arXiv subjects

Simon C. Harris

Publications and source records attributed to Simon C. Harris.

At least 19 recordsLinked to original sources

arXiv2026

Sample genealogies within a Brownian excursion

In this article, we apply It\^o's excursion theory to find the joint law of minima of a Brownian excursion conditioned to go above level $1$ over the intervals defined by $k$ independent identically distributed (i.i.d.) points of intersection of the excursion with level $a \in (0,1]$. This approach offers an intuitive way to find the joint law of coalescent times of an i.i.d. sample of k particles alive at time $a$ in Aldous' continuum random tree of height $1$. This can be thought of as ``sampling of the limit" of critical Galton-Watson trees conditioned to survive a large time, agreeing with some special cases obtained in [4] and [5] which instead consider ``the limit of sampling" from critical Galton-Watson trees.

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arXiv2026

Uniform sampling of multitype continuous-time Bienaym\'e-Galton-Watson trees

We study the genealogy of a sample of $k$ individuals taken uniformly without replacement from a continuous-time multitype Bienaym\'e--Galton--Watson process at fixed times. Our results are quite general, requiring only that the process be non-simple and conservative, and that every type has a positive probability to ``eventually lead to'' all other types within the population. The corresponding single-type case has recently been studied by Johnston (2019), Harris, Johnston, and Roberts (2020), and Harris, Johnston, and Pardo (2024). Our approach is based on a $k$-spine decomposition and a suitable change of measure under which the distinguished spines form a uniform sample at time $T$, while the population size is subject to $k$-size biasing and exponential discounting. This construction preserves a branching Markov property and yields an explicit description of the genealogical tree at fixed times. In particular, we characterise spine splitting times, offspring distributions, and type-dependent ancestral structures, revealing rich interactions between types that are absent in the single-type setting. The present results form the basis of a forthcoming series of papers in which limiting genealogical behaviour is analysed under various asymptotic regimes and more general sampling schemes by the authors, see Angtuncio et al. (2026b), (2026c) and (2026d).

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arXiv2025

Sampling schemes of multitype continuous-time Bienaym\'e-Galton-Watson trees and limiting critical genealogies

We study the genealogies of samples of $k$ distinguished particles drawn from the population alive at some fixed time in a continuous-time multitype Bienaym\'e-Galton-Watson (MBGW) process under two different type dependent sampling schemes: uniform sampling without replacement within types given a fixed type configuration, and sampling according to type-dependent weights. These schemes complement the uniform sampling at fixed time $T$ considered in Angtuncio, Pardo, C. Harris (2026a) which did not distinguish between sampled types. Under each scheme for a fixed sampling time $T$, we characterise the associated times of most recent common ancestors, ancestral offspring distributions, and type-dependent ancestral structure of the sample genealogy. In addition, under the assumption that the MBGW process is critical with finite second moments, we show that, conditional on survival of the population, a large time limiting sample genealogy emerges which is robust to the sampling scheme used. We identify this universal genealogy to have the same tree structure as the single-type case in C. Harris, Johnston, Roberts (2020), and we describe its ancestral type behaviour over scaled-times - this essentially being decoupled from the tree structure except at the times of ancestral splitting events.

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arXiv2024

The longest branches in a non-Markovian phylogenetic tree

Consider a Bellman--Harris-type branching process, in which individuals evolve independently of one another, giving birth after a random time $T$ to a random number $L$ of children. In this article, we study the asymptotic behaviour of the length of the longest branches of this branching process at time $t$, both pendant branches (corresponding to individuals still alive at time $t$) and interior branches (corresponding to individuals dead before time $t$).

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arXiv2024

The coalescent structure of Galton-Watson trees in varying environments

We investigate the genealogy of a sample of $k\geq1$ particles chosen uniformly without replacement from a population alive at large times in a critical discrete-time Galton-Watson process in a varying environment (GWVE). We will show that subject to an explicit deterministic time-change involving only the mean and variances of the varying offspring distributions, the sample genealogy always converges to the same universal genealogical structure; it has the same tree topology as Kingman's coalescent, and the coalescent times of the $k-1$ pairwise mergers look like a mixture of independent identically distributed times. Our approach uses $k$ distinguished \emph{spine} particles and a suitable change of measure under which (a) the spines form a uniform sample without replacement, as required, but additionally (b) there is $k$-size biasing and discounting according to the population size. Our work significantly extends the spine techniques developed in Harris, Johnston, and Roberts \emph{[Annals Applied Probability, 2020]} for genealogies of uniform samples of size $k$ in near-critical continuous-time Galton-Watson processes, as well as a two-spine GWVE construction in Cardona and Palau \emph{[Bernoulli, 2021]}. Our results complement recent works by Kersting \emph{[Proc. Steklov Inst. Maths., 2022]} and Boenkost, Foutel-Rodier, and Schertzer \emph{[arXiv:2207.11612]}.

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arXiv2023

Long edges in Galton-Watson trees

In this article, we will establish a number of results concerning the limiting behaviour of the longest edges in the genealogical tree generated by a continuous-time Galton-Watson (GW) process. Separately, we consider the large time behaviour of the longest pendant edges, the longest (strictly) interior edges, and the longest of all the edges. These results extend the special case of long pendant edges of birth-death processes established in Bocharov, Harris, Kominek, Mooers, and Steel [1] .

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arXiv2023

Universality classes for the coalescent structure of heavy-tailed Galton-Watson trees

Consider a population evolving as a critical continuous-time Galton-Watson (GW) tree. Conditional on the population surviving until a large time $T$, sample $k$ individuals uniformly at random (without replacement) from amongst those alive at time $T$. What is the genealogy of this sample of individuals? In cases where the offspring distribution has finite variance, the probabilistic properties of the joint ancestry of these $k$ particles are well understood, as seen in \cite{HJR20, J19}. In the present article, we study the joint ancestry of a sample of $k$ particles under the following regime: the offspring distribution has mean $1$ (critical) and the tails of the offspring distribution are \emph{heavy} in that $α\in (1,2]$ is the supremum over indices $β$ such that the $β^{\text{th}}$ moment is finite. We show that for each $α$, after rescaling time by $1/T$, there is a universal stochastic process describing the joint coalescent structure of the $k$ distinct particles. The special case $α= 2$ generalises the known case of sampling from critical GW trees with finite variance where only pairwise mergers are observed and the genealogical tree is, roughly speaking, some kind of mixture of time-changed Kingman coalescents. The cases $α\in (1,2)$ introduce new universal limiting partition-valued stochastic processes with interesting probabilistic structures which have representations connected to the Lauricella function and the Dirichlet distribution, and whose coalescent structures exhibit multiple-mergers of family lines. Moreover, in the case $α\in (1,2)$, we show that the coalescent events of the ancestry of the $k$ particles are associated with birth events that produce giant numbers of offspring of the same order of magnitude as the entire population size, and we compute the joint law of the ancestry together with the sizes of these giant births.

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arXiv2022

Many-to-few for non-local branching Markov process

We provide a many-to-few formula in the general setting of non-local branching Markov processes. This formula allows one to compute expectations of k-fold sums over functions of the population at k different times. The result generalises [14] to the non-local setting, as introduced in [11] and [8]. As an application, we consider the case when the branching process is critical, and conditioned to survive for a large time. In this setting, we prove a general formula for the limiting law of the death time of the most recent common ancestor of two particles selected uniformly from the population at two different times, as t tends to infinity. Moreover, we describe the limiting law of the population sizes at two different times, in the same asymptotic regime.

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arXiv2022

Yaglom limit for critical neutron transport

We consider the classical Yaglom limit theorem for a branching Markov process $X = (X_t, t \ge 0)$, with non-local branching mechanism in the setting that the mean semigroup is critical, i.e. its leading eigenvalue is zero. In particular, we show that there exists a constant $c(f)$ such that \[ {\rm Law}\left(\frac{\langle f, X_t\rangle}{t} \bigg| \langle 1, X_t\rangle > 0 \right) \to {\mathbf e}_{c(f)}, \qquad t \to \infty, \] where ${\mathbf e}_{c(f)}$ is an exponential random variable with rate $c(f)$ and the convergence is in distribution. As part of the proof, we also show that the probability of survival decays inversely proportionally to time. Although Yaglom limit theorems have recently been handled in the setting of branching Brownian motion in a bounded domain and superprocesses, \cite{Ellen, Yanxia}, these results do not allow for non-local branching, which complicates the analysis. Our approach and the main novelty of this work is based around a precise result for the scaled asymptotics for the $k$-th martingale moments of $X$ (rather than the Yaglom limit itself). We then illustrate our results in the setting of neutron transport, for which the non-locality is essential, complementing recent developments in this domain \cite{SNTE, SNTEII, SNTEIII, MCNTE, MultiNTE}.

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arXiv2021

Monte-Carlo Methods for the Neutron Transport Equation

This paper continues our treatment of the Neutron Transport Equation (NTE) building on the work in [arXiv:1809.00827v2], [arXiv:1810.01779v4] and [arXiv:1901.00220v3], which describes the flux of neutrons through inhomogeneous fissile medium. Our aim is to analyse existing and novel Monte Carlo (MC) algorithms, aimed at simulating the lead eigenvalue associated with the underlying model. This quantity is of principal importance in the nuclear regulatory industry for which the NTE must be solved on complicated inhomogenous domains corresponding to nuclear reactor cores, irradiative hospital equipment, food irradiation equipment and so on. We include a complexity analysis of such MC algorithms, noting that no such undertaking has previously appeared in the literature. The new MC algorithms offer a variety of advantages and disadvantages of accuracy vs cost, as well as the possibility of more convenient computational parallelisation.

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arXiv2020

Stochastic Methods for the Neutron Transport Equation II: Almost sure growth

The neutron transport equation (NTE) describes the flux of neutrons across a planar cross-section in an inhomogeneous fissile medium when the process of nuclear fission is active. Classical work on the NTE emerges from the applied mathematics literature in the 1950s through the work of R. Dautray and collaborators, [7, 8, 19]. The NTE also has a probabilistic representation through the semigroup of the underlying physical process when envisaged as a stochastic process; cf. [7, 17, 18, 20]. More recently, [6] and [16] have continued the probabilistic analysis of the NTE, introducing more recent ideas from the theory of spatial branching processes and quasi-stationary distributions. In this paper, we continue in the same vein and look at a fundamental description of stochastic growth in the supercritical regime. Our main result provides a significant improvement on the last known contribution to growth properties of the physical process in [20], bringing neutron transport theory in line with modern branching process theory such as [14, 12].

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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.

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arXiv2019

Multi-species neutron transport equation

The Neutron Transport Equation (NTE) describes the flux of neutrons through inhomogeneous fissile medium. Whilst well treated in the nuclear physics literature (cf. [9, 27]), the NTE has had a somewhat scattered treatment in mathematical literature with a variety of different approaches (cf. [8, 25]). Within a probabilistic framework it has somewhat undeservingly received little attention in recent years; nonetheless, probabilistic treatments can be found see for example [19, 26, 24, 29, 4, 3]. In this article our aim is threefold. First we want to introduce a slightly more general setting for the NTE, which gives a more complete picture of the different species of particle and radioactive fluxes that are involved in fission. Second we consolidate the classical c0-semigroup approach to solving the NTE with the method of stochastic representation which involves expectation semigroups. Third we provide the leading asymptotic of our multi-species NTE, which will turn out to be crucial for further stochastic analysis of the NTE in forthcoming work [6, 5]. The methodology used in this paper harmonises the culture of expectation semigroup analysis from the theory of stochastic processes against c0-semigroup theory from functional analysis. In this respect, our presentation is thus part review of existing theory and part presentation of new research results based on generalisation of existing results.

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arXiv2016

Branching Brownian motion with absorption and the all-time minimum of branching Brownian motion with drift

We study a dyadic branching Brownian motion on the real line with absorption at 0, drift $μ\in \mathbb{R}$ and started from a single particle at position $x>0.$ When $μ$ is large enough so that the process has a positive probability of survival, we consider $K(t),$ the number of individuals absorbed at 0 by time $t$ and for $s\ge 0$ the functions $ω_s(x):= \mathbb{E}^x[s^{K(\infty)}].$ We show that $ω_s<\infty$ if and only of $s\in[0,s_0]$ for some $s_0>1$ and we study the properties of these functions. Furthermore, for $s=0, ω(x) := ω_0(x) =\mathbb{P}^x(K(\infty)=0)$ is the cumulative distribution function of the all time minimum of the branching Brownian motion with drift started at 0 without absorption. We give three descriptions of the family $ω_s, s\in [0,s_0]$ through a single pair of functions, as the two extremal solutions of the Kolmogorov-Petrovskii-Piskunov (KPP) traveling wave equation on the half-line, through a martingale representation and as an explicit series expansion. We also obtain a precise result concerning the tail behavior of $K(\infty)$. In addition, in the regime where $K(\infty)>0$ almost surely, we show that $u(x,t) := \mathbb{P}^x(K(t)=0)$ suitably centered converges to the KPP critical travelling wave on the whole real line.

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arXiv2016

Limiting Distribution of the Rightmost Particle in Catalytic Branching Brownian Motion

We study the model of binary branching Brownian motion with spatially-inhomogeneous branching rate $βδ_0(\cdot)$, where $δ_0(\cdot)$ is the Dirac delta function and $β$ is some positive constant. We show that the distribution of the rightmost particle centred about $\fracβ{2}t$ converges to a mixture of Gumbel distributions according to a martingale limit. Our results form a natural extension to S. Lalley and T. Sellke [6] for the degenerate case of catalytic branching.

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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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arXiv2014

The many-to-few lemma and multiple spines

We develop a simple and intuitive identity for calculating expectations of weighted $k$-fold sums over particles in branching processes, generalising the well-known many-to-one lemma.

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arXiv2014

Growth rates of the population in a branching Brownian motion with an inhomogeneous breeding potential

We consider a branching particle system where each particle moves as an independent Brownian motion and breeds at a rate proportional to its distance from the origin raised to the power $p$, for $p\in[0,2)$. The asymptotic behaviour of the right-most particle for this system is already known; in this article we give large deviations probabilities for particles following "difficult" paths, growth rates along "easy" paths, the total population growth rate, and we derive the optimal paths which particles must follow to achieve this growth rate.

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