arXiv subjects

James Norris

Publications and source records attributed to James Norris.

16 recordsLinked to original sources

arXiv2025

Interlacing sequences resulting from an interval split-merge dynamics and the induced probability measures

We study sequences of partitions of the unit interval into subintervals, starting from the trivial partition, in which each partition is obtained from the one before by splitting its subintervals in two, according to a given rule, and then merging pairs of subintervals at the break points of the old partition. The $n$th partition then comprises $n+1$ subintervals with $n$ break points, which inherently possess an interlacing property. The empirical distribution of these points reveals a surprisingly rich structure, even when the splitting rule is completely deterministic. We consider both deterministic and randomized splitting rules and we study from multiple angles the limiting behavior of the empirical distribution of the break points.

math.PR↗

arXiv2022

Diffusion in small time in incomplete sub-Riemannian manifolds

For incomplete sub-Riemannian manifolds, and for an associated second-order hypoelliptic operator, which need not be symmetric, we identify two alternative conditions for the validity of Gaussian-type upper bounds on heat kernels and transition probabilities, with optimal constant in the exponent. Under similar conditions, we obtain the small-time logarithmic asymptotics of the heat kernel, and show concentration of diffusion bridge measures near a path of minimal energy. The first condition requires that we consider points whose distance apart is no greater than the sum of their distances to infinity. The second condition requires only that the operator not be too asymmetric.

math.PR↗

arXiv2021

Scaling limits for planar aggregation with subcritical fluctuations

We study scaling limits of a family of planar random growth processes in which clusters grow by the successive aggregation of small particles. In these models, clusters are encoded as a composition of conformal maps and the location of each successive particle is distributed according to the density of harmonic measure on the cluster boundary, raised to some power. We show that, when this power lies within a particular range, the macroscopic shape of the cluster converges to a disk, but that as the power approaches the edge of this range the fluctuations approach a critical point, which is a limit of stability. The methodology developed in this paper provides a blueprint for analysing more general random growth models, such as the Hastings-Levitov family.

math.PR↗

arXiv2021

Stability of regularized Hastings-Levitov aggregation in the subcritical regime

We prove bulk scaling limits and fluctuation scaling limits for a two-parameter class ALE$(α,η)$ of continuum planar aggregation models. The class includes regularized versions of the Hastings--Levitov family HL$(α)$ and continuum versions of the family of dielectric breakdown models, where the local attachment intensity for new particles is specified as a negative power $-η$ of the density of arc length with respect to harmonic measure. The limit dynamics follow solutions of a certain Loewner--Kufarev equation, where the driving measure is made to depend on the solution and on the parameter $ζ=α+η$. Our results are subject to a subcriticality condition $ζ\le1$: this includes HL$(α)$ for $α\le1$ and also the case $α=2,η=-1$ corresponding to a continuum Eden model. Hastings and Levitov predicted a change in behaviour for HL$(α)$ at $α=1$, consistent with our results. In the regularized regime considered, the fluctuations around the scaling limit are shown to be Gaussian, with independent Ornstein--Uhlenbeck processes driving each Fourier mode, which are seen to be stable if and only if $ζ\le1$.

math.PR↗

arXiv2018

Small-time fluctuations for the bridge of a sub-Riemannian diffusion

We consider small-time asymptotics for diffusion processes conditioned by their initial and final positions, under the assumption that the diffusivity has a sub-Riemannian structure, not necessarily of constant rank. We show that, if the endpoints are joined by a unique path of minimal energy, and lie outside the sub-Riemannian cut locus, then the fluctuations of the conditioned diffusion from the minimal energy path, suitably rescaled, converge to a Gaussian limit. The Gaussian limit is characterized in terms of the bicharacteristic flow, and also in terms of a second variation of the energy functional at the minimal path, the formulation of which is new in this context.

math.PR↗

arXiv2017

Yang-Mills measure and the master field on the sphere

We study the Yang--Mills measure on the sphere with unitary structure group. In the limit where the structure group has high dimension, we show that the traces of loop holonomies converge in probability to a deterministic limit, which is known as the master field on the sphere. The values of the master field on simple loops are expressed in terms of the solution of a variational problem. We show that, given its values on simple loops, the master field is characterized on all loops of finite length by a system of differential equations, known as the Makeenko--Migdal equations. We obtain a number of further properties of the master field. On specializing to families of simple loops, our results identify the high-dimensional limit, in non-commutative distribution, of the Brownian loop in the group of unitary matrices.

math.PR↗

arXiv2016

A consistency estimate for Kac's model of elastic collisions in a dilute gas

An explicit estimate is derived for Kac's mean-field model of colliding hard spheres, which compares, in a Wasserstein distance, the empirical velocity distributions for two versions of the model based on different numbers of particles. For suitable initial data, with high probability, the two processes agree to within a tolerance of order $N^{-1/d}$, where $N$ is the smaller particle number and $d$ is the dimension, provided that $d\ge3$. From this estimate we can deduce that the spatially homogeneous Boltzmann equation is well posed in a class of measure-valued processes and provides a good approximation to the Kac process when the number of particles is large. We also prove in an appendix a basic lemma on the total variation of time-integrals of time-dependent signed measures.

math.PR↗

arXiv2015

Weak convergence of the localized disturbance flow to the coalescing Brownian flow

We define a new state-space for the coalescing Brownian flow, also known as the Brownian web, on the circle. The elements of this space are families of order-preserving maps of the circle, depending continuously on two time parameters and having a certain weak flow property. The space is equipped with a complete separable metric. A larger state-space, allowing jumps in time, is also introduced, and equipped with a Skorokhod-type metric, also complete and separable. We prove that the coalescing Brownian flow is the weak limit in this larger space of a family of flows which evolve by jumps, each jump arising from a small localized disturbance of the circle. A local version of this result is also obtained, in which the weak limit law is that of the coalescing Brownian flow on the line. Our set-up is well adapted to time-reversal and our weak limit result provides a new proof of time-reversibility of the coalescing Brownian flow. We also identify a martingale associated with the coalescing Brownian flow on the circle and use this to make a direct calculation of the Laplace transform of the time to complete coalescence.

math.PR↗

arXiv2015

Influence in product spaces

The theory of influence and sharp threshold is a key tool in probability and probabilistic combinatorics, with numerous applications. One significant aspect of the theory is directed at identifying the level of generality of the product probability space that accommodates the event under study. We derive the influence inequality for a completely general product space, by establishing a relationship to the Lebesgue cube studied by Bourgain, Kahn, Kalai, Katznelson, and Linial (BKKKL) in 1992. This resolves one of the assertions of BKKKL. Our conclusion is valid also in the setting of the generalized influences of Keller.

math.PR↗

arXiv2014

Measure solutions for the Smoluchowski coagulation-diffusion equation

A notion of measure solution is formulated for a coagulation-diffusion equation, which is the natural counterpart of Smoluchowski's coagulation equation in a spatially inhomogeneous setting. Some general properties of such solutions are established. Sufficient conditions are identified on the diffusivity, coagulation rates and initial data for existence, uniqueness and mass conservation of solutions. These conditions impose no form of monotonicity on the coagulation kernel, which may depend on complex characteristics of the particles. They also allow singular behaviour in both diffusivity and coagulation rates for small particles. The general results apply to the Einstein-Smoluchowski model for colloidal particles suspended in a fluid.

math.AP↗

arXiv2014

Surprise probabilities in Markov chains

In a Markov chain started at a state $x$, the hitting time $τ(y)$ is the first time that the chain reaches another state $y$. We study the probability $\mathbf{P}_x(τ(y) = t)$ that the first visit to $y$ occurs precisely at a given time $t$. Informally speaking, the event that a new state is visited at a large time $t$ may be considered a "surprise". We prove the following three bounds: 1) In any Markov chain with $n$ states, $\mathbf{P}_x(τ(y) = t) \le \frac{n}{t}$. 2) In a reversible chain with $n$ states, $\mathbf{P}_x(τ(y) = t) \le \frac{\sqrt{2n}}{t}$ for $t \ge 4n + 4$. 3) For random walk on a simple graph with $n \ge 2$ vertices, $\mathbf{P}_x(τ(y) = t) \le \frac{4e \log n}{t}$. We construct examples showing that these bounds are close to optimal. The main feature of our bounds is that they require very little knowledge of the structure of the Markov chain. To prove the bound for random walk on graphs, we establish the following estimate conjectured by Aldous, Ding and Oveis-Gharan (private communication): For random walk on an $n$-vertex graph, for every initial vertex $x$, \[ \sum_y \left( \sup_{t \ge 0} p^t(x, y) \right) = O(\log n). \]

math.PR↗

arXiv2012

Diffusivity of a random walk on random walks

We consider a random walk $(Z^{(1)}_n, ..., Z^{(K+1)}_n) \in \mathbb{Z}^{K+1}$ with the constraint that each coordinate of the walk is at distance one from the following one. In this paper, we show that this random walk is slowed down by a variance factor $σ_K^2 = \frac{2}{K+2}$ with respect to the case of the classical simple random walk without constraint.

math.PR↗

arXiv2011

Hastings-Levitov aggregation in the small-particle limit

We establish some scaling limits for a model of planar aggregation. The model is described by the composition of a sequence of independent and identically distributed random conformal maps, each corresponding to the addition of one particle. We study the limit of small particle size and rapid aggregation. The process of growing clusters converges, in the sense of Caratheodory, to an inflating disc. A more refined analysis reveals, within the cluster, a tree structure of branching fingers, whose radial component increases deterministically with time. The arguments of any finite sample of fingers, tracked inwards, perform coalescing Brownian motions. The arguments of any finite sample of gaps between the fingers, tracked outwards, also perform coalescing Brownian motions. These properties are closely related to the evolution of harmonic measure on the boundary of the cluster, which is shown to converge to the Brownian web.

math.PR↗

arXiv2008

Planar aggregation and the coalescing Brownian flow

We study a scaling limit associated to a model of planar aggregation. The model is obtained by composing certain independent random conformal maps. The evolution of harmonic measure on the boundary of the cluster is shown to converge to the coalescing Brownian flow.

math.PR↗

arXiv2004

Strong approximation for the supermarket model

We prove three strong approximation theorems for the `supermarket' or `join the shortest queue' model -- a law of large numbers, a jump process approximation and a central limit theorem. The estimates are carried through rather explicitly. This allows us to estimate each of the infinitely many components of the process in its own scale and to exhibit a cut-off in the set of active components which grows slowly with the number of servers.

math.PR↗

arXiv2004

Essential edges in Poisson random hypergraphs

Consider a random hypergraph on a set of N vertices in which, for k between 1 and N, a Poisson(N beta_k) number of hyperedges is scattered randomly over all subsets of size k. We collapse the hypergraph by running the following algorithm to exhaustion: pick a vertex having a 1-edge and remove it; collapse the hyperedges over that vertex onto their remaining vertices; repeat until there are no 1-edges left. We call the vertices removed in this process "identifiable". Also any hyperedge all of whose vertices are removed is called "identifiable". We say that a hyperedge is "essential" if its removal prior to collapse would have reduced the number of identifiable vertices. The limiting proportions, as N tends to infinity, of identifiable vertices and hyperedges were obtained by Darling and Norris. In this paper, we establish the limiting proportion of essential hyperedges. We also discuss, in the case of a random graph, the relation of essential edges to the 2-core of the graph, the maximal sub-graph with minimal vertex degree 2.

math.PR↗