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Branching random walk in the presence of a hard wall

Rishideep Roy

Abstract

We consider a branching random walk on a $d$-ary tree of height $n$ ($n \in \mathbb{N}$), under the presence of a hard wall which restricts each value to be positive, where $d$ is a natural number satisfying $d\geqslant2$. The question of behaviour of Gaussian processes with long range interactions, for example the discrete Gaussian free field, under the condition that it is positive on a large subset of {\color{blue}vertices}, and a relation with the expected maximum of the processes has been observed. We find the probability of the event that the branching random {\color{blue}walk} is positive at every vertex in the $n^{th}$ generation, and show that the conditional expectation of the Gaussian variable at a typical vertex, under positivity, is less than the expected maximum by order of $\log n$.

Branching random walk in the presence of a hard wall

Abstract

We consider a branching random walk on a -ary tree of height (), under the presence of a hard wall which restricts each value to be positive, where is a natural number satisfying . The question of behaviour of Gaussian processes with long range interactions, for example the discrete Gaussian free field, under the condition that it is positive on a large subset of {\color{blue}vertices}, and a relation with the expected maximum of the processes has been observed. We find the probability of the event that the branching random {\color{blue}walk} is positive at every vertex in the generation, and show that the conditional expectation of the Gaussian variable at a typical vertex, under positivity, is less than the expected maximum by order of .

Paper Structure

This paper contains 6 sections, 11 theorems, 44 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Left tail of maximum of BRW
  3. Switching Sign Branching Random Walk
  4. Estimates on left tail and positivity
  5. Expected value of a typical vertex under positivity
  6. Acknowledgement

Key Result

Theorem 1.1

There exists $\lambda'=\frac{\sqrt{2}\log n}{\sqrt{\log d}}+O(1)$, such that for $n$ sufficiently large we have, for $K_1, K_2, K_3 > 0$ independent of $n$, and $K_4 = \frac{1}{c\sigma^2_{d,n}\log d}$,

Figures (3)

  • Figure 1: BRW on binary tree
  • Figure 2: Node of the varying part of SSBRW
  • Figure 3: SSBRW on Binary tree

Theorems & Definitions (23)

  • Theorem 1.1: Positivity probability
  • Theorem 1.2: Expected value
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • ...and 13 more