THE WIRED MINIMAL SPANNING FOREST ON THE POISSON-WEIGHTED INFINITE TREE

Asaf Nachmias; Pengfei Tang

doi:10.1214/23-AAP2027

THE WIRED MINIMAL SPANNING FOREST ON THE POISSON-WEIGHTED INFINITE TREE

Asaf Nachmias, Pengfei Tang

School of Mathematical Sciences

Tel Aviv University

Research output: Contribution to journal › Article › peer-review

Abstract

We study the spectral and diffusive properties of the wired minimal spanning forest (WMSF) on the Poisson-weighted infinite tree (PWIT). Let M be the tree containing the root in the WMSF on the PWIT and (Y_n)_n≥0 be a simple random walk on M starting from the root. We show that almost surely M has P[Y_2n = Y₀] = n^-3/4+o(1) and dist(Y₀, Y_n) = n^1/4+o(1) with high probability. That is, the spectral dimension of M is 3/2 and its typical displacement exponent is 1/4, almost surely. These confirm Addario–Berry’s predictions (Addario-Berry (2013)).

Original language	English
Pages (from-to)	2415-2446
Number of pages	32
Journal	Annals of Applied Probability
Volume	34
Issue number	2
DOIs	https://doi.org/10.1214/23-AAP2027
State	Published - Apr 2024

Keywords

Minimal spanning tree
Poisson-weighted infinite tree
local limit
spectral dimension
wired minimal spanning forest

All Science Journal Classification (ASJC) codes

Statistics and Probability
Statistics, Probability and Uncertainty

Access to Document

10.1214/23-AAP2027

Cite this

@article{ba2a787a4dcd4dff92f5c27bb0ea6676,

title = "THE WIRED MINIMAL SPANNING FOREST ON THE POISSON-WEIGHTED INFINITE TREE",

abstract = "We study the spectral and diffusive properties of the wired minimal spanning forest (WMSF) on the Poisson-weighted infinite tree (PWIT). Let M be the tree containing the root in the WMSF on the PWIT and (Yn)n≥0 be a simple random walk on M starting from the root. We show that almost surely M has P[Y2n = Y0] = n-3/4+o(1) and dist(Y0, Yn) = n1/4+o(1) with high probability. That is, the spectral dimension of M is 3/2 and its typical displacement exponent is 1/4, almost surely. These confirm Addario–Berry{\textquoteright}s predictions (Addario-Berry (2013)).",

keywords = "Minimal spanning tree, Poisson-weighted infinite tree, local limit, spectral dimension, wired minimal spanning forest",

author = "Asaf Nachmias and Pengfei Tang",

note = "Publisher Copyright: {\textcopyright} Institute of Mathematical Statistics, 2024.",

year = "2024",

month = apr,

doi = "10.1214/23-AAP2027",

language = "الإنجليزيّة",

volume = "34",

pages = "2415--2446",

journal = "Annals of Applied Probability",

issn = "1050-5164",

publisher = "Institute of Mathematical Statistics",

number = "2",

}

TY - JOUR

T1 - THE WIRED MINIMAL SPANNING FOREST ON THE POISSON-WEIGHTED INFINITE TREE

AU - Nachmias, Asaf

AU - Tang, Pengfei

N1 - Publisher Copyright: © Institute of Mathematical Statistics, 2024.

PY - 2024/4

Y1 - 2024/4

N2 - We study the spectral and diffusive properties of the wired minimal spanning forest (WMSF) on the Poisson-weighted infinite tree (PWIT). Let M be the tree containing the root in the WMSF on the PWIT and (Yn)n≥0 be a simple random walk on M starting from the root. We show that almost surely M has P[Y2n = Y0] = n-3/4+o(1) and dist(Y0, Yn) = n1/4+o(1) with high probability. That is, the spectral dimension of M is 3/2 and its typical displacement exponent is 1/4, almost surely. These confirm Addario–Berry’s predictions (Addario-Berry (2013)).

AB - We study the spectral and diffusive properties of the wired minimal spanning forest (WMSF) on the Poisson-weighted infinite tree (PWIT). Let M be the tree containing the root in the WMSF on the PWIT and (Yn)n≥0 be a simple random walk on M starting from the root. We show that almost surely M has P[Y2n = Y0] = n-3/4+o(1) and dist(Y0, Yn) = n1/4+o(1) with high probability. That is, the spectral dimension of M is 3/2 and its typical displacement exponent is 1/4, almost surely. These confirm Addario–Berry’s predictions (Addario-Berry (2013)).

KW - Minimal spanning tree

KW - Poisson-weighted infinite tree

KW - local limit

KW - spectral dimension

KW - wired minimal spanning forest

UR - http://www.scopus.com/inward/record.url?scp=85189834728&partnerID=8YFLogxK

U2 - 10.1214/23-AAP2027

DO - 10.1214/23-AAP2027

M3 - مقالة

SN - 1050-5164

VL - 34

SP - 2415

EP - 2446

JO - Annals of Applied Probability

JF - Annals of Applied Probability

IS - 2

ER -

THE WIRED MINIMAL SPANNING FOREST ON THE POISSON-WEIGHTED INFINITE TREE

Abstract

Keywords

All Science Journal Classification (ASJC) codes

Access to Document

Other files and links

Cite this