1,720,990 research outputs found
Recommended from our members
Lipschitz Embeddings of Random Objects and Related Topics
Get PDFMore than twenty years ago Peter Winkler introduced a fascinating class of dependent or “co-ordinate” percolation models with his compatible sequences and clairvoyant demon scheduling problems. These, and other problems in this class, can be interpreted either as problems of embedding one sequence into another according to certain rules or as oriented percolation problems in two dimensions where the sites are open or closed according to random variables on the co-ordinate axes. In most situations, these problems are not tractable by the usual tools of independent Bernoulli percolation, and new methods are required. We study several problems in this class and their natural extensions.We develop a new multi-scale framework flexible enough to solve a number of problems involving embedding random sequences into random sequences. A natural question in this class was considered by Grimmett, Liggett and Richthammer; they asked whether there exists an increasing -Lipschitz embedding from one i.i.d. Bernoulli sequence into an independent copy with positive probability. We give a positive answer for large enough . A closely related problem is to show thattwo independent Poisson processes on the real line are almost surely roughly isometric (or quasi-isometric). Our approach also applies in this case answering a conjecture of Szegedy and of Peled. We also obtain a new proof for Winkler's compatible sequence problem. All these results are obtained as corollaries to an abstract embedding result that can potentially be applied to a number of one-dimensional embedding questions.We build upon the central idea of the multi-scale construction in the above work to apply it to a different problem. On the complete graph with vertices consider two independent discretetime random walks and , choosing their steps uniformly at random. We say that it is possible to {\textbf{schedule}} a pair of trajectories and , if by delaying their jump times one can keep both walks at distinct vertices forever. It was conjectured by Winkler that for large enough the set of pairs of trajectories that can be scheduled has positive measure. Noga Alon translated this problem to the language of coordinate percolation. In this representation Winkler'sconjecture is equivalent to the existence of an infinite open cluster for large enough . With a multi-scale construction we provide a positive answer for sufficiently large.The questions of Lipschitz embedding and rough isometry of random sequences have natural higher dimensional analogues. We consider the higher dimensional analogue of the Lipschitz embedding problem. We show that for sufficiently large and two independent collections of i.i.d.\ Bernoulli random variables and , almost surely there exists an -Lipschitz embedding of into . The argument is again multi-scale using similar ideas, but this is technically much more challenging because of the more complicated geometry in two dimensions.This presents an added difficulty in extending the argument in one dimension to show that copies of two dimensional Poisson processes are almost surely rough isometric. A key ingredient is to show that one can map measurable sets to smaller measurable sets in a bi-Lipschitz manner. This motivates the final problem we consider. We show that for 0<\gamma, \gamma'<1 and for measurable subsets of the unit square with Lebesgue measure there exist bi-Lipschitz maps with bounded Lipschitz constant(uniformly over all such sets) which are identity on the boundary and increases the Lebesgue measure of the set to at least
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
A Peano curve from mated geodesic trees in the directed landscape
Get PDFA number of interesting examples of random space filling curves have been
constructed in two dimensional statistical mechanics models. Examples of the
above include SLE(), arising as the scaling limit of the Peano curve between
the uniform spanning tree of and its dual (Lawler-Schramm-Werner
'04), or SLE(), which describes the interface between two mated
trees in -Liouville Quantum gravity (Duplantier-Miller-Sheffield '14).
In this work, we construct a Peano curve from the geodesic tree in a fixed
direction and its dual in the directed landscape, the putative universal
space-time scaling limit object in the KPZ universality class. The geodesic
tree and its dual were constructed in Bhatia '23, where it was shown that they
have the same law up to reflection; indeed, the dual tree is the geodesic tree
of the dual landscape as constructed there. We show that the interface between
the two trees is a space filling curve that is naturally parametrized by the
area and encodes the geometry of the two trees. We study the regularity and
fractal properties of this Peano curve, exploiting simultaneously the
symmetries of the directed landscape and probabilistic estimates obtained in
the planar exponential last passage percolation, which is known to converge to
the directed landscape in the scaling limit.Comment: 49 pages, 15 figures; minor edit
Variations on the Author
Get PDF“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Small deviation estimates and small ball probabilities for geodesics in last passage percolation
No full textFor the exactly solvable model of exponential last passage percolation on ℤ2, consider the geodesic Γn joining (0, 0) and (n, n) for large n. It is well known that the transversal fluctuation of Γn around the line x = y is n2/3+o(1) with high probability. We obtain the exponent governing the decay of the small ball probability for Γn and establish that for small δ, the probability that Γn is contained in a strip of width δn2/3 around the diagonal is exp(−Θ(δ−3/2)) uniformly in high n. We also obtain optimal small deviation estimates for the one point distribution of the geodesic showing that for t/2n bounded away from 0 and 1, we have ℙ(∣x(t) − y(t)∣ ≤ δn2/3) = Θ(δ) uniformly in high n, where (x(t), y(t)) is the unique point where Γn intersects the line x + y = t. Our methods are expected to go through for other exactly solvable models of planar last passage percolation and also, upon taking the n → ∞ limit, expected to provide analogous estimates for geodesics in the directed landscape
Appropriate Similarity Measures for Author Cocitation Analysis
Get PDFWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
Sharp deviation bounds for midpoint and endpoint of geodesics in exponential last passage percolation
Get PDFFor exponential last passage percolation on the plane we analyse the
probability that the point-to-line geodesic exhibits an atypically large
transversal fluctuation at the endpoint as well as the probability that the
point-to-point geodesic exhibits an atypically large transversal fluctuation at
the halfway point. In particular, we show that , the probability that
the point-to-line geodesic from the origin to the line ends at
satisfies that
for large and
, the probability that the geodesic from the origin to
the point passes through the point , satisfies
for large. The
latter result solves a special case of a conjecture from Liu (PTRF, 2022).Comment: 19 pages, 4 figure
First Passage Percolation on Hyperbolic groups
Get PDFWe study first passage percolation (FPP) on a Gromov-hyperbolic group
with boundary equipped with the Patterson-Sullivan measure .
We associate an i.i.d.\ collection of random passage times to each edge of a
Cayley graph of , and investigate classical questions about the asymptotics
of first passage time as well as the geometry of geodesics in the FPP metric.
Under suitable conditions on the passage time distribution, we show that the
`velocity' exists in -almost every direction , and is
almost surely constant by ergodicity of the action on . For
every , we also show almost sure coalescence of any two
geodesic rays directed towards . Finally, we show that the variance of the
first passage time grows linearly with word distance along word geodesic rays
in every fixed boundary direction. This provides an affirmative answer to a
conjecture of Benjamini, Tessera, and Zeitouni.Comment: v2 53 pages, 3 figures. Final version incorporating referee comments.
To appear in Advances in Mat
- …
