arXiv subjects

Samuel G. G. Johnston

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

At least 19 recordsLinked to original sources

arXiv2026

Free probability via entropic optimal transport

Let $μ$ and $ν$ be compactly supported probability measures on $\mathbb{R}$, and let $μ\boxplus ν$ denote their additive free convolution. We show that for all sufficiently large real $z$, \begin{equation} \int_{-\infty}^\infty \log(z - x) \, (μ\boxplus ν)(\mathrm{d}x) = \sup_Π \left\{ \mathbf{E}_Π[\log(z - (X+Y))] - H(Π\mid μ\otimes ν) \right\}, \end{equation} where the supremum is taken over all couplings $Π$ of $μ$ and $ν$. Analogous formulas hold for multiplicative free convolution $μ\boxtimes ν$ and free compression $[μ]_τ$. In this way, integrals of a log-potential against free convolutions can be expressed as entropic optimal transport problems. The corresponding optimal couplings admit explicit formulas, from which the standard $R$- and $S$-transform descriptions of additive and multiplicative free convolution are recovered. In particular, the optimizer $Π_z$ encodes the subordination equations via couplings of random variables. Our approach is based on a large deviation principle on the symmetric group, combined with the quadrature method of Marcus--Spielman--Srivastava.

math.PR↗

arXiv2026

Behaviour near explosion in continuous-time Galton-Watson trees

We study continuous-time Galton--Watson trees whose offspring generating function takes the form \begin{align} \label{eq:gen} f(s) = s - (1-s)^αL(1-s), \end{align} where $α\in (0,1)$ and where $L:[0,1] \to [0,\infty)$ is slowly varying at zero. Processes with this offspring generating function explode in finite time. We observe that, conditional on explosion at time $T$, at each earlier time $t < T$ there is a unique particle $ξ_t$ alive at time $t$ who is an ancestor of all but finitely many particles at the explosion time. We call $(ξ_t)_{t \in [0,T)}$ the spine to explosion, and show that the births off the spine admit a Poissonian description, where they become both more frequent and larger as $t \uparrow T$. We undertake a careful study of the size of the population leading up to explosion, showing in particular that near explosion, the rescaled population is approximately gamma distributed. More generally, conditional on explosion at time $T$, define a stochastic process $Z^\varepsilon := (Z_t^\varepsilon)_{t \in \mathbb{R}}$ by setting \begin{align*} Z_t^\varepsilon := E(\varepsilon e^{-t})N_{T-\varepsilon e^{-t}}, \qquad E(t) = \mathbf{P}(N_t = \infty). \end{align*} We show that as $\varepsilon \downarrow 0$, $Z^\varepsilon$ converges in finite-dimensional distributions to a stationary continuous-state branching process with immigration whose branching mechanism is subcritical and whose immigration mechanism corresponds to spine events. Finally, we show that this limiting process has a Markovian time reversal, and that the coalescent process associated with it is a stochastically time-changed Beta$(2-α,α)$-coalescent.

math.PR↗

arXiv2026

The macroscopic shape of Gelfand-Tsetlin patterns and free probability

A compression is a function $F:\mathbb{R}\times[0,1]\to[0,1]$ such that each $F(\cdot,τ)$ is the distribution function of a measure of mass $τ$, while each $F(x,\cdot)$ is increasing and $1$-Lipschitz. Compressions are continuum analogues of Gelfand--Tsetlin patterns: if $(t_{k,j})_{ 0 \leq j \leq k \leq n}$ is a Gelfand--Tsetlin pattern, setting $F(t_{k,j},k/n) = j/n$ and interpolating creates a compression. For a differentiable compression $F$, we define the compression entropy \begin{align*} \mathcal{H}[F] :=\int_{-\infty}^\infty\int_0^1 F_x \left\{-\log F_x+\log\sin(πF_τ)+1-\logπ\right\} \mathrm{d}τ\mathrm{d}x. \end{align*} If $μ$ is absolutely continuous and compactly supported, we prove \begin{equation*} \sup\left\{\mathcal{H}[F]:F \text{ compression}, F(\cdot,1)\text{ is the distribution function of }μ\right\} =χ[μ], \end{equation*} where $χ[μ]$ is Voiculescu's free entropy. By identifying the Euler--Lagrange equations for $\mathcal{H}[F]$ with a Burgers equation for Cauchy transforms, we show that the supremum is attained uniquely by the free compression of free probability theory. We also view Gelfand--Tsetlin patterns as Ginzburg--Landau $\nablaϕ$-interface models with a hard-core interaction, and compute the surface tension: \begin{equation*} σ(u_1,u_2) =-\log(u_1+u_2)-\log\sin\left(π\frac{u_1}{u_1+u_2}\right)-1+\logπ. \end{equation*} Finally, we prove that uniform $n$-dimensional Gelfand--Tsetlin patterns with deterministic bottom rows converging to $μ$ satisfy a large deviation principle with speed $n^2$ and rate function \begin{equation*} I_μ[F]=-\mathcal{H}[F]+χ[μ]. \end{equation*} These results resolve a conjecture of Shlyakhtenko and Tao stating that the Euler--Lagrange equations for free compression arise from the statistical mechanics of interlacing point processes.

math.PR↗

arXiv2026

On creating convexity in high dimensions

Given a subset $A$ of $\mathbb{R}^n$, we define \begin{align*} \mathrm{conv}_k(A) := \left\{ λ_1 s_1 + \cdots + λ_k s_k : λ_i \in [0,1], \sum_{i=1}^k λ_i = 1 , s_i \in A \right\} \end{align*} to be the set of vectors in $\mathbb{R}^n$ that can be written as a $k$-fold convex combination of vectors in $A$. Let $γ_n$ denote the standard Gaussian measure on $\mathbb{R}^n$. We show that for every $\varepsilon > 0$, there exists a subset $A$ of $\mathbb{R}^n$ with Gaussian measure $γ_n(A) \geq 1- \varepsilon$ such that for all $k = O_\varepsilon(\sqrt{\log \log(n)})$, $\mathrm{conv}_k(A)$ contains no convex set $K$ of Gaussian measure $γ_n(K) \geq \varepsilon$. This result acts as a complement to the recent affirmative resolution of Talagrand's convexity conjecture by Hua, Song, and Tudose, which states that a universal dilation of the threefold Minkowski sum $A+A+A$ of a large set $A$ guarantees a large convex subset. Our approach utilises concentration properties of random copulas and the application of optimal transport techniques to the empirical coordinate measures of vectors in high dimensions.

math.MG↗

arXiv2026

The largest fragment in self-similar fragmentation processes of positive index

We study a self-similar fragmentation process with dislocation measure $ν$ and self-similarity index $α> 0$. Let $e^{-m_t}$ denote the size of the largest fragment at time $t \geq 0$. For dislocation measures satisfying a regularity condition of the form $ν(1 - s_1 > δ) = δ^{-θ} \ell(1/δ)$ with $θ\in [0,1)$ and slowly varying $\ell$, we prove almost sure convergence \[ \lim_{t \to \infty} (m_t - g(t)) = 0, \] where $g(t) = (\log t - (1 - θ) \log \log t + f(t))/α$, and $f(t) = o(\log \log t)$ is a lower order correction that can be described explicitly in terms of $\ell$ and $θ$. Our results sharpen substantially the best prior result on general self-similar fragmentation processes, due to Bertoin, which states that $m_t = (1+o(1)) \log (t)/α$.

math.PR↗

arXiv2026

Log-concavity and concentration bounds for a single gap between GUE eigenvalues

We observe that the distribution of the eigenvalues of an $N$-by-$N$ GUE random matrix is log-concave on $\mathbb{R}^N$, and that the same is true for the law of a single gap between two consecutive eigenvalues. We use this observation to prove several concentration bounds for the semicircle-renormalised eigengaps, improving on bounds recently obtained in [Tao (2024). On the distribution of eigenvalues of GUE and its minors at fixed index. [arXiv:2412.10889]].

math.PR↗

arXiv2025

Explicit correlation functions for the six-vertex model in the free-fermion regime

In this article, we show that, in the free-fermion regime of the six-vertex model, all $k$-point correlation functions of vertex types admit a determinantal representation: \begin{align*} \mathbb{P}\Bigg( \bigcap_{p=1}^k \{ \text{vertex at } v^p \text{ has type } t_p \} \Bigg) = \left( \prod_{p=1}^k a_{t_p} \right) \det\big[ L(x^i,y^j) \big]_{i,j=1}^{2k}, \end{align*} where $t_1,\ldots,t_k \in \{1,\ldots,6\}$ label the six possible vertex types, and $\{a_t : 1 \leq t \leq 6\}$ are the corresponding six-vertex weights. For each $1 \leq p \leq k$, the four points $x^{2p-1}, x^{2p}, y^{2p-1}, y^{2p} \in (\mathbb{Z}/2)^2$ are $t_p$-dependent choices among the midpoints of the edges incident to $v^p$. The correlation kernel $L$ has the contour integral representation \begin{align*} L(x,y) = \oint_{|w_1|=1} \oint_{|w_2|=1} \frac{dw_1}{2πi\, w_1}\, \frac{dw_2}{2πi\, w_2}\, w_1^{\,y_1 - x_1}\, w_2^{\,y_2 - x_2}\, h\big(c(x),c(y);w_1,w_2\big), \end{align*} where $h\big(c(x),c(y);w_1,w_2\big)$ is a simple rational function of $(w_1,w_2)$ that depends on $x$ and $y$ only through their orientations $c(x)$ and $c(y)$. Our proof is fully self-contained: we construct a determinantal point process on $\mathbb{Z}^2$ and identify the six-vertex model as its pushforward under an explicit mapping.

math.PR↗

arXiv2025

Continuous Kasteleyn theory for the bead model

Consider the semi-discrete torus $\mathbb{T}_n := [0,1) \times \{0,1,\ldots,n-1\}$ representing $n$ unit length strings running in parallel. A bead configuration on $\mathbb{T}_n$ is a point process on $\mathbb{T}_n$ with the property that between every two consecutive points on the same string, there lies a point on each of the neighbouring strings. In this article we develop a continuous version of Kasteleyn theory to show that partition functions for bead configurations on $\mathbb{T}_n$ may be expressed in terms of Fredholm determinants of certain operators on $\mathbb{T}_n$. We obtain an explicit formula for the volumes of bead configurations on $\mathbb{T}_n$. The asymptotics of this formula confirm a recent prediction in the free probability literature. Thereafter we study random bead configurations on $\mathbb{T}_n$, showing that they have a determinantal structure which can be connected with exclusion processes. We use this machinery to construct a new probabilistic representation of TASEP on the ring.

math.PR↗

arXiv2025

The Brownian marble

Let $R:(0,\infty) \to [0,\infty)$ be a measurable function. Consider coalescing Brownian motions started from every point in the subset $\{ (0,x) : x \in \mathbb{R} \}$ of $[0,\infty) \times \mathbb{R}$ (with $[0,\infty)$ denoting time and $\mathbb{R}$ denoting space) and proceeding according to the following rule: the interval $\{t\} \times [L_t,U_t]$ between two consecutive Brownian motions instantaneously fragments' at rate $R(U_t - L_t)$. At a fragmentation event at a time $t$, we initiate new coalescing Brownian motions from each of the points $\{ (t,x) : x \in [L_t,U_t]\}$. The resulting process, which we call the $R$-marble, is easily constructed when $R$ is bounded, and may be considered a random subset of the Brownian web. Under mild conditions, we show that it is possible to construct the $R$-marble when $R$ is unbounded as a limit as $n \to \infty$ of $R_n$-marbles where $R_n(g) = R(g) \wedge n$. The behaviour of this limiting process is mainly determined by the shape of $R$ near zero. The most interesting case occurs when the limit $\lim_{g \downarrow 0} g^2 R(g) = λ$ exists in $(0,\infty)$, in which case we find a phase transition. For $λ\geq 6$, the limiting object is indistinguishable from the Brownian web, whereas if $λ< 6$, then the limiting object is a nontrivial stochastic process with large gaps. When $R(g) = λ/g^2$, the $R$-marble is a self-similar stochastic process which we refer to as the \emph{Brownian marble with parameter $λ> 0$}. We give an explicit description of the spacetime correlations of the Brownian marble, which can be described in terms of an object we call the Brownian vein; a spatial version of a recurrent extension of a killed Bessel-$3$ process.

math.PR↗

arXiv2025

The Integrable Snake Model

A pure snake configuration is a bijection $σ:\mathbb{Z}^2 \to \mathbb{Z}^2$ containing no two-cycles and such that for each $x \in \mathbb{Z}^2$ we have $σ(x) \in \{ x , x+ \mathbf{e}^1, x+\mathbf{e}^2 , x- \mathbf{e}^2 \}.$ The non-trivial cycles of a pure snake configuration may be regarded as a collection of non-intersecting paths in $\mathbb{Z}^2$ that may travel right, up, or down (but not left) from a given vertex. Pure snake configurations are a generalisation of lozenge tilings, which are in natural correspondence with paths that only travel right or up. We introduce a partition function on a finite version of this model and study the probabilistic properties of random pure snake configurations chosen according to their contribution to this partition function. Under a suitable weighting, the model is integrable in the sense that we have access to explicit formulas for its partition function and correlation function. We utilise the integrable structure of this model in several applications through its various scaling limits, such as to prove a traffic representation of ASEP on the ring, generalising the analogous result for TASEP by the first author.

math.PR↗

arXiv2025

On the limiting Horn inequalities

The Horn inequalities characterise the possible spectra of triples of $n$-by-$n$ Hermitian matrices $A+B=C$. We study integral inequalities that arise as limits of Horn inequalities as $n \to \infty$. These inequalities are parametrised by the points of an infinite-dimensional convex body, the asymptotic Horn system $\mathscr{H}[0,1]$, which can be regarded as a topological closure of the countable set of Horn inequalities for all finite $n$. We prove three main results. The first shows that arbitrary points of $\mathscr{H}[0,1]$ can be well approximated by specific sets of finite-dimensional Horn inequalities. Our second main result shows that $\mathscr{H}[0,1]$ has a remarkable self-characterisation property. That is, membership in $\mathscr{H}[0,1]$ is determined by the very inequalities corresponding to the points of $\mathscr{H}[0,1]$ itself. To illuminate this phenomenon, we sketch a general theory of sets that characterise themselves in the sense that they parametrise their own membership criteria, and we consider the question of what further information would be needed in order for this self-characterisation property to determine the Horn inequalities uniquely. Our third main result is a quantitative result on the redundancy of the Horn inequalities in an infinite-dimensional setting. Concretely, the Horn inequalities for finite $n$ are indexed by certain sets $T^n_r$ with $1 \le r \le n-1$; we show that if $(n_k)_{k \ge 1}$ and $(r_k)_{k \ge 1}$ are any sequences such that $(r_k / n_k)_{k \ge 1}$ is a dense subset of $(0,1)$, then the Horn inequalities indexed by the sets $T^{n_k}_{r_k}$ are sufficient to imply all of the others.

math.FA↗

arXiv2024

Random planar trees and the Jacobian conjecture

We develop a probabilistic approach to the celebrated Jacobian conjecture, which states that any Keller map (i.e. any polynomial mapping $F\colon \mathbb{C}^n \to \mathbb{C}^n$ whose Jacobian determinant is a nonzero constant) has a compositional inverse which is also a polynomial. The Jacobian conjecture may be formulated in terms of a problem involving labellings of rooted trees; we give a new probabilistic derivation of this formulation using multi-type branching processes. Thereafter, we develop a simple and novel approach to the Jacobian conjecture in terms of a problem involving shuffling subtrees of $d$-Catalan trees, i.e. planar $d$-ary trees. We also show that, if one can construct a certain Markov chain on large $d$-Catalan trees which updates its value by randomly shuffling certain nearby subtrees, and in such a way that the stationary distribution of this chain is uniform, then the Jacobian conjecture is true. Finally, we use the local limit theory of large random trees to show that the subtree shuffling conjecture is true in a certain asymptotic sense, and thereafter use our machinery to prove an approximate version of the Jacobian conjecture, stating that inverses of Keller maps have small power series coefficients for their high degree terms.

math.CO↗

arXiv2023

A fast Berry-Esseen theorem under minimal density assumptions

Let $X_1,\ldots,X_N$ be i.i.d.\ random variables distributed like $X$. Suppose that the first $k \geq 3$ moments $\{ \mathbb{E}[X^j] : j = 1,\ldots,k\}$ of $X$ agree with that of the standard Gaussian distribution, that $\mathbb{E}[|X|^{k+1}] < \infty$, and that there is a subinterval of $\mathbb{R}$ of width $w$ over which the law of $X$ has a density of at least $h$. Then we show that \begin{align} \label{eq:bnew} \sup_{s \in \mathbb{R}} \left| \mathbb{P} \left( \frac{X_1 + \ldots + X_N}{ \sqrt{N} } \leq s \right) - \int_{-\infty}^s \frac{ e^{ - u^2/2} \mathrm{d} u }{ \sqrt{2 π}} \right| \leq 3 \left\{ \frac{\mathbb{E}[|X|^{k+1}]}{ N^{ \frac{k-1}{2}} } + e^{ - c hw^3 N/\mathbb{E}[|X|^{k+1}] } \right\}, \end{align} where $c > 0$ is universal. By setting $k=3$, we see that in particular all symmetric random variables with densities and finite fourth moment satisfy a Berry-Esseen inequality with a bound of the order $1/N$. Thereafter, we study the Berry-Esseen theorem as it pertains to perturbations of the Bernoulli law with a small density component, showing by means of a reverse inequality that the power $hw^3$ in the exponential term is asymptotically sharp.

math.PR↗

arXiv2023

Beads on the torus via scaling limits of dimer matchings

In a previous article, we develop a continuous version of Kasteleyn theory to study the bead model on the torus. These are the point processes on the semi-discrete torus $\mathbb{T}_n := [0,1) \times \{0,1,\ldots,n-1\}$ (thought of as $n$ unit length strings wrapped around a doughnut) with the property that between every two consecutive points on same string, there lies a point on the neighbouring strings. In this companion article, we obtain the main results of the previous article via an alternative route, using scaling limits of dimer models as opposed to the continuous Kasteleyn theory. In any case, we hope that the article may serve as a gentle introduction to Kasteleyn theory on the torus.

math.PR↗

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.

math.PR↗

arXiv2022

Faà di Bruno's formula and inversion of power series

Faà di Bruno's formula gives an expression for the derivatives of the composition of two real-valued functions. In this paper we prove a multivariate and synthesized version of Faà di Bruno's formula in higher dimensions, providing a combinatorial expression for the derivatives of chain compositions $F^{(1)} \circ \ldots \circ F^{(m)}$ of functions $F^{(l)} : \mathbb{R}^N \to \mathbb{R}^N$ in terms of sums over labelled trees. We give several applications of this formula, including a new involution formula for the inversion of multivariate power series. We use this framework to outline a combinatorial approach to studying the invertibility of polynomial mappings, giving a purely combinatorial restatement of the Jacobian conjecture. Our methods extend naturally to the non-commutative case, where we prove a free version of Faà di Bruno's formula for multivariate power series in free indeterminates, and use this formula as a tool for obtaining a new inversion formula for free power series.

math.CO↗

arXiv2022

Ancestral reproductive bias in branching processes

Consider a branching process with a homogeneous reproduction law. Sampling a single cell uniformly from the population at a time $T > 0$ and looking along the sampled cell's ancestral lineage, we find that the reproduction law is heterogeneous - the expected reproductive output of ancestral cells on the lineage from time $0$ to time $T$ continuously increases. This `inspection paradox' is due to sampling bias, that cells with a larger number of offspring are more likely to have one of their descendants sampled by virtue of their prolificity, and the bias's strength grows with the random population size and/or the sampling time $T$. Our main result explicitly characterises the evolution of reproduction rates and sizes along the sampled ancestral lineage as a mixture of Poisson processes, which simplifies in special cases. The ancestral bias helps to explain recently observed variation in mutation rates along lineages of the developing human embryo.

q-bio.PE↗

arXiv2022

A degree bound for strongly nilpotent polynomial automorphisms

Let $k$ be a field of characteristic zero. Let $F = X + H$ be a polynomial mapping from $k^n \to k^n$, where $X$ is the identity mapping and $H$ has only degree two terms and higher. We say that the Jacobian matrix $JH$ of $H$ is strongly nilpotent with index $p$ if for all $X^{(1)},\ldots,X^{(p)} \in k^n$ we have \begin{align*} JH(X^{(1)})\ldots JH (X^{(p)}) = 0. \end{align*} Every $F$ of this form is a polynomial automorphism, i.e. there is a second polynomial mapping $F^{-1}$ such that $F \circ F^{-1} = F^{-1} \circ F = X$. We prove that the degree of the inverse $F^{-1}$ satisfies \begin{align*} deg(F^{-1}) \leq deg(F)^p, \end{align*} improving in the strongly nilpotent case on the well known degree bound $deg(F^{-1}) \leq deg(F)^n$ for general polynomial automorphisms.

math.AG↗