Existence and Coexistence in First-Passage Percolation

Ahlberg, Daniel

doi:10.1007/978-3-030-60754-8_1

Cited by 4 publications

(15 citation statements)

References 23 publications

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“…Our proof of Theorem 1.1 relies on the other hand on a covariance formula derived in [1], and requires no precise model specific estimates, which we emphasise by working outside of the exactly solvable setting. The covariance formula alluded to concerns the function Q t :" ErT 0 T t s defined for t P r0, 1s.…”

Section: Discussionmentioning

confidence: 99%

“…The remainder of the paper is organized as follows. In Section 2 we describe a covariance formula from our companion paper [1]. In Section 3 we derive lower and upper bounds on the influence of a vertex in terms of the probability that the vertex belongs to the geodesic both at time 0 and at time t. These bounds are then combined with the covariance formula in the proof of Theorem 1.1 Section 4.…”

Section: Discussionmentioning

confidence: 99%

“…While the methods of [6] are not specific to the Gaussian setting, they are applied to eigenvalues and the corresponding vectors of random matrices, which is a well understood topic. In a companion paper [1], our goal has been to show that the equivalence between superconcentration and chaos is part of a more general principle, achieved by establishing the connection in the context of first-passage percolation. In the current paper we proceed in this vein, and determine the precise location of the transition from stability to chaos in non-Gaussian and non-integrable models of last-passage percolation.…”

Section: Department Of Mathematics Stockholm University Swedenmentioning

confidence: 99%

“…More recently, the precise location of the transition from stability to chaos has been established for the top eigenvector of Wigner matrices by Bordenave-Lugosi-Zhivotovskiy [6] and in the context of Brownian last-passage percolation by Ganguly-Hammond [10,11]. 1…”

Section: Introduction 1motivation and Backgroundmentioning

confidence: 99%

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Competing first passage percolation on random graphs with finite variance degrees

Ahlberg

Deijfen

Janson

2019

Random Struct Algorithms

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We study the growth of two competing infection types on graphs generated by the configuration model with a given degree sequence. Starting from two vertices chosen uniformly at random, the infection types spread via the edges in the graph in that an uninfected vertex becomes type 1 (2) infected at rate 1 ( 2 ) times the number of nearest neighbors of type 1 (2). Assuming (essentially) that the degree of a randomly chosen vertex has finite second moment, we show that if 1 = 2 , then the fraction of vertices that are ultimately infected by type 1 converges to a continuous random variable V ∈ (0, 1), as the number of vertices tends to infinity. Both infection types hence occupy a positive (random) fraction of the vertices. If 1 ≠ 2 , on the other hand, then the type with the larger intensity occupies all but a vanishing fraction of the vertices. Our results apply also to a uniformly chosen simple graph with the given degree sequence.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Department Of Mathematics Stockholm University Swedenmentioning

confidence: 99%

Section: Introduction 1motivation and Backgroundmentioning

confidence: 99%

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Competing first passage percolation on random graphs with finite variance degrees

Ahlberg

Deijfen

Janson

2019

Random Struct Algorithms

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“…One important problem is the behavior of the model under perturbations: how does the structure of the metric space change under resampling of the random environment? The reader is directed to, for example, [2,3,16], for work on this question.…”

Section: Introduction 1backgroundmentioning

confidence: 99%

Transitions for exceptional times in dynamical first-passage percolation

Damron

Hanson

Harper

et al. 2022

Probab. Theory Relat. Fields

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We study first-passage percolation (FPP) on the square lattice. The model is defined using i.i.d. nonnegative random edge-weights (t e ) associated to the nearest neighbor edges of Z 2 . The passage time between vertices x and y, T (x, y), is the minimal total weight of any lattice path from x to y. The growth rate of T (x, y) depends on the value of F (0) = P(t e = 0): if F (0) < 1/2 then T (x, y) grows linearly in |x − y|, but if F (0) > 1/2 then it is stochastically bounded. In the critical case, where F (0) = 1/2, T (x, y) can be bounded or unbounded depending on the behavior of the distribution function F of t e near 0. In this paper, we consider the critical case in which T (x, y) is unbounded and prove the existence of an incipient infinite cluster (IIC) type measure, constructed by conditioning the environment on the event that the passage time from 0 to a far distance remains bounded. This IIC measure is a natural candidate for the distribution of the weights at a typical exceptional time in dynamical FPP. A major part of the analysis involves characterizing the limiting behavior of independent nonnegative random variables conditioned to have small sum. We give conditions on random variables that ensure that such limits are trivial, and several examples that exhibit nontrivial limits.

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Annihilating Branching Brownian Motion

Ahlberg,

Angel,

Kolesnik

2024

International Mathematics Research Notices

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We study an interacting system of competing particles on the real line. Two populations of positive and negative particles evolve according to branching Brownian motion. When opposing particles meet, their charges neutralize and the particles annihilate, as in an inert chemical reaction. We show that, with positive probability, the two populations coexist and that, on this event, the interface is asymptotically linear with a random slope. A variety of generalizations and open problems are discussed.

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Existence and Coexistence in First-Passage Percolation

Cited by 4 publications

References 23 publications

Competing first passage percolation on random graphs with finite variance degrees

Competing first passage percolation on random graphs with finite variance degrees

Transitions for exceptional times in dynamical first-passage percolation

Annihilating Branching Brownian Motion

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