arXiv subjects

Steven P. Lalley

Publications and source records attributed to Steven P. Lalley.

At least 19 recordsLinked to original sources

arXiv2023

Efficiency in Collective Decision-Making via Quadratic Transfers

Consider the following collective choice problem: a group of budget constrained agents must choose one of several alternatives. Is there a budget balanced mechanism that: i) does not depend on the specific characteristics of the group, ii) does not require unaffordable transfers, and iii) implements utilitarianism if the agents' preferences are quasilinear and their private information? We study the following procedure: every agent can express any intensity of support or opposition to each alternative, by transferring to the rest of the agents wealth equal to the square of the intensity expressed; and the outcome is determined by the sums of the expressed intensities. We prove that as the group grows large, in every equilibrium of this quadratic-transfers mechanism, each agent's transfer converges to zero, and the probability that the efficient outcome is chosen converges to one.

econ.TH↗

arXiv2020

Occupation densities of Ensembles of Branching Random Walks

We study the limiting occupation density process for a large number of critical and driftless branching random walks. We show that the rescaled occupation densities of $\lfloor sN\rfloor$ branching random walks, viewed as a function-valued, increasing process $\{g_{s}^{N}\}_{s\ge 0}$, converges weakly to a pure jump process in the Skorohod space $\mathbb D([0, +\infty), \mathcal C_{0}(\mathbb R))$, as $N\to\infty$. Moreover, the jumps of the limiting process consist of i.i.d. copies of an Integrated super-Brownian Excursion (ISE) density, rescaled and weighted by the jump sizes in a real-valued stable-1/2 subordinator.

math.PR↗

arXiv2020

Local geometry of random geodesics on negatively curved surfaces

It is shown that the tessellation of a compact, negatively curved surface induced by a typical long geodesic segment, when properly scaled, looks locally like a Poisson line process. This implies that the global statistics of the tessellation -- for instance, the fraction of triangles -- approach those of the limiting Poisson line process.

math.GT↗

arXiv2019

Nash Equilbria for Quadratic Voting

Voters making a binary decision purchase votes from a centralized clearing house, paying the square of the number of votes purchased. The net payoff to an agent with utility $u$ who purchases $v$ votes is $Ψ(S_{n+1})u-v^{2}$, where $Ψ$ is a monotone function taking values between -1 and +1 and $S_{n+1}$ is the sum of all votes purchased by the $n+1$ voters participating in the election. The utilities of the voters are assumed to arise by random sampling from a probability distribution $F_{U}$ with compact support; each voter knows her own utility, but not those of the other voters, although she does know the sampling distribution $F_{U}$. Nash equilibria for this game are described. These results imply that the expected inefficiency of any Nash equilibrium decays like $1/n$.

cs.GT↗

arXiv2018

Phase Transition for the Chase-Escape Model on 2D Lattices

Chase-Escape is a simple stochastic model that describes a predator-prey interaction. In this model, there are two types of particles, red and blue. Red particles colonize adjacent empty sites at an exponential rate $λ_{R}$, whereas blue particles take over adjacent red sites at exponential rate $λ_{B}$, but can never colonize empty sites directly. Numerical simulations suggest that there is a critical value $p_{c}$ for the relative growth rate $p:=λ_{R}/λ_{B}$. When $p p_{c}$ mutual survival occurs with positive probability. In particular, $p_{c} \approx 0.50$ for the square lattice case ($\mathbb Z^{2}$). Our simulations provide a plausible explanation for the critical value. Near the critical value, the set of occupied sites exhibits a fractal nature, and the hole sizes approximately follow a power-law distribution.

cond-mat.dis-nn↗

arXiv2015

Critical Branching Brownian Motion with Killing

We obtain sharp asymptotic estimates for hitting probabilities of a critical branching Brownian motion in one dimension with killing at 0 We also obtain sharp asymptotic formulas for the tail probabilities of the number of particles killed at 0. In the special case of double-or-nothing branching, we give exact formulas for both the hitting probabilities, in terms of elliptic functions, and the distribution of the number of killed particles.

math.PR↗

arXiv2014

On the Maximal Displacement of a Critical Branching Random Walk

We consider a branching random walk initiated by a single particle at location 0 in which particles alternately reproduce according to the law of a Galton-Watson process and disperse according to the law of a driftless random walk on the integers. When the offspring distribution has mean 1 the branching process is critical, and therefore dies out with probability 1. We prove that if the particle jump distribution has mean zero, positive finite variance $η^{2}$, and finite $4+\varepsilon$ moment, and if the offspring distribution has positive variance $σ^{2}$ and finite third moment then the distribution of the rightmost position $M$ reached by a particle of the branching random walk satisfies $P\{M \geq x\}\sim 6η^{2}/ (σ^{2}x^{2})$ as $x \rightarrow \infty$. We also prove a conditional limit theorem for the distribution of the rightmost particle location at time $n$ given that the process survives for $n$ generations.

math.PR↗

arXiv2014

A phase transition for measure-valued SIR epidemic processes

We consider measure-valued processes $X=(X_t)$ that solve the following martingale problem: for a given initial measure $X_0$, and for all smooth, compactly supported test functions $φ$, \begin{eqnarray*}X_t(φ)=X_0(φ)+\frac{1}{2}\int _0^tX_s(Δφ)\,ds+θ\int_0^tX_s(φ)\,ds\\{}-\int_0^tX_s(L_sφ)\,ds+M_t(φ).\end{eqnarray*} Here $L_s(x)$ is the local time density process associated with $X$, and $M_t(φ)$ is a martingale with quadratic variation $[M(φ)]_t=\int_0^tX_s(φ^2)\,ds$. Such processes arise as scaling limits of SIR epidemic models. We show that there exist critical values $θ_c(d)\in(0,\infty)$ for dimensions $d=2,3$ such that if $θ>θ_c(d)$, then the solution survives forever with positive probability, but if $θ<θ_c(d)$, then the solution dies out in finite time with probability 1. For $d=1$ we prove that the solution dies out almost surely for all values of $θ$. We also show that in dimensions $d=2,3$ the process dies out locally almost surely for any value of $θ$; that is, for any compact set $K$, the process $X_t(K)=0$ eventually.

math.PR↗

arXiv2013

Exchangeable Markov Processes on $[k]^{\zz{N}}$ with Cadlag Sample Paths

Any exchangeable Markov processes on $[k]^{\mathbb{N}}$ with cadlag sample paths projects to a Markov process on the simplex whose sample paths are cadlag and of locally bounded variation. Furthermore, any such process has a de Finetti-type description as a mixture of i.i.d. copies of time-inhomogeneous Markov processes on $[k]$. In the Feller case, these time-inhomogeneous Markov processes have a relatively simple structure; however, in the non-Feller case a greater variety of behaviors is possible since the transition law of the underlying Markov process on $[k]^{\zz{N}}$ can depend in a non-trivial way on the exchangeable $σ$-algebra of the process.

math.PR↗

arXiv2013

Maximal Displacement of Critical Branching Symmetric Stable Processes

We consider a critical continuous-time branching process (a Yule process) in which the individuals independently execute symmetric $α-$stable random motions on the real line starting at their birth points. Because the branching process is critical, it will eventually die out, and so there is a well-defined maximal location $M$ ever visited by an individual particle of the process. We prove that the distribution of $M$ satisfies the asymptotic relation $P\{M\geq x \}\sim (2/α)^{1/2}x^{-α/2}$ as $x \rightarrow \infty$.

math.PR↗

arXiv2013

Statistical regularities of self-intersection counts for geodesics on negatively curved surfaces

Let $Υ$ be a compact, negatively curved surface. From the (finite) set of all closed geodesics on $Υ$ of length $\leq L$, choose one, say $γ_{L}$, at random and let $N (γ_{L})$ be the number of its self-intersections. It is known that there is a positive constant $κ$ depending on the metric such that $N (γ_{L})/L^{2} \rightarrow κ$ in probability as $L\rightarrow \infty$. The main results of this paper concern the size of typical fluctuations of $N (γ_{L})$ about $κL^{2}$. It is proved that if the metric has constant curvature -1 then typical fluctuations are of order $L$, in particular, $(N (γ_{L})-κL^{2})/L$ converges weakly to a nondegenerate probability distribution. In contrast, it is also proved that if the metric has variable negative curvature then fluctuations of $N (γ_{L})$ are of order $L^{3/2}$, in particular, $(N (γ_{L})-κL^{2})/L^{3/2}$ converges weakly to a Gaussian distribution. Similar results are proved for generic geodesics, that is, geodesics whose initial tangent vectors are chosen randomly according to normalized Liouville measure.

math.DS↗

arXiv2012

Convergence Rates of Markov Chains on Spaces of Partitions

We study the convergence rate to stationarity for a class of exchangeable partition-valued Markov chains called cut-and-paste chains. The law governing the transitions of a cut-and-paste chain are determined by products of i.i.d. stochastic matrices, which describe the chain induced on the simplex by taking asymptotic frequencies. Using this representation, we establish upper bounds for the mixing times of ergodic cut-and-paste chains, and under certain conditions on the distribution of the governing random matrices we show that the "cutoff phenomenon" holds.

math.PR↗

arXiv2012

Random Walk on a Co-Compact Fuchsian Group

It is proved that the Green's function of a symmetric finite range random walk on a co-compact Fuchsian group decays exponentially in distance at the radius of convergence R. It is also shown that Ancona's inequalities extend to R, and therefore that the Martin boundary for R-potentials coincides with the natural geometric boundary S^1, and that the Martin kernel is uniformly Hölder continuous. Finally, it is proved that this implies a local limit theorem for the transition probabilities.

math.PR↗

arXiv2011

Random Walk on a Surface Group: Boundary Behavior of the Green's Function at the Spectral Radius

It is proved that the Green's function of the simple random walk on a surface group of large genus decays exponentially at the spectral radius. It is also shown that Ancona's inequalities extend to the spectral radius R, and therefore that the Martin boundary for R-potentials coincides with the natural geometric boundary S^1. This implies that the Green's function obeys a power law with exponent 1/2 at the spectral radius.

math.PR↗

arXiv2011

Self-intersections in combinatorial topology: statistical structure

Oriented closed curves on an orientable surface with boundary are described up to continuous deformation by reduced cyclic words in the generators of the fundamental group and their inverses. By self-intersection number one means the minimum number of transversal self-intersection points of representatives of the class. We prove that if a class is chosen at random from among all classes of $m$ letters, then for large $m$ the distribution of the self-intersection number approaches the Gaussian distribution.

math.GT↗

arXiv2009

Self-Intersections of Random Geodesics on Negatively Curved Surfaces

We study the fluctuations of self-intersection counts of random geodesic segments of length $t$ on a compact, negatively curved surface in the limit of large $t$. If the initial direction vector of the geodesic is chosen according to the \emph{Liouville measure}, then it is not difficult to show that the number $N (t)$ of self-intersections by time $t$ grows like $κt^{2}$, where $κ=κ_{M}$ is a positive constant depending on the surface $M$. We show that (for a smooth modification of $N (t)$) the fluctuations are of size $t$, and the limit distribution is a weak limit of Gaussian quadratic forms. We also show that the fluctuations of \emph{localized} self-intersection counts (that is, only self-intersections in a fixed subset of $M$ are counted) are typically of size $t^{3/2}$, and the limit distribution is Gaussian.

math.DS↗

arXiv2009

Spatial Epidemics and Local Times for Critical Branching Random Walks in Dimensions 2 and 3

The behavior at criticality of spatial SIR (susceptible/infected/recovered) epidemic models in dimensions two and three is investigated. In these models, finite populations of size N are situated at the vertices of the integer lattice, and infectious contacts are limited to individuals at the same or at neighboring sites. Susceptible individuals, once infected, remain contagious for one unit of time and then recover, after which they are immune to further infection. It is shown that the measure-valued processes associated with these epidemics, suitably scaled, converge, in the large-N limit, either to a standard Dawson-Watanabe process (super-Brownian motion) or to a Dawson-Watanabe process with location-dependent killing, depending on the size of the the initially infected set. A key element of the argument is a proof of Adler's 1993 conjecture that the local time processes associated with branching random walks converge to the local time density process associated with the limiting super-Brownian motion.

math.PR↗

arXiv2007

Critical scaling of stochastic epidemic models

In the simple mean-field SIS and SIR epidemic models, infection is transmitted from infectious to susceptible members of a finite population by independent $p-$coin tosses. Spatial variants of these models are proposed, in which finite populations of size $N$ are situated at the sites of a lattice and infectious contacts are limited to individuals at neighboring sites. Scaling laws for both the mean-field and spatial models are given when the infection parameter $p$ is such that the epidemics are critical. It is shown that in all cases there is a critical threshold for the numbers initially infected: below the threshold, the epidemic evolves in essentially the same manner as its branching envelope, but at the threshold evolves like a branching process with a size-dependent drift.

math.PR↗