Geometry of the vacant set left by random walk on random graphs, Wright's constants, and critical random graphs with prescribed degrees

Bhamidi, Shankar; Sen, Sanchayan

doi:10.1002/rsa.20880

Cited by 15 publications

(31 citation statements)

References 66 publications

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“…(See also [22], where a similar result for critical percolation on the supercritical configuration model was derived as an application of a more general universality principle.) The next result gives a variant of [26,Theorem 2.2]. This result follows from arguments similar to those used in [26].…”

Section: Geometry Of Critical Random Graphsmentioning

confidence: 67%

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Geometry of the minimal spanning tree of a random 3-regular graph

Addario‐Berry

Sen

2021

Probab. Theory Relat. Fields

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The global structure of the minimal spanning tree (MST) is expected to be universal for a large class of underlying random discrete structures. However, very little is known about the intrinsic geometry of MSTs of most standard models, and so far the scaling limit of the MST viewed as a metric measure space has only been identified in the case of the complete graph (Addario-Berry et al. in Ann Probab 45(5):3075-3144, 2017). In this work, we show that the MST constructed by assigning i.i.d. continuous edge weights to either the random (simple) 3-regular graph or the 3-regular configuration model on n vertices, endowed with the tree distance scaled by n −1/3 and the uniform probability measure on the vertices, converges in distribution with respect to Gromov-Hausdorff-Prokhorov topology to a random compact metric measure space. Further, this limiting space has the same law as the scaling limit of the MST of the complete graph identified in up to a scaling factor of 6 1/3 . Our proof relies on a novel argument that proceeds via a comparison between a 3-regular configuration model and the largest component in the critical Erdős-Rényi random graph. The techniques of this paper can be used to establish the scaling limit of the MST in the setting of general random graphs with given degree sequences provided two additional technical conditions are verified.

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Section: Geometry Of Critical Random Graphsmentioning

confidence: 67%

“…The next result gives a variant of [26,Theorem 2.2]. This result follows from arguments similar to those used in [26]. A sketch of proof is given in Sect.…”

Section: Geometry Of Critical Random Graphsmentioning

confidence: 77%

Geometry of the minimal spanning tree of a random 3-regular graph

Addario‐Berry

Sen

2021

Probab. Theory Relat. Fields

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“…(1) In the last few years, a host of random graph models have been shown to belong to the Erdős-Rényi or more precisely, the multiplicative coalescent universality class [6,7,9,10,12,13,20,25,29,35]. This includes the configuration model [14,34], a large subclass of the inhomogeneous random graph models as formulated in [16], and the so-called bounded size rules [36].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

A probabilistic approach to the leader problem in random graphs

Addario‐Berry

Bhamidi

Sen

2020

Random Struct Algorithms

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We study the fixation time of the identity of the leader, that is, the most massive component, in the general setting of Aldous's multiplicative coalescent, which in an asymptotic sense describes the evolution of the component sizes of a wide array of near-critical coalescent processes, including the classical Erdős-Rényi process. We show tightness of the fixation time in the "Brownian" regime, explicitly determining the median value of the fixation time to within an optimal O(1) window. This generalizes Łuczak's result for the Erdős-Rényi random graph using completely different techniques. In the heavy-tailed case, in which the limit of the component sizes can be encoded using a thinned pure-jump Lévy process, we prove that only one-sided tightness holds. This shows a genuine difference in the possible behavior in the two regimes.

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“…The continuum scaling limit of the metric structure of critical random graphs is studied in the "BMPD" universality class in [1,12,13] and in the "Lévy processes without replacement" universality class in [14].…”

Section: 43mentioning

confidence: 99%

Rigid representations of the multiplicative coalescent with linear deletion

Martin¹,

Ráth²

2017

Electron. J. Probab.

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We introduce the multiplicative coalescent with linear deletion, a continuoustime Markov process describing the evolution of a collection of blocks. Any two blocks of sizes x and y merge at rate xy, and any block of size x is deleted with rate λx (where λ ≥ 0 is a fixed parameter). This process arises for example in connection with a variety of random-graph models which exhibit self-organised criticality. We focus on results describing states of the process in terms of collections of excursion lengths of random functions. For the case λ = 0 (the coalescent without deletion) we revisit and generalise previous works by authors including Aldous, Limic, Armendariz, Uribe Bravo, and Broutin and Marckert, in which the coalescence is related to a "tilt" of a random function, which increases with time; for λ > 0 we find a novel representation in which this tilt is complemented by a "shift" mechanism which produces the deletion of blocks. We describe and illustrate other representations which, like the tilt-and-shift representation, are "rigid", in the sense that the coalescent process is constructed as a projection of some process which has all of its randomness in its initial state. We explain some applications of these constructions to models including mean-field forest-fire and frozen-percolation processes.RIGID REPRESENTATIONS OF THE MULTIPLICATIVE COALESCENT WITH DELETION 2 Proposition 1.2 (Aldous). There exists a MC process (m t , t ∈ R) such that for each t, the marginal distribution m t is the same as that of E ↓ (h t ) where h t ∼ BMPD(t).This process is known as the standard multiplicative coalescent. In her PhD thesis, Armendáriz [7,8] showed that in fact the whole process can be constructed as a function of a single realisation of Brownian motion. Proposition 1.3 (Armendáriz). Let h 0 ∼ BMPD(0), and define h t (x) = h 0 (x) + tx for all t ∈ R (so that h t ∼ BMPD(t) for all t). Then the process E ↓ (h t ), t ∈ R is the standard MC.

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Geometry of the vacant set left by random walk on random graphs, Wright's constants, and critical random graphs with prescribed degrees

Cited by 15 publications

References 66 publications

Geometry of the minimal spanning tree of a random 3-regular graph

Geometry of the minimal spanning tree of a random 3-regular graph

A probabilistic approach to the leader problem in random graphs

Rigid representations of the multiplicative coalescent with linear deletion

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