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Greg Markowsky

Publications and source records attributed to Greg Markowsky.

At least 19 recordsLinked to original sources
arXiv2025

Complex analytic proofs of two probabilistic theorems

In this paper, we use purely complex analytic techniques to prove two results of the first author which were hitherto given only probabilistic proofs. A general form of the Phragm\'en-Lindel\"of principle states that if the $p$\textsuperscript{th} Hardy norm of the conformal map from the disk to a simply connected domain is finite, then an analytic function on that domain is either bounded by its supremum on the boundary or else goes to $\ff$ along some sequence more rapidly than $e^{|z|^{p}}$. We will prove this and discuss a number of special cases. We also derive a series expansion for the Green's function of a disk, and show how it leads to an infinite product identity. The celebrated infinite product expansions for sine and cosine are realized as special cases.

math.CV↗
arXiv2025

A collection of results relating the geometry of plane domains and the exit time of planar Brownian motion, II

This paper is the sequel to another with the same name (Buttigieg et al., Comput. Methods Funct. Theory, 2023), and is concerned with results of the same type. We deduce a result on the moments of the exit time of Brownian motion from domains whose boundary curve is replaced by a dashed line, and from domains arising from a periodic tiling of the plane. We also give a construction of a type of domain which is similar to a wedge domain, but the behaviour of whose exit time moments answer several questions that had been speculated upon.

math.PR↗
arXiv2025

An improved bound for strongly regular graphs with smallest eigenvalue $-m$

In 1979, Neumaier gave a bound on $\lambda$ in terms of $m$ and $\mu$, where $-m$ is the smallest eigenvalue of a primitive strongly regular graph, unless the graph in question belongs to one of the two infinite families of strongly regular graphs. We improve this result. We also indicate how our methods can be used to give an alternate derivation of Bruck's Completion Theorem for orthogonal arrays.

math.CO↗
arXiv2024

A method of solution for the inverse problem for $h$-functions of planar Brownian motion

Given a planar domain $D$, the harmonic measure distribution function $h_D(r)$, with base point $z$, is the harmonic measure with pole at $z$ of the parts of the boundary which are within a distance $r$ of $z$. Equivalently it is the probability Brownian motion started from $z$ first strikes the boundary within a distance $r$ from $z$. We call $h_D$ the $h$-function of $D$, this function captures geometrical aspects of the domain, such as connectivity, or curvature of the boundary. This paper is concerned with the inverse problem: given a suitable function $h$, does there exist a domain $D$ such that $h = h_D$? To answer this, we first extend the concept of a $h$-function of a domain to one of a stopping time $\tau$ . By using the conformal invariance of Brownian motion we solve the inverse problem for that of a stopping time. The associated stopping time will be the projection of a hitting time of the real line. If this projection corresponds to the hitting time of a domain $D$, then this technique solves the original inverse problem. We have found a large family of examples such that the associated stopping time is that of a hitting time.

math.PR↗
arXiv2024

Stochastic Domination of Exit Times for Random Walks and Brownian Motion with Drift

In this note, by an elementary use of Girsanov's transform we show that the exit time for either a biased random walk or a drifted Brownian motion on a symmetric interval is stochastically monotone with respect to the drift parameter. In the random walk case, this gives an alternative proof of a recent result of E. Pek\"oz and R. Righter in 2024. Our arguments in both discrete and continuous cases are parallel to each other. We also outline a simple SDE proof for the Brownian case based on a standard comparison theorem.

math.PR↗
arXiv2024

On the exponential integrability of the derivative of intersection and self-intersection local time for Brownian motion and related processes

We show that the derivative of the intersection and self-intersection local times of alpha-stable processes are exponentially integrable for certain parameter values. This includes the Brownian motion case. We also discuss related results present in the literature for fractional Brownian motion, and in particular give a counter-example to a result in [Guo, J., Hu, Y., and Xiao, Y., Higher-order derivative of intersection local time for two independent fractional Brownian motions, Journal of Theoretical Probability 32, (2019), pp. 1190-1201] related to this question.

math.PR↗
arXiv2023

Non-geometric distance-regular graphs of diameter at least $3$ with smallest eigenvalue at least $-3$

In this paper, we classify non-geometric distance-regular graphs of diameter at least $3$ with smallest eigenvalue at least $-3$. This is progress towards what is hoped to be an eventual complete classification of distance-regular graphs with smallest eigenvalue at least $-3$, analogous to existing classification results available in the case that the smallest eigenvalue is at least $-2$.

math.CO↗
arXiv2023

Comments on the infinitely divisibility of the Conway--Maxwell--Poisson distribution

In an elegant recent paper \cite{geng2022conway}, Geng and Xia settled the question of the infinite divisibility of the Conway--Maxwell--Poisson distribution, using in large part several results from complex analysis. In this note we show how these complex analytic methods can be circumvented, thereby giving a proof of their result which is completely elementary.

math.PR↗
arXiv2021

The transformation of edge-regular and pseudo strongly regular graphs under graph operations

The graph $G$ is said to be strongly regular with parameters $(n,k,\lambda,\mu)$ if the following conditions hold: (1) each vertex has $k$ neighbours; (2) any two adjacent vertices of $G$ have $\lambda$ common neighbours; (3) any two non-adjacent vertices of $G$ have $\mu$ common neighbours. In this paper we study two weaker notions of strongly regular graphs. A graph satisfying the conditions $(1)$ and $(2)$ is called an edge-regular graph with parameters $(n,k,\lambda)$. We call a graph satisfying the conditions $(1)$ and $(3)$ a pseudo strongly regular graph with parameters $(n,k,\mu)$. In this paper we study the impact of various graph operations on edge regular graphs and pseudo strongly regular graphs.

math.CO↗
arXiv2021

A collection of results relating the geometry of plane domains and the exit time of planar Brownian motion

We prove a number of results relating exit times of planar Brownian with the geometric properties of the domains in question. Included are proofs of the conformal invariance of moduli of rectangles and annuli using Brownian motion; similarly probabilistic proofs of some recent results of Karafyllia on harmonic measure on starlike domains; examples of domains and their complements which are simultaneously large when measured by the moments of exit time of Brownian motion, and examples of domains and their complements which are simultaneously small; and proofs of several identities involving the Cauchy distribution using the optional stopping theorem.

math.PR↗
arXiv2021

Binary sequences with a Ces\`aro limit

The Ces\`aro limit - the asymptotic average of a sequence of real numbers - is an operator of fundamental importance in probability, statistics and mathematical analysis. To better understand sequences with Ces\`aro limits, this paper considers the space $\mathcal{F}$ comprised of all binary sequences with a Ces\`aro limit, and the associated functional $\nu: \mathcal{F} \rightarrow [0,1]$ mapping each such sequence to its Ces\`aro limit. The basic properties of $\mathcal{F}$ and $\nu$ are enumerated, and chains (totally ordered sets) in $\mathcal{F}$ on which $\nu$ is countably additive are studied in detail. The main result of the paper concerns a structural property of the pair $(\mathcal{F},\nu)$, specifically that $\mathcal{F}$ can be factored (in a certain sense) to produce a monotone class on which $\nu$ is countably additive. In the process, a slight generalisation and clarification of the monotone class theorem for Boolean algebras is proved.

math.FA↗
arXiv2021

Spectrum of Strongly Regular Graphs under Graph Operators

In this paper, we show that if G is strongly regular then the Gallai graph and the anti-Gallai graph of G are edge-regular. We also identify conditions under which the Gallai and anti-Gallai graphs are themselves strongly regular, as well as conditions under which they are 2-connected. We include also a number of concrete examples and a discussion of spectral properties of the Gallai and anti-Gallai graphs.

math.CO↗
arXiv2021

On the duration of stays of Brownian motion in domains in Euclidean space

Let $T_D$ denote the first exit time of a Brownian motion from a domain $D$ in ${\mathbb R}^n$. Given domains $U,W \subseteq {\mathbb R}^n$ containing the origin, we investigate the cases in which we are more likely to have fast exits from $U$ than $W$, meaning ${\bf P}(T_U {\bf P}(T_W t) > {\bf P}(T_W>t)$ for $t$ large. This result, which applies only in two dimensions, shows that the unit disk has the lowest probability of long stays amongst all Schlicht domains.

math.PR↗
arXiv2021

Several consequences of adding or removing an edge from an electric network

In certain instances an electric network transforms in natural ways by the addition or removal of an edge. This can have interesting consequences for random walks, in light of the known relationships between electric resistance and random walks. We exhibit several instances in which this can be used to prove facts or simplify calculations. In particular, a new proof is given for the formula for the expected return time of a random walk on a graph. We also show how hitting times can be calculated in certain instances when a network differs from a highly symmetric one by one edge.

math.CO↗
arXiv2021

A theory of integration for Ces\`aro limits

The Ces\`aro limit - the asymptotic average of a sequence of real numbers - is an operator of fundamental importance in probability, statistics and analysis. Surprisingly, spaces of sequences with Ces\`aro limits have not previously been studied. This paper introduces spaces of such sequences, denoted $K_p(\mathcal{A})$, with the Ces\`aro limit acting as a kind of integral. The space $\mathcal{F}$ comprised of all binary sequences with a Ces\`aro limit is studied first, along with the associated functional $\nu: \mathcal{F} \rightarrow [0,1]$ mapping each such sequence to its Ces\`aro limit. It is shown that $\mathcal{F}$ can be factored to produce a monotone class on which $\nu$ induces a countably additive set function. The space $K_p(\mathcal{A})$ is then defined, and a quotient denoted $\mathcal{K}_p(\mathcal{A})$ is shown to be isometrically isomorphic, under certain conditions, to the function space $\mathcal{L}_p(\mathbb{N},\mathcal{A},\nu)$, where $\mathcal{A}$ is a field of sets isomorphic to a subset of $\mathcal{F}$, and $\nu$ is a finitely additive measure induced by the functional mentioned above. The Ces\`aro limit of an element of $K_p(\mathcal{A})$ is shown to be equal to its integral. The complete $\mathcal{L}_p(\mathbb{N},\mathcal{A},\nu)$ spaces (and by implication, the $\mathcal{K}_p(\mathcal{A})$ spaces isomorphic to them) are characterised, and a sufficient condition for these spaces to be separable is identified.

math.CA↗
arXiv2021

On the finiteness of moments of the exit time of planar Brownian motion from comb domains

A comb domain is defined to be the entire complex plain with a collection of vertical slits, symmetric over the real axis, removed. In this paper, we consider the question of determining whether the exit time of planar Brownian motion from such a domain has finite $p$-th moment. This question has been addressed before in relation to starlike domains, but these previous results do not apply to comb domains. Our main result is a sufficient condition on the location of the slits which ensures that the $p$-th moment of the exit time is finite. Several auxiliary results are also presented, including a construction of a comb domain whose exit time has infinite $p$-th moment for all $p \geq 1/2$.

math.PR↗
arXiv2020

Existence, renormalization, and regularity properties of higher order derivatives of self-intersection local time of fractional Brownian motion

In a recent paper by Yu (arXiv:2008.05633, 2020), higher order derivatives of self-intersection local time of fractional Brownian motion were defined, and existence over certain regions of the Hurst parameter $H$ was proved. Utilizing the Wiener chaos expansion, we provide new proofs of Yu's results, and show how a Varadhan-type renormalization can be used to extend the range of convergence for the even derivatives.

math.PR↗