{(k)}$.
  • Figure 3: Illustration for Lemma \ref{['lem:xy_distance']}. On the tree $\Gamma^{(k)}$, two points corresponding to $x,y$ has common ancestor of height $h$. In $\mathbb Z^d$, $x,y$ fall into a cube congruent to $\Delta_h$.
  • Figure 4: In this picture we list $(u_i)$ for $\Gamma^{(2)}$ from left to right. Here the colored paths correspond to $Y^{(2)}_2=(1,1), Y^{(2)}_6=(0,1), Y^{(2)}_{7}=(0,0)$.
  • Figure 5: Illustration for $h^m_{ii'}$, $u_i$, and $u_{i'}$.
  • ...and 2 more figures

Theorems & Definitions (31)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 1.4
  • Theorem 1.5
  • Remark 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Definition 2.1
  • Lemma 2.2
  • ...and 21 more