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Greg Markowsky

Publications and source records attributed to Greg Markowsky.

At least 37 records · Page 2Linked to original sources
arXiv2020

Remarks on results by Müger and Tuset on the moments of polynomials

Let $f(x)$ be a non-zero polynomial with complex coefficients, and $M_p = \int_{0}^1 f(x)^p dx$ for $p$ a positive integer. In a recent paper, Müger and Tuset showed that $\limsup_{p \to \infty} |M_p|^{1/p} > 0$, and conjectured that this limit is equal to the maximum amongst the critical values of $f$ together with the values $|f(0)|$ and $|f(1)|$. We give an example that shows that this conjecture is false. It also may be natural to guess that $\limsup_{p \to \infty} |M_p|^{1/p}$ is equal to the maximum of $|f(x)|$ on $[0,1]$. However, we give a counterexample to this as well. We also provide a few more guesses as to the behaviour of the quantity $\limsup_{p \to \infty} |M_p|^{1/p}$.

math.CV↗
arXiv2020

A note on the moments of sequences of complex numbers

We give a short proof that the limsup of the p-th root of the modulus of the p-th moment of a sequence of complex numbers is equal to the modulus of the maximum of the sequence.This strengthens known results, and provides an analog to a recent result concerning moments of complex polynomials.

math.CV↗
arXiv2020

On the Dirichlet eigenvalue problem and the conformal Skorokhod embedding problem

In a recent work by Gross, the following problem was stated and solved: given a measure $μ$ with finite second moment, find a simply connected domain $U$ in $\CC$ such that the real part of a Brownian motion stopped when it leaves $U$ is distributed as $μ$. The construction developed by Gross yields a domain which is symmetric with respect to the real axis, but it has been noted by other authors that other domains are also possible, in particular there are a number of examples which have the property that a vertical ray starting at a point in the domain lies entirely within the domain. In this paper we give a new solution to the problem posed by Gross, and show that these other cases noted before are special cases of this method. We further show that the domain generated by this method has the property that it always has the minimal rate (as defined in terms of the spectrum of the Laplacian operator) among all possible domains corresponding to a fixed distribution $μ$, which gives a partial solution to a question posed by Mariano and Panzo. We show that the domain is unique, provided certain conditions are imposed, and use this to give several examples. We also describe a method for identifying the boundary curve of the domain, and discuss several other related topics.

math.PR↗
arXiv2020

Remarks on the speeds of a class of random walks on the integers

In recent years, there has been an interest in deriving certain important probabilistic results as consequences of deterministic ones; see for instance \cite{beig} and \cite{acc}. In this work, we continue on this path by deducing a well known equivalence between the speed of random walks on the integers and the growth of the size of their ranges. This result is an immediate consequence of the Kesten-Spitzer-Whitman theorem, and by appearances is probabilistic in nature, but we will show that it follows easily from an elementary deterministic result. We also investigate the common property of recurrent random walks of having speed zero, and show by example that this property need not be shared by deterministic sequences. However, if we consider the inter-arrival times (times at which the sequence is equal to 0) then we find a sufficient deterministic condition for a sequence to have zero speed, and show that this can be used to derive several probabilistic results.

math.PR↗
arXiv2020

Maximizing the $p$-th moment of exit time of planar Brownian motion from a given domain

In this paper we address the question of finding the point which maximizes the $p$-th moment of the exit time of planar Brownian motion from a given domain. We present a geometrical method of excluding parts of the domain from consideration which makes use of a coupling argument and the conformal invariance of Brownian motion. In many cases the maximizing point can be localized to a relatively small region. Several illustrative examples are presented.

math.PR↗
arXiv2020

On the probability of fast exits and long stays of planar Brownian motion in simply connected domains

Let $T^D$ denote the first exit time of a planar Brownian motion from a domain $D$. Given two simply connected planar domains $U,W \neq \SC$ containing $0$, we investigate the cases in which we are more likely to have fast exits (meaning for instance ${\bf P}(T^U {\bf P}(T^W t) > {\bf P}(T^W>t)$ for $t$ large). We prove several results on these questions. In particular, we show that the primary factor in the probability of fast exits is the proximity of the boundary to the origin, while for long stays an important factor is the moments of the exit time. The complex analytic theory that motivated our inquiry is also discussed.

math.PR↗
arXiv2019

Remarks on Gross' technique for obtaining a conformal Skorohod embedding of planar Brownian motion

In a recent work by Gross, it was proved that, given a distribution $μ$ with zero mean and finite second moment, we can find a simply connected domain $Ω$ such that if $Z_{t}$ is a standard planar BM, then $\mathcal{R}e(Z_{τ_Ω})$ has the distribution $μ$. In this note, we extend his method to prove that if $μ$ has a finite $L^{p}$ moment then the exit time $τ_Ω$ has a finite moment of order $\frac{p}{2}$.

math.PR↗
arXiv2019

A note on invariance of the Cauchy and related distributions

It is known that if $f$ is an analytic self map of the complex upper half-plane which also maps $\mathbb{R}\cup\{\infty\}$ to itself, and $f(i)=i$, then $f$ preserves the Cauchy distribution. This note concerns three results related to the above fact.

math.PR↗
arXiv2019

On the Cheeger constant for distance-regular graphs

The Cheeger constant of a graph is the smallest possible ratio between the size of a subgraph and the size of its boundary. It is well known that this constant must be at least $\frac{λ_1}{2}$, where $λ_1$ is the smallest positive eigenvalue of the Laplacian matrix. The subject of this paper is a conjecture of the authors that for distance-regular graphs the Cheeger constant is at most $λ_1$. In particular, we prove the conjecture for the known infinite families of distance-regular graphs, distance-regular graphs of diameter 2 (the strongly regular graphs), several classes of imprimitive distance-regular graphs, and most distance-regular graphs with small valency.

math.CO↗
arXiv2019

On the nonexistence of pseudo-generalized quadrangles

In this paper we consider the question of when a strongly regular graph with parameters $((s+1)(st+1),s(t+1),s-1,t+1)$ can exist. These parameters arise when the graph is derived from a generalized quadrangle, but there are other examples which do not arise in this manner, and we term these {\it pseudo-generalized quadrangles}. If the graph is a generalized quadrangle then $t \leq s^2$ and $s \leq t^2$, while for pseudo-generalized quadrangles we still have the former bound but not the latter. Previously, Neumaier has proved a bound for $s$ which is cubic in $t$, but we improve this to one which is quadratic. The proof involves a careful analysis of cliques and cocliques in the graph. This improved bound eliminates many potential parameter sets which were otherwise feasible.

math.CO↗
arXiv2017

On the planar Brownian Green's function for stopping times

It has been known for some time that the Green's function of a planar domain can be defined in terms of the exit time of Brownian motion, and this definition has been extended to stopping times more general than exit times. In this paper, we extend the notion of conformal invariance of Green's function to analytic functions which are not injective, and use this extension to calculate the Green's function for a stopping time defined by the winding of Brownian motion. These considerations lead to a new proof of the Riemann mapping theorem. We also show how this invariance can be used to deduce several identities, including the standard infinite product representations of several trigonometric functions.

math.PR↗
arXiv2017

Projecting the distribution of planar Browian motion at a stopping time through an analytic function

A method is given of deriving the distribution of planar Brownian motion evaluated at certain stopping times using analytic functions. This method relies upon a generalization of the standard conformal invariance of harmonic measure. A number of examples are given, including several in which the stopping time in question is not the exit time of a domain. It is also shown how appropriate choices of domains and stopping times can lead to new proofs of identities, including Euler's Basel sum and a generalization of Leibniz's formula for $π$.

math.PR↗
arXiv2017

Sum rules for effective resistances in infinite graphs

Extending work of Foster, Doyle, and others, we show how the Foster Theorems, a family of results concerning effective resistances on finite graphs, can in certain cases be extended to infinite graphs. A family of sum rules is then obtained, which allows one to easily calculate the sum of the resistances over all paths of a given length. The results are illustrated with some of the most common grids in the plane, including the square, triangular, and hexagonal grids.

math.PR↗