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A branching random walk in the presence of a hard wall

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  22 May 2023

Rishideep Roy Open the ORCID record for Rishideep Roy [Opens in a new window]
Rishideep Roy*
Affiliation:
IIM Bangalore
*
*Postal address: IIM Bangalore, Bannerghatta Road, Bangalore 560076, India. Email rishideeproy@gmail.com
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Abstract

We consider a branching random walk on a d-ary tree of height n ($n \in \mathbb{N}$), in the presence of a hard wall which restricts each value to be positive, where d is a natural number satisfying $d\geqslant2$. We consider the 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 vertices. We observe a relation with the expected maximum of the processes. We find the probability of the event that the branching random walk is positive at every vertex in the nth 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 $\log n$.

Keywords

Branching random walkextrema of Gaussian processeslog-correlated fieldsentropic repulsion

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G15: Gaussian processes 60G70: Extreme value theory; extremal processes

Information

Type
Original Article
Information
Journal of Applied Probability , Volume 61 , Issue 1 , March 2024 , pp. 1 - 17
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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