arXiv subjects

Eric Cator

Publications and source records attributed to Eric Cator.

33 records · Page 2Linked to original sources

arXiv2016

Self-averaging sequences which fail to converge

We consider self-averaging sequences in which each term is a weighted average over previous terms. For several sequences of this kind it is known that they do not converge to a limit. These sequences share the property that $n$th term is mainly based on terms around a fixed fraction of $n$. We give a probabilistic interpretation to such sequences and give weak conditions under which it is natural to expect non-convergence. Our methods are illustrated by application to the group Russian roulette problem.

math.PR↗

arXiv2015

Conditioned multi-type Galton-Watson trees

We consider multi-type Galton Watson trees, and find the distribution of these trees when conditioning on very general types of recursive events. It turns out that the conditioned tree is again a multi-type Galton Watson tree, possibly with more types and with offspring distributions, depending on the type of the father node and on the height of the father node. These distributions are given explicitly. We give some interesting examples for the kind of conditioning we can handle, showing that our methods have a wide range of applications.

math.PR↗

arXiv2014

Constructing and searching conditioned Galton-Watson trees

We investigate conditioning Galton-Watson trees on general recursive-type events, such as the event that the tree survives until a specific level. It turns out that the conditioned tree is again a type of Galton-Watson tree, with different types of offspring and a level-dependent offspring distribution, which will all be given explicitly. As an interesting application of these results, we will calculate the expected cost of searching a tree until reaching a given level.

math.PR↗

arXiv2014

Space-time stationary solutions for the Burgers equation

We construct space-time stationary solutions of the 1D Burgers equation with random forcing in the absence of periodicity or any other compactness assumptions. More precisely, for the forcing given by a homogeneous Poissonian point field in space-time we prove that there is a unique global solution with any prescribed average velocity. These global solutions serve as one-point random attractors for the infinite-dimensional dynamical system associated to solutions to the Cauchy problem. The probability distribution of the global solutions defines a stationary distribution for the corresponding Markov process. We describe a broad class of initial Cauchy data for which the distribution of the Markov process converges to the above stationary distribution. Our construction of the global solutions is based on a study of the field of action-minimizing curves. We prove that for an arbitrary value of the average velocity, with probability 1 there exists a unique field of action-minimizing curves initiated at all of the Poissonian points. Moreover action-minimizing curves corresponding to different starting points merge with each other in finite time.

math.PR↗

arXiv2014

Constraints on determinism: Bell versus Conway-Kochen

Bell's Theorem from 1964 and the (Strong) Free Will Theorem of Conway and Kochen from 2009 both exclude deterministic hidden variable theories (or, in modern parlance, `ontological models') that are compatible with some small fragment of quantum mechanics, admit `free' settings of the archetypal Alice & Bob experiment, and satisfy a locality condition called Parameter Independence. We clarify the relationship between these theorems by giving reformulations of both that exactly pinpoint their resemblance and their differences. Our reformulation imposes determinism in what we see as the only consistent way, in which the `ontological state' initially determines both the settings and the outcome of the experiment. The usual status of the settings as `free' parameters is subsequently recovered from independence assumptions on the pertinent (random) variables. Our reformulation also clarifies the role of the settings in Bell's later generalization of his theorem to stochastic hidden variable theories.

quant-ph↗

arXiv2013

On the local fluctuations of last-passage percolation models

Using the fact that the Airy process describes the limiting fluctuations of the Hammersley last-passage percolation model, we prove that it behaves locally like a Brownian motion. Our method is quite straightforward, and it is based on a certain monotonicity and good control over the equilibrium measures of the Hammersley model (local comparison).

math.PR↗

arXiv2011

Influence of the initial condition in equilibrium last-passage percolation models

In this paper we consider an equilibrium last-passage percolation model on an environment given by a compound two-dimensional Poisson process. We prove an $\LL^2$-formula relating the initial measure with the last-passage percolation time. This formula turns out to be a useful tool to analyze the fluctuations of the last-passage times along non-characteristic directions.

math.PR↗

arXiv2011

Hydrodynamical Methods in Last Passage Percolation Models

These lecture notes are written as reference material for the Advanced Course "Hydrodynamical Methods in Last Passage Percolation Models", given at the 28th Coloquio Brasileiro de Matematica at IMPA, Rio de Janeiro, July 2011.

math.PR↗

arXiv2010

Busemann functions and the speed of a second class particle in the rarefaction fan

In this paper we will show how the results found in Cator and Pimentel 2009, about the Busemann functions in last-passage percolation, can be used to calculate the asymptotic distribution of the speed of a single second class particle starting from an arbitrary deterministic configuration which has a rarefaction fan, in either the totally asymetric exclusion process, or the Hammersley interacting particle process. The method will be to use the well known last-passage percolation description of the exclusion process and of the Hammersley process, and then the well known connection between second class particles and competition interfaces.

math.PR↗

arXiv2010

Busemann functions and Equilibrium measures in last passage percolation models

The interplay between two-dimensional percolation growth models and one-dimensional particle processes has been a fruitful source of interesting mathematical phenomena. In this paper we develop a connection between the construction of Busemann functions in the Hammersley last-passage percolation model with i.i.d. random weights, and the existence, ergodicity and uniqueness of equilibrium (or time-invariant) measures for the related (multi-class) interacting fluid system. As we shall see, in the classical Hammersley model, where each point has weight one, this approach brings a new and rather geometrical solution of the longest increasing subsequence problem, as well as a central limit theorem for the Busemann function.

math.PR↗

arXiv2009

Technical report: Adaptivity and optimality of the monotone least squares estimator for four different models

In this paper we will consider the estimation of a monotone regression (or density) function in a fixed point by the least squares (Grenander) estimator. We will show that this estimator is fully adaptive, in the sense that the attained rate is given by a functional relation using the underlying function $f_0$, and not by some smoothness parameter, and that this rate is optimal when considering the class of all monotone functions, in the sense that there exists a sequence of alternative monotone functions $f_1$, such that no other estimator can attain a better rate for both $f_0$ and $f_1$. We also show that under mild conditions the estimator attains the same rate in $L^q$ sense, and we give general conditions for which we can calculate a (non-standard) limiting distribution for the estimator.

math.ST↗

arXiv2006

Second class particles and cube root asymptotics for Hammersley's process

We show that, for a stationary version of Hammersley's process, with Poisson sources on the positive x-axis and Poisson sinks on the positive y-axis, the variance of the length of a longest weakly North--East path $L(t,t)$ from $(0,0)$ to $(t,t)$ is equal to $2\mathbb {E}(t-X(t))_+$, where $X(t)$ is the location of a second class particle at time $t$. This implies that both $\mathbb {E}(t-X(t))_+$ and the variance of $L(t,t)$ are of order $t^{2/3}$. Proofs are based on the relation between the flux and the path of a second class particle, continuing the approach of Cator and Groeneboom [Ann. Probab. 33 (2005) 879--903].

math.PR↗

arXiv2006

Behavior of a second class particle in Hammersley's process

In the case of a rarefaction fan in a non-stationary Hammersley process, we explicitly calculate the asymptotic behavior of the process as we move out along a ray, and the asymptotic distribution of the angle within the rarefaction fan of a second class particle and a dual second class particle. Furthermore, we consider a stationary Hammersley process and use the previous results to show that trajectories of a second class particle and a dual second class particles touch with probability one, and we give some information on the area enclosed by the two trajectories, up until the first intersection point. This is linked to the area of influence of an added Poisson point in the plane.

math.PR↗

arXiv2006

Cube root fluctuations for the corner growth model associated to the exclusion process

We study the last-passage growth model on the planar integer lattice with exponential weights. With boundary conditions that represent the equilibrium exclusion process as seen from a particle right after its jump we prove that the variance of the last-passage time in a characteristic direction is of order t^{2/3}. With more general boundary conditions that include the rarefaction fan case we show that the last-passage time fluctuations are still of order t^{1/3}, and also that the transversal fluctuations of the maximal path have order t^{2/3}. We adapt and then build on a recent study of Hammersley's process by Cator and Groeneboom, and also utilize the competition interface introduced by Ferrari, Martin and Pimentel. The arguments are entirely probabilistic, and no use is made of the combinatorics of Young tableaux or methods of asymptotic analysis.

math.PR↗

arXiv2005

Hammersley's process with sources and sinks

We show that, for a stationary version of Hammersley's process, with Poisson ``sources'' on the positive x-axis, and Poisson ``sinks'' on the positive y-axis, an isolated second-class particle, located at the origin at time zero, moves asymptotically, with probability 1, along the characteristic of a conservation equation for Hammersley's process. This allows us to show that Hammersley's process without sinks or sources, as defined by Aldous and Diaconis [Probab. Theory Related Fields 10 (1995) 199-213] converges locally in distribution to a Poisson process, a result first proved in Aldous and Diaconis (1995) by using the ergodic decomposition theorem and a construction of Hammersley's process as a one-dimensional point process, developing as a function of (continuous) time on the whole real line. As a corollary we get the result that EL(t,t)/t converges to 2, as t\to\infty, where L(t,t) is the length of a longest North-East path from (0,0) to (t,t). The proofs of these facts need neither the ergodic decomposition theorem nor the subadditive ergodic theorem. We also prove a version of Burke's theorem for the stationary process with sources and sinks and briefly discuss the relation of these results with the theory of longest increasing subsequences of random permutations.

math.PR↗