Forests on wired regular trees

Ray, Gourab; Xiao, Ben

doi:10.48550/arxiv.2108.04287

Cited by 2 publications

(2 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Limits of the RCM giving rise to measures which are supported on forests are usually referred to as arboreal gases. Most of the literature on arboreal gas models is focused on unrooted forests; see, e.g., [11,12,13,34,18] for some very recent mathematical work in this direction. In [12, Appendix A] the authors consider what they call arboreal gas with an external field and they notice that this can be interpreted as a marginal of a measure over rooted spanning forests.…”

Section: Relations With the Random-cluster Modelmentioning

confidence: 99%

Loop-erased partitioning of a graph: mean-field analysis

Avena

Gaudillière

Milanesi

et al. 2022

Electron. J. Probab.

View full text Add to dashboard Cite

Given a weighted finite graph G, we consider a random partition of its vertex set induced by a measure on spanning rooted forests on G. The latter is a generalized parametric version of the classical Uniform Spanning Tree measure which can be sampled using loop-erased random walks stopped at a random independent exponential time of parameter q > 0. The related random trees-identifying the blocks of the partition-tend to cluster nodes visited by the random walk on time scale 1/q.We explore the emerging macroscopic structure by analyzing two-point correlations, as a function of the tuning parameter q. To this aim, it is defined an interaction potential between pair of vertices, as the probability that they do not belong to the same block. This interaction potential can be seen as an affinity measure for "densely connected nodes" and capture well-separated regions in network models presenting non-homogeneous landscapes. In this spirit, we compute this potential and its scaling limits on a complete graph and on a non-homogeneous weighted version with community structure. For such geometries we show phase-transitions in the behavior of the random partition as a function of the tuning parameter and the edge weights. Moreover, as a corollary of our main results, we infer the right scaling of the parameters that give rise to the emergence of "giant" blocks.

show abstract

Section: Relations With the Random-cluster Modelmentioning

confidence: 99%

Loop-erased partitioning of a graph: mean-field analysis

Avena

Gaudillière

Milanesi

et al. 2022

Electron. J. Probab.

View full text Add to dashboard Cite

show abstract

“…Rather, its supercritical phase behaves like a critical model off the giant. This can again be given a more precise formulation on the complete graph K N and on the wired regular tree, where detailed results are known [38,50,53,61]. In particular, the exact cluster distribution can be determined: on K N in the supercritical phase there is a unique giant tree, and an unbounded number of trees of size Θ(N 2/3 ).…”

Section: Nonetheless Many Open Questions Beckonmentioning

confidence: 99%

Spin systems with hyperbolic symmetry: a survey

Bauerschmidt,

Helmuth

2021

Preprint

View full text Add to dashboard Cite

Spin systems with hyperbolic symmetry originated as simplified models for the Anderson metal-insulator transition, and were subsequently found to exactly describe probabilistic models of linearly reinforced walks and random forests. In this survey we introduce these models, discuss their origins and main features, some existing tools available for their study, recent probabilistic results, and relations to other well-studied probabilistic models. Along the way we discuss some of the (many) open questions that remain.

show abstract

Forests on wired regular trees

Cited by 2 publications

References 6 publications

Loop-erased partitioning of a graph: mean-field analysis

Loop-erased partitioning of a graph: mean-field analysis

Spin systems with hyperbolic symmetry: a survey

Contact Info

Product

Resources

About