arXiv subjects

Daniel Ahlberg

Publications and source records attributed to Daniel Ahlberg.

At least 19 recordsLinked to original sources

arXiv2026

Surviving from the tip of a cone in competing first-passage percolation

In two-type first passage percolation on $\mathbb{Z}^2$, two entities compete to capture the sites of the lattice. The entities spread between nearest neighbor sites at times specified by random passage times associated with the edges. We consider the case when both types have the same passage time distribution, with one type starting at the origin and the other from an infinite cone with tip at the origin and pointing in direction $\theta$. Itai Benjamini has suggested that the type starting at the origin can grow unboundedly if and only if the slope of the cone is strictly smaller than $\pi/2$, so that the cone does not fill a whole half-plane. The main result is that this is correct for any $\theta$ such that the asymptotic shape of the one-type process has a tangent line with direction $\theta$. The proofs are based on a description of infinite time-minimizing paths in terms of Busemann functions together with local modification arguments.

math.PR↗

arXiv2026

Noise sensitivity in last-passage percolation

The study of noise sensitivity of Boolean functions was initiated in a seminal paper of Benjamini, Kalai and Schramm, published in 1999. While this study has revealed fascinating phenomena in the context of Bernoulli percolation, few results have been obtained regarding other random spatial processes. In this paper we prove the first instance of noise sensitivity for a spatial growth process associated to the Kardar-Parisi-Zhang class of universality. More specifically, we show that travel times in geometric last-passage percolation are noise sensitive with respect to a perturbation acting on a Bernoulli encoding of the geometric weights. Our method of proof includes a generalisation of the celebrated Benjamini-Kalai-Schramm noise sensitivity/influence theorem, and precise bounds on the probability of a given vertex being on a geodesic, which we believe to be of independent interest.

math.PR↗

arXiv2025

Gaussian fluctuations in mean field stable matchings

Two sets of objects of size $n$ are to be matched to each other based on i.i.d. costs associated to every pair of objects. Objects prefer to be matched as cheaply as possible, and a matching is said to be stable if there is no pair of objects that would prefer to match to each other rather than to their current partners. Properties of such matchings are analysed for cost distributions with a density $\rho$ satisfying $\rho(x)/(dx^{d-1})\to 1$ as $x\to 0^+$, where the number $d$ is known as the pseudo-dimension. For $d>0$, the typical matching cost is shown to be of order $n^{-1/d}$, with an explicit distributional limit. For $d>1$ the total matching cost is shown to be of order $n^{1-1/d}$, and to obey a law of large numbers. For $d>2$ the fluctuations of the total matching cost are shown to be of order $n^{1/2-1/d}$, and to obey a central limit theorem.

math.PR↗

arXiv2024

Mean field stable matchings

Consider the complete bipartite graph on $n+n$ vertices where the edges are equipped with i.i.d. exponential costs. A matching of the vertices is stable if it does not contain any pair of vertices where the connecting edge is cheaper than both matching costs. There exists a unique stable matching obtained by iteratively pairing vertices with small edge costs. We show that the total cost $C_{n,n}$ of this matching is of order $\log n$ with bounded variance, and that $C_{n,n}-\log n$ converges to a Gumbel distribution. We also show that the typical cost of an edge in the matching is of order $1/n$, with an explicit density on this scale, and analyze the rank of a typical edge. These results parallel those of Aldous for the minimal cost matching in the same setting. We then consider the sensitivity of the matching and the matching cost to perturbations of the underlying edge costs. The matching itself is shown to be robust in the sense that two matchings based on largely identical edge costs will have a substantial overlap. The matching cost however is shown to be noise sensitive, as a result of the fact that the most expensive edges will with high probability be replaced after resampling. Our proofs also apply to the complete (unipartite) graph and the results in this case are qualitatively similar.

math.PR↗

arXiv2023

Annihilating branching Brownian motion

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.

math.PR↗

arXiv2023

Is 'being above the median' a noise sensitive property?

Assign independent weights to the edges of the square lattice, from the uniform distribution on $\{a,b\}$ for some $0<a<b<\infty$. The weighted graph induces a random metric on $\mathbb{Z}^2$. Let $T_n$ denote the distance between $(0,0)$ and $(n,0)$ in this metric. The distribution of $T_n$ has a well-defined median. Itai Benjamini asked in 2011 if the sequence of Boolean functions encoding whether $T_n$ exceeds its median is noise sensitive? In this paper we present the first progress on Benjamini's problem. More precisely, we study the minimal weight along any path crossing an $n\times n$-square horizontally and whose vertical fluctuation is smaller than $n^{1/22}$, and show that for this observable, 'being above the median' is a noise sensitive property.

math.PR↗

arXiv2023

Chaos, concentration and multiple valleys in first-passage percolation

A decade and a half ago Chatterjee established the first rigorous connection between anomalous fluctuations and a chaotic behaviour of the ground state in certain Gaussian disordered systems. The purpose of this paper is to show that Chatterjee's work gives evidence of a more general principle, by establishing an analogous connection between fluctuations and chaos in the context of first-passage percolation. The notion of `chaos' here refers to the sensitivity of the time-minimising path between two points when exposed to a slight perturbation. More precisely, we resample a small proportion of the edge weights, and find that a vanishing fraction of the edges on the time-minimising path still belongs to the time-minimising path obtained after resampling. We also identify the point at which the system transitions from being stable to being chaotic in terms of the variance of the system. Finally we show that the chaotic behaviour implies the existence of a large number of almost-optimal paths that are almost disjoint from the time-minimising path, a phenomenon known as 'multiple valleys'.

math.PR↗

arXiv2023

From stability to chaos in last-passage percolation

We study the transition from stability to chaos in a dynamic last passage percolation model on $\mathbb{Z}^d$ with random weights at the vertices. Given an initial weight configuration at time $0$, we perturb the model over time in such a way that the weight configuration at time $t$ is obtained by resampling each weight independently with probability $t$. On the cube $[0,n]^d$, we study geodesics, that is, weight-maximizing up-right paths from $(0,0, \dots, 0)$ to $(n,n, \dots, n)$, and their passage time $T$. Under mild conditions on the weight distribution, we prove a phase transition between stability and chaos at $t \asymp \frac{1}{n}\mathrm{Var}(T)$. Indeed, as $n$ grows large, for small values of $t$, the passage times at time $0$ and time $t$ are highly correlated, while for large values of $t$, the geodesics become almost disjoint.

math.PR↗

arXiv2022

The number of geodesics in planar first-passage percolation grows sublinearly

We study a random perturbation of the Euclidean plane, and show that it is unlikely that the distance-minimizing path between the two points can be extended into an infinite distance-minimizing path. More precisely, we study a large class of planar first-passage percolation models and show that the probability that a given site is visited by an infinite geodesic starting at the origin tends to zero uniformly with the distance. In particular, this show that the collection of infinite geodesics starting at the origin covers a negligible fraction of the plane. This provides the first progress on the `highways and byways' problem, posed by Hammersley and Welsh in the 1960s.

math.PR↗

arXiv2022

Multi-colour competition with reinforcement

We study a system of interacting urns where balls of different colour/type compete for their survival, and annihilate upon contact. For competition between two types, the underlying graph (finite and connected), determining the interaction between the urns, is known to be irrelevant for the possibility of coexistence, whereas for $K\ge3$ types the structure of the graph does affect the possibility of coexistence. We show that when the underlying graph is a cycle, competition between $K\ge3$ types almost surely has a single survivor, thus establishing a conjecture of Griffiths, Janson, Morris and the first author. Along the way, we give a detailed description of an auto-annihilative process on the cycle, which can be perceived as an expression of the geometry of a M\"obius strip in a discrete setting.

math.PR↗

arXiv2021

On the rate of convergence in quenched Voronoi percolation

Position $n$ points uniformly at random in the unit square $S$, and consider the Voronoi tessellation of $S$ corresponding to the set $\eta$ of points. Toss a fair coin for each cell in the tessellation to determine whether to colour the cell red or blue. Let $H_S$ denote the event that there exists a red horizontal crossing of $S$ in the resulting colouring. In 1999, Benjamini, Kalai and Schramm conjectured that knowing the tessellation, but not the colouring, asymptotically gives no information as to whether the event $H_S$ will occur or not. More precisely, since $H_S$ occurs with probability $1/2$, by symmetry, they conjectured that the conditional probabilities $\mathbb{P}(H_S|\eta)$ converge in probability to 1/2, as $n\to\infty$. This conjecture was settled in 2016 by Ahlberg, Griffiths, Morris and Tassion. In this paper we derive a stronger bound on the rate at which $\mathbb{P}(H_S|\eta)$ approaches its mean. As a consequence we strengthen the convergence in probability to almost sure convergence.

math.PR↗

arXiv2020

To fixate or not to fixate in two-type annihilating branching random walks

We study a model of competition between two types evolving as branching random walks on $\mathbb{Z}^d$. The two types are represented by red and blue balls respectively, with the rule that balls of different colour annihilate upon contact. We consider initial configurations in which the sites of $\mathbb{Z}^d$ contain one ball each, which are independently coloured red with probability $p$ and blue otherwise. We address the question of \emph{fixation}, referring to the sites eventually settling for a given colour, or not. Under a mild moment condition on the branching rule, we prove that the process will fixate almost surely for $p\neq 1/2$, and that every site will change colour infinitely often almost surely for the balanced initial condition $p=1/2$.

math.PR↗

arXiv2020

Existence and coexistence in first-passage percolation

We consider first-passage percolation with i.i.d. non-negative weights coming from some continuous distribution under a moment condition. We review recent results in the study of geodesics in first-passage percolation and study their implications for the multi-type Richardson model. In two dimensions this establishes a dual relation between the existence of infinite geodesics and coexistence among competing types. The argument amounts to making precise the heuristic that infinite geodesics can be thought of as `highways to infinity'. We explain the limitations of the current techniques by presenting a partial result in dimensions higher than two.

math.PR↗

arXiv2020

Tertiles and the time constant

We consider planar first-passage percolation and show that the time constant can be bounded by multiples of the first and second tertiles of the weight distribution. As a consequence we obtain a counter-example to a problem proposed by Alm and Deijfen.

math.PR↗

arXiv2019

Random coalescing geodesics in first-passage percolation

We continue the study of infinite geodesics in planar first-passage percolation, pioneered by Newman in the mid 1990s. Building on more recent work of Hoffman, and Damron and Hanson, we develop an ergodic theory for infinite geodesics via the study of what we shall call `random coalescing geodesics'. Random coalescing geodesics have a range of nice asymptotic properties, such as asymptotic directions and linear Busemann functions. We show that random coalescing geodesics are (in some sense) dense in the space of geodesics. This allows us to extrapolate properties from random coalescing geodesics to obtain statements on all infinite geodesics. As an application of this theory we solve the `midpoint problem' of Benjamini, Kalai and Schramm and address a question of Furstenberg on the existence of bigeodesics.

math.PR↗

arXiv2018

Noise sensitivity and Voronoi percolation

In this paper we study noise sensitivity and threshold phenomena for Poisson Voronoi percolation on $\mathbb{R}^2$. In the setting of Boolean functions, both threshold phenomena and noise sensitivity can be understood via the study of randomized algorithms. Together with a simple discretization argument, such techniques apply also to the continuum setting. Via the study of a suitable algorithm we show that box-crossing events in Voronoi percolation are noise sensitive and present a threshold phenomenon with polynomial window. We also study the effect of other kinds of perturbations, and emphasize the fact that the techniques we use apply for a broad range of models.

math.PR↗

arXiv2018

The two-type Richardson model in the half-plane

The two-type Richardson model describes the growth of two competing infection types on the two or higher dimensional integer lattice. For types that spread with the same intensity, it is known that there is a positive probability for infinite coexistence, while for types with different intensities, it is conjectured that infinite coexistence is not possible. In this paper we study the two-type Richardson model in the upper half-plane $\mathbb{Z}\times\mathbb{Z}_+$, and prove that coexistence of two types starting on the horizontal axis has positive probability if and only if the types have the same intensity.

math.PR↗

arXiv2017

Competing first passage percolation on random graphs with finite variance degrees

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 $λ_1$ ($λ_2$) 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 $λ_1=λ_2$, then the fraction of vertices that are ultimately infected by type 1 converges to a continuous random variable $V\in(0,1)$, as the number of vertices tends to infinity. Both infection types hence occupy a positive (random) fraction of the vertices. If $λ_1\neq λ_2$, 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.

math.PR↗