arXiv subjects

Frankie Higgs

Publications and source records attributed to Frankie Higgs.

5 recordsLinked to original sources

arXiv2026

Tip growth in a strongly concentrated aggregation model follows local geodesics

We analyse the aggregate Loewner evolution (ALE), introduced in 2018 by Sola, Turner and Viklund to generalise versions of diffusion limited aggregation (DLA) in the plane using complex analysis. They showed convergence of the ALE for certain parameters to a single growing slit. Started from a non-trivial initial configuration of $k$ needles and the same parameters, we show that the small-particle scaling limit of ALE is the Laplacian path model, introduced by Carleson and Makarov in 2002, in which the tips grow along geodesics towards $\infty$. Our proof involves analysis of Loewner's equation near its singular points, and we extend martingale methods to the backward equation, where what we have to control is non-adapted. Most conformal growth models introduce an extra regularisation factor to deal with the singularities in Loewner's equation at the sharp tips and right-angle bases of slit particles. As an intermediate step we prove a limit result for a model with no such regularisation factor, developing methods which should prove useful in analysing other weakly-regularised models with non-trivial limits.

math.PR↗

arXiv2025

Random coverage from within with variable radii, and Johnson-Mehl cover times

Given a compact planar region $A$, let $τ_A$ be the (random) time it takes for the Johnson-Mehl tessellation of $A$ to be complete, i.e. the time it takes for $A$ to be fully covered by a spatial birth-growth process in $A$ with seeds arriving as a unit-intensity Poisson point process in $A \times [0,\infty)$, where upon arrival each seed grows at unit rate in all directions. We show that if $\partial A$ is smooth or polygonal then $\Pr [ πτ_{sA}^3 - 6 \log s - 4 \log \log s \leq x]$ tends to $\exp(- (\frac{81}{4π})^{1/3} |A|e^{-x/3} -(\frac{9}{2π^2})^{1/3} |\partial A| e^{-x/6})$ in the large-$s$ limit; the second term in the exponent is due to boundary effects, the importance of which was not recognized in earlier work on this model. We present similar results in higher dimensions (where boundary effects dominate). These results are derived using new results on the asymptotic probability of covering $A$ with a high-intensity spherical Poisson Boolean model restricted to $A$ with grains having iid small random radii, which generalize recent work of the first author that dealt only with grains of deterministic radius.

math.PR↗

arXiv2025

Exponential decay for the random connection model using asymptotic transitivity

We prove that the probability the cluster of the origin in a subcritical Poisson random connection model (RCM) has size at least $n$ decays exponentially as $n$ increases, under minimal assumptions. We extend a recent method of Vanneuville (arXiv:2304.12110) from Bernoulli percolation on vertex-transitive graphs to the RCM. The key idea is that the subcritical RCM can be constructed by site percolation on a very high-intensity RCM. The latter RCM becomes ``almost vertex-transitive'' in a certain sense at very high intensities, which is a new method that we expect to be useful for other problems. We obtain the result for connection functions with unbounded support, a setting in which it was not previously known.

math.PR↗

arXiv2025

Covering one point process with another

Let $X_1,X_2, \ldots $ and $Y_1, Y_2, \ldots$ be i.i.d. random uniform points in a bounded domain $A \subset \mathbb{R}^2$ with smooth or polygonal boundary. Given $n,m,k \in \mathbb{N}$, define the {\em two-sample $k$-coverage threshold} $R_{n,m,k}$ to be the smallest $r$ such that each point of $ \{Y_1,\ldots,Y_m\}$ is covered at least $k$ times by the disks of radius $r$ centred on $X_1,\ldots,X_n$. We obtain the limiting distribution of $R_{n,m,k}$ as $n \to \infty$ with $m= m(n) \sim τn$ for some constant $τ>0$, with $k $ fixed. If $A$ has unit area, then $n πR_{n,m(n),1}^2 - \log n$ is asymptotically Gumbel distributed with scale parameter $1$ and location parameter $\log τ$. For $k >2$, we find that $n πR_{n,m(n),k}^2 - \log n - (2k-3) \log \log n$ is asymptotically Gumbel with scale parameter $2$ and a more complicated location parameter involving the perimeter of $A$; boundary effects dominate when $k >2$. For $k=2$ the limiting cdf is a two-component extreme value distribution with scale parameters 1 and 2. We also give analogous results for higher dimensions, where the boundary effects dominate for all $k$.

math.PR↗

arXiv2021

SLE scaling limits for a Laplacian random growth model

We consider a model of planar random aggregation from the ALE$(0,η)$ family where particles are attached preferentially in areas of low harmonic measure. We find that the model undergoes a phase transition in negative $η$, where for sufficiently large values the attachment distribution of each particle becomes atomic in the small particle limit, with each particle attaching to one of the two points at the base of the previous particle. This complements the result of Sola, Turner and Viklund for large positive $η$, where the attachment distribution condenses to a single atom at the tip of the previous particle. As a result of this condensation of the attachment distributions we deduce that in the limit as the particle size tends to zero the ALE cluster converges to a Schramm--Loewner evolution with parameter $κ= 4$ (SLE$_4$). We also conjecture that using other particle shapes from a certain family, we have a similar SLE scaling result, and can obtain SLE$_κ$ for any $κ\geq 4$.

math.PR↗