arXiv subjects

Matthew I. Roberts

Publications and source records attributed to Matthew I. Roberts.

At least 19 recordsLinked to original sources

arXiv2026

Accessibility Percolation with Rough Mount Fuji labels

Consider an infinite, rooted, connected graph where each vertex is labelled with an independent and identically distributed Uniform(0,1) random variable, plus a parameter $\theta$ times its distance from the root $\rho$. That is, we label vertex $v$ with $X_v = U_v + \theta d(\rho,v)$. We say that accessibility percolation occurs if there is an infinite path started from $\rho$ along which the vertex labels are increasing. When the graph is a Bienaym\'e-Galton-Watson tree, we give an exact characterisation of the critical value $\theta_c$ such that there is accessibility percolation with positive probability if and only if $\theta>\theta_c$. We also give more explicit bounds on the value of $\theta_c$. The lower bound holds for a much more general class of trees. When the graph is the lattice $\mathbb{Z}^n$ for $n\ge 2$, we show that there is a non-trivial phase transition and give some first bounds on $\theta_c$. To do this we introduce a novel coupling with oriented percolation.

math.PR↗

arXiv2024

Asymptotics for the growth of the infinite-parent Spatial Lambda-Fleming-Viot model

The infinite-parent spatial Lambda-Fleming-Viot (SLFV) process is a model of random growth, in which a set evolves by the addition of balls according to points of an underlying Poisson point process, and which was recently introduced to study genetic diversity in spatially expanding populations. In this article, we give asymptotics for the location and depth of the moving interface, and identify the exact asymptotic scale of the transverse fluctuations of geodesics. Our proofs are based on a new representation of the infinite-parent SLFV in terms of chains of reproduction events, and on the study of the properties of a typical geodesic. Moreover, we show that our representation coincides with the alternative definitions of the process considered in the literature, subject to a simple condition on the initial state. Our results represent a novel development in the study of stochastic growth models, and also have consequences for the study of genetic diversity in expanding populations.

math.PR↗

arXiv2021

The probability of unusually large components for critical percolation on random $d$-regular graphs

Let $d\ge 3$ be a fixed integer, $p\in (0,1)$, and let $n\geq 1$ be a positive integer such that $dn$ is even. Let $\mathbb{G}(n, d, p)$ be a (random) graph on $n$ vertices obtained by drawing uniformly at random a $d$-regular (simple) graph on $[n]$ and then performing independent $p$-bond percolation on it, i.e. we independently retain each edge with probability $p$ and delete it with probability $1-p$. Let $|\mathcal{C}_{\text{max}}|$ be the size of the largest component in $\mathbb{G}(n, d, p)$. We show that, when $p$ is of the form $p=(d-1)^{-1}(1+λn^{-1/3})$ for $λ\in \mathbb{R}$, and $A$ is large, \begin{align*} \mathbb{P}(|\mathcal{C}_{\text{max}}|>An^{2/3})\asymp A^{-3/2}e^{-\frac{A^3(d-1)(d-2)}{8d^2}+\frac{λA^2(d-2)^2}{2d(d-1)}-\frac{λ^2 A(d-1)}{2(d-2)}}. \end{align*} This improves on a result of Nachmias and Peres. We also give an analogous asymptotic for the probability that a particular vertex is in a component of size larger than $An^{2/3}$.

math.PR↗

arXiv2021

A spatially-dependent fragmentation process

We define a spatially-dependent fragmentation process, which involves rectangles breaking up into progressively smaller pieces at rates that depend on their shape. Long, thin rectangles are more likely to break quickly, and are also more likely to split along their longest side. We are interested in how the system evolves over time: how many fragments are there of different shapes and sizes, and how did they reach that state? Our theorem gives an almost sure growth rate along paths, which does not match the growth rate in expectation - there are paths where the expected number of fragments of that shape and size is exponentially large, but in reality no such fragments exist at large times almost surely.

math.PR↗

arXiv2021

Genealogy and spatial distribution of the $N$-particle branching random walk with polynomial tails

The $N$-particle branching random walk is a discrete time branching particle system with selection. We have $N$ particles located on the real line at all times. At every time step each particle is replaced by two offspring, and each offspring particle makes a jump of non-negative size from its parent's location, independently from the other jumps, according to a given jump distribution. Then only the $N$ rightmost particles survive; the other particles are removed from the system to keep the population size constant. Inspired by work of J. Bérard and P. Maillard, we examine the long term behaviour of this particle system in the case where the jump distribution has regularly varying tails and the number of particles is large. We prove that at a typical large time the genealogy of the population is given by a star-shaped coalescent, and that almost the whole population is near the leftmost particle on the relevant space scale.

math.PR↗

arXiv2021

A Gaussian particle distribution for branching Brownian motion with an inhomogeneous branching rate

Motivated by the goal of understanding the evolution of populations undergoing selection, we consider branching Brownian motion in which particles independently move according to one-dimensional Brownian motion with drift, each particle may either split into two or die, and the difference between the birth and death rates is a linear function of the position of the particle. We show that, under certain assumptions, after a sufficiently long time, the empirical distribution of the positions of the particles is approximately Gaussian. This provides mathematically rigorous justification for results in the biology literature indicating that the distribution of the fitness levels of individuals in a population over time evolves like a Gaussian traveling wave.

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arXiv2021

The number of ends of critical branching random walks

We investigate the number of topological ends of the trace of branching random walk (BRW) on a graph, giving a sufficient condition for the trace to have infinitely many ends. We then describe some interesting examples of non-symmetric BRWs with just one end.

math.PR↗

arXiv2021

Unusually large components in near-critical Erdős-Rényi graphs via ballot theorems

We consider the near-critical Erdős-Rényi random graph $G(n,p)$ and provide a new probabilistic proof of the fact that, when $p$ is of the form $p=p(n)=1/n+λ/n^{4/3}$ and $A$ is large, \[\mathbb{P}(|\mathcal{C}_{\max}|>An^{2/3})\asymp A^{-3/2}e^{-\frac{A^3}{8}+\frac{λA^2}{2}-\frac{λ^2A}{2}}\] where $\mathcal{C}_{\max}$ is the largest connected component of the graph. Our result allows $A$ and $λ$ to depend on $n$. While this result is already known, our proof relies only on conceptual and adaptable tools such as ballot theorems, whereas the existing proof relies on a combinatorial formula specific to Erdős-Rényi graphs, together with analytic estimates.

math.PR↗

arXiv2019

Cover time for branching random walks on regular trees

Let $T$ be the regular tree in which every vertex has exactly $d\ge 3$ neighbours. Run a branching random walk on $T$, in which at each time step every particle gives birth to a random number of children with mean $d$ and finite variance, and each of these children moves independently to a uniformly chosen neighbour of its parent. We show that, starting with one particle at some vertex $0$ and conditionally on survival of the process, the time it takes for every vertex within distance $r$ of $0$ to be hit by a particle of the branching random walk is almost surely $r + \frac{2}{\log(3/2)}\log\log r + o(\log\log r)$.

math.PR↗

arXiv2019

Noise sensitivity and exceptional times of transience for a simple symmetric random walk in one dimension

We define a dynamical simple symmetric random walk in one dimension, and show that there almost surely exist exceptional times at which the walk tends to infinity. This is in contrast to the usual dynamical simple symmetric random walk in one dimension, for which such exceptional times are known not to exist. In fact we show that the set of exceptional times has Hausdorff dimension $1/2$ almost surely, and give bounds on the rate at which the walk diverges at such times. We also show noise sensitivity of the event that our random walk is positive after $n$ steps. In fact this event is maximally noise sensitive, in the sense that it is quantitatively noise sensitive for any sequence $\varepsilon_n$ such that $n\varepsilon_n\to\infty$. This is again in contrast to the usual random walk, for which the corresponding event is known to be noise stable.

math.PR↗

arXiv2019

The coalescent structure of continuous-time Galton-Watson trees

Take a continuous-time Galton-Watson tree. If the system survives until a large time $T$, then choose $k$ particles uniformly from those alive. What does the ancestral tree drawn out by these $k$ particles look like? Some special cases are known but we give a more complete answer. We concentrate on near-critical cases where the mean number of offspring is $1+μ/T$ for some $μ\in\mathbb{R}$, and show that a scaling limit exists as $T\to\infty$. Viewed backwards in time, the resulting coalescent process is topologically equivalent to Kingman's coalescent, but the times of coalescence have an interesting and highly non-trivial structure. The randomly fluctuating population size, as opposed to constant size populations where the Kingman coalescent more usually arises, have a pronounced effect on both the results and the method of proof required. We give explicit formulas for the distribution of the coalescent times, as well as a construction of the genealogical tree involving a mixture of independent and identically distributed random variables. In general subcritical and supercritical cases it is not possible to give such explicit formulas, but we highlight the special case of birth-death processes.

math.PR↗

arXiv2017

The probability of unusually large components in the near-critical Erdős-Rényi graph

The largest components of the critical Erdős-Rényi graph, $G(n,p)$ with $p=1/n$, have size of order $n^{2/3}$ with high probability. We give detailed asymptotics for the probability that there is an unusually large component, i.e. of size $an^{2/3}$ for large $a$. Our results, which extend work of Pittel, allow $a$ to depend upon $n$ and also hold for a range of values of $p$ around $1/n$. We also provide asymptotics for the distribution of the size of the component containing a particular vertex.

math.PR↗

arXiv2017

Exceptional times of the critical dynamical Erdős-Rényi graph

In this paper we introduce a network model which evolves in time, and study its largest connected component. We consider a process of graphs $(G_t:t\in [0,1])$, where initially we start with a critical Erdős-Rényi graph ER(n, 1/n), and then evolve forwards in time by resampling each edge independently at rate 1. We show that the size of the largest connected component that appears during the time interval $[0, 1]$ is of order $n^{2/3} log^{1/3} n$ with high probability. This is in contrast to the largest component in the static critical Erdős-Rényi graph, which is of order $n^{2/3}$.

math.PR↗

arXiv2017

Mixing time bounds via bottleneck sequences

We provide new upper bounds for mixing times of general finite Markov chains. We use these bounds to show that the total variation mixing time is robust under rough isometry for bounded degree graphs that are roughly isometric to trees.

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arXiv2016

Intermittency for branching random walk in Pareto environment

We consider a branching random walk on the lattice, where the branching rates are given by an i.i.d. Pareto random potential. We describe the process, including a detailed shape theorem, in terms of a system of growing lilypads. As an application we show that the branching random walk is intermittent, in the sense that most particles are concentrated on one very small island with large potential. Moreover, we compare the branching random walk to the parabolic Anderson model and observe that although the two systems show similarities, the mechanisms that control the growth are fundamentally different.

math.PR↗

arXiv2016

Scaling limit and ageing for branching random walk in Pareto environment

We consider a branching random walk on the lattice, where the branching rates are given by an i.i.d. Pareto random potential. We show that the system of particles, rescaled in an appropriate way, converges in distribution to a scaling limit that is interesting in its own right. We describe the limit object as a growing collection of "lilypads" built on a Poisson point process in $\mathbb{R}^d$. As an application of our main theorem, we show that the maximizer of the system displays the ageing property.

math.PR↗

arXiv2016

One-point localization for branching random walk in Pareto environment

We consider a branching random walk on the lattice, where the branching rates are given by an i.i.d. Pareto random potential. We show a very strong form of intermittency, where with high probability most of the mass of the system is concentrated in a single site with high potential. The analogous one-point localization is already known for the parabolic Anderson model, which describes the expected number of particles in the same system. In our case, we rely on very fine estimates for the behaviour of particles near a good point. This complements our earlier results that in the rescaled picture most of the mass is concentrated on a small island.

math.PR↗

arXiv2015

Vanishing corrections for the position in a linear model of FKPP fronts

Take the linearised FKPP equation \[\partial_t h =\partial^2_x h +h\] with boundary condition $h(m(t),t)=0$. Depending on the behaviour of the initial condition $h_0(x)=h(x,0)$ we obtain the asymptotics - up to a $o(1)$ term $r(t)$ - of the absorbing boundary $m(t)$ such that $ω(x):=\lim_t h(x+m(t) ,t)$ exists and is non-trivial. In particular, as in Bramson's results for the non-linear FKPP equation, we recover the celebrated $-(3/2)\log t$ correction for initial conditions decaying faster than $x^νe^{-x}$ for some $ν<-2$. Furthermore, when we are in this regime, the main result of the present work is the identification (to first order) of the $r(t)$ term which ensures the fastest convergence to $ω(x)$. When $h_0(x)$ decays faster than $x^νe^{-x}$ for some $ν<-3$, we show that $r(t)$ must be chosen to be $-3\sqrt{π/t}$ which is precisely the term predicted heuristically by Ebert-van Saarloos in the non-linear case. When the initial condition decays as $x^νe^{-x}$ for some $ν\in [-3,-2)$, we show that even though we are still in the regime where Bramson's correction is $-(3/2)\log t$, the Ebert-van Saarloos correction has to be modified. Similar results were recently obtained by Henderson using an analytical approach and only for compactly supported initial conditions.

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