arXiv subjects

Samuel G. G. Johnston

Publications and source records attributed to Samuel G. G. Johnston.

27 records · Page 2Linked to original sources

arXiv2022

Multitype $Λ$-coalescents

Consider a multitype coalescent process in which each block has a colour in $\{1,\ldots,d\}$. Individual blocks may change colour, and some number of blocks of various colours may merge to form a new block of some colour. We show that if the law of a multitype coalescent process is invariant under permutations of blocks of the same colour, has consistent Markovian projections, and has asychronous mergers, then it is a multitype $Λ$-coalescent: a process in which single blocks may change colour, two blocks of like colour may merge to form a single block of that colour, or large mergers across various colours happen at rates governed by a $d$-tuple of measures on the unit cube $[0,1]^d$. We go on to identify when such processes come down from infinity. Our framework generalises Pitman's celebrated classification theorem for singletype coalescent processes, and provides a unifying setting for numerous examples that have appeared in the literature including the seed-bank model, the island model and the coalescent structure of continuous-state branching processes.

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arXiv2021

Projections of the uniform distribution on the cube -- a large deviation perspective

Let $Θ^{(n)}$ be a random vector uniformly distributed on the unit sphere $\mathbb S^{n-1}$ in $\mathbb R^n$. Consider the projection of the uniform distribution on the cube $[-1,1]^n$ to the line spanned by $Θ^{(n)}$. The projected distribution is the random probability measure $μ_{Θ^{(n)}}$ on $\mathbb R$ given by \[ μ_{Θ^{(n)}}(A) := \frac 1 {2^n} \int_{[-1,1]^n} \mathbb 1\{\langle u, Θ^{(n)} \rangle \in A\} du, \] for Borel subets $A$ of $\mathbb{R}$. It is well known that, with probability $1$, the sequence of random probability measures $μ_{Θ^{(n)}}$ converges weakly to the centered Gaussian distribution with variance $1/3$. We prove a large deviation principle for the sequence $μ_{Θ^{(n)}}$ on the space of probability measures on $\mathbb R$ with speed $n$. The (good) rate function is explicitly given by $I(ν(α)) := - \frac{1}{2} \log ( 1 - \|α\|_2^2)$ whenever $ν(α)$ is the law of a random variable of the form \begin{align*} \sqrt{1 - \|α\|_2^2 } \frac{Z}{\sqrt 3} + \sum_{ k = 1}^\infty α_k U_k, \end{align*} where $Z$ is standard Gaussian independent of $U_1,U_2,\ldots$ which are i.i.d. $\text{Unif}[-1,1]$, and $α_1 \geq α_2 \geq \ldots $ is a non-increasing sequence of non-negative reals with $\|α\|_2<1$. We obtain a similar result for random projections of the uniform distribution on the discrete cube $\{-1,+1\}^n$.

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arXiv2021

The coalescent structure of uniform and Poisson samples from multitype branching processes

We introduce a Poissonization method to study the coalescent structure of uniform samples from branching processes. This method relies on the simple observation that a uniform sample of size $k$ taken from a random set with positive Lebesgue measure may be represented as a mixture of Poisson samples with rate $λ$ and mixing measure $k \mathrm{d} λ/ λ$. We develop a multitype analogue of this mixture representation, and use it to characterise the coalescent structure of multitype continuous-state branching processes in terms of random multitype forests. Thereafter we study the small time asymptotics of these random forests, establishing a correspondence between multitype continuous-state branching proesses and multitype $Λ$-coalescents.

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arXiv2021

Sharp concentration for the largest and smallest fragment in a $k$-regular self-similar fragmentation

We study the asymptotics of the $k$-regular self-similar fragmentation process. For $α> 0$ and an integer $k \geq 2$, this is the Markov process $(I_t)_{t \geq 0}$ in which each $I_t$ is a union of open subsets of $[0,1)$, and independently each subinterval of $I_t$ of size $u$ breaks into $k$ equally sized pieces at rate $u^α$. Let $k^{ - m_t}$ and $k^{ - M_t}$ be the respective sizes of the largest and smallest fragments in $I_t$. By relating $(I_t)_{t \geq 0}$ to a branching random walk, we find that there exist explicit deterministic functions $g(t)$ and $h(t)$ such that $|m_t - g(t)| \leq 1$ and $|M_t - h(t)| \leq 1$ for all sufficiently large $t$. Furthermore, for each $n$, we study the final time at which fragments of size $k^{-n}$ exist. In particular, by relating our branching random walk to a certain point process, we show that, after suitable rescaling, the laws of these times converge to a Gumbel distribution as $n \to \infty$.

math.PR↗

arXiv2020

A Maxwell principle for generalized Orlicz balls

In [A dozen de {F}inetti-style results in search of a theory, Ann. Inst. H. Poincaré Probab. Statist. 23(2)(1987), 397--423], Diaconis and Freedman studied low-dimensional projections of random vectors from the Euclidean unit sphere and the simplex in high dimensions, noting that the individual coordinates of these random vectors look like Gaussian and exponential random variables respectively. In subsequent works, Rachev and Rüschendorf and Naor and Romik unified these results by establishing a connection between $\ell_p^N$ balls and a $p$-generalized Gaussian distribution. In this paper, we study similar questions in a significantly generalized and unifying setting, looking at low-dimensional projections of random vectors uniformly distributed on sets of the form \[B_{ϕ,t}^N := \Big\{(s_1,\ldots,s_N)\in\mathbb{R}^N : \sum_{ i =1}^Nϕ(s_i)\leq t N\Big\},\] where $ϕ:\mathbb{R}\to [0,\infty]$ is a potential (including the case of Orlicz functions). Our method is different from both Rachev-Rüschendorf and Naor-Romik, based on a large deviation perspective in the form of quantitative versions of Cramér's theorem and the Gibbs conditioning principle, providing a natural framework beyond the $p$-generalized Gaussian distribution while simultaneously unraveling the role this distribution plays in relation to the geometry of $\ell_p^N$ balls. We find that there is a critical parameter $t_{\mathrm{crit}}$ at which there is a phase transition in the behaviour of the projections: for $t > t_{\mathrm{crit}}$ the coordinates of random points sampled from $B_{ϕ,t}^N$ behave like uniform random variables, but for $t \leq t_{\mathrm{crit}}$ the Gibbs conditioning principle comes into play, and here there is a parameter $β_t>0$ (the inverse temperature) such that the coordinates are approximately distributed according to a density proportional to $e^{ -β_tϕ(s)}$.

math.PR↗

arXiv2019

Berry-Esseen bounds for random projections of $\ell_p^n$-balls

In this work we study the rate of convergence in the central limit theorem for the Euclidean norm of random orthogonal projections of vectors chosen at random from an $\ell_p^n$-ball which has been obtained in [Alonso-Gutiérrez, Prochno, Thäle: Gaussian fluctuations for high-dimensional random projections of $\ell_p^n$-balls, Bernoulli 25(4A), 2019, 3139--3174]. More precisely, for any $n\in\mathbb N$ let $E_n$ be a random subspace of dimension $k_n\in\{1,\ldots,n\}$, $P_{E_n}$ the orthogonal projection onto $E_n$, and $X_n$ be a random point in the unit ball of $\ell_p^n$. We prove a Berry-Esseen theorem for $\|P_{E_n}X_n\|_2$ under the condition that $k_n\to\infty$. This answers in the affirmative a conjecture of Alonso-Gutiérrez, Prochno, and Thäle who obtained a rate of convergence under the additional condition that $k_n/n^{2/3}\to\infty$ as $n\to\infty$. In addition, we study the Gaussian fluctuations and Berry-Esseen bounds in a $3$-fold randomized setting where the dimension of the Grassmannian is also chosen randomly. Comparing deterministic and randomized subspace dimensions leads to a quite interesting observation regarding the central limit behavior. In this work we also discuss the rate of convergence in the central limit theorem of [Kabluchko, Prochno, Thäle: High-dimensional limit theorems for random vectors in $\ell_p^n$-balls, Commun. Contemp. Math. (2019)] for general $\ell_q$-norms of non-projected vectors chosen at random in an $\ell_p^n$-ball.

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arXiv2019

Scaling limits for non-intersecting polymers and Whittaker measures

We study the partition functions associated with non-intersecting polymers in a random environment. By considering paths in series and in parallel, the partition functions carry natural notions of subadditivity, allowing the effective study of their asymptotics. For a certain choice of random environment, the geometric RSK correspondence provides an explicit representation of the partition functions in terms of a stochastic interface. Formally this leads to a variational description of the macroscopic behaviour of the interface and hence the free energy of the associated non-intersecting polymer model. At zero temperature we relate this variational description to the Marchenko-Pastur distribution, and give a new derivation of the surface tension of the bead model.

math.PR↗

arXiv2019

The genealogy of Galton-Watson trees

Take a continuous-time Galton-Watson tree and pick $k$ distinct particles uniformly from those alive at a time $T$. What does their genealogical tree look like? The case $k=2$ has been studied by several authors, and the near-critical asymptotics for general $k$ appear in Harris, Johnston and Roberts (2018). Here we give the full picture.

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.

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