Sublinear Average-Case Shortest Paths in Weighted Unit-Disk Graphs

We consider the problem of computing shortest paths in weighted unit-disk graphs in constant dimension $d$. Although the single-source and all-pairs variants of this problem are well-studied in the plane case, no non-trivial exact distance oracles for unit-disk graphs have been known to date, even for $d=2$. The classical result of Sedgewick and Vitter [Algorithmica '86] shows that for weighted unit-disk graphs in the plane the $A^*$ search has average-case performance superior to that of a standard shortest path algorithm, e.g., Dijkstra's algorithm. Specifically, if the $n$ corresponding points of a weighted unit-disk graph $G$ are picked from a unit square uniformly at random, and the connectivity radius is $r\in (0,1)$, $A^*$ finds a shortest path in $G$ in $O(n)$ expected time when $r=\Omega(\sqrt{\log n/n})$, even though $G$ has $\Theta((nr)^2)$ edges in expectation. In other words, the work done by the algorithm is in expectation proportional to the number of vertices and not the number of edges. In this paper, we break this natural barrier and show even stronger sublinear time results. We propose a new heuristic approach to computing point-to-point exact shortest paths in unit-disk graphs. We analyze the average-case behavior of our heuristic using the same random graph model as used by Sedgewick and Vitter and prove it superior to $A^*$. Specifically, we show that, if we are able to report the set of all $k$ points of $G$ from an arbitrary rectangular region of the plane in $O(k + t(n))$ time, then a shortest path between arbitrary two points of such a random graph on the plane can be found in $O(1/r^2 + t(n))$ expected time. In particular, the state-of-the-art range reporting data structures imply a sublinear expected bound for all $r=\Omega(\sqrt{\log n/n})$ and $O(\sqrt{n})$ expected bound for $r=\Omega(n^{-1/4})$ after only near-linear preprocessing of the point set.Comment: Full version of a SoCG'21 paper. Abstract truncated to meet arxiv requirement

Angle between two random segments

The study of "random segments" is a classic issue in geometrical probability, whose complexity depends on how it is defined. But in apparently simple models, the random behavior is not immediate. In the present manuscript the following setting is considered. Let four independent random points that follow a uniform distribution on the unit disk. Two random segments are built with them, which always are inside of the disk. We compute the density function of the angle between these two random segments when they intersect each other. This type of problem tends to be complex due to the high stochastic dependency that exists between the elements that form them. The expression obtained is in terms of integrals, however it allows us to understand the behavior of the distribution of the random angle between the two random segments

Sylvester's question and the Random Acceleration Process

Let n points be chosen randomly and independently in the unit disk. "Sylvester's question" concerns the probability p_n that they are the vertices of a convex n-sided polygon. Here we establish the link with another problem. We show that for large n this polygon, when suitably parametrized by a function r(phi) of the polar angle phi, satisfies the equation of the random acceleration process (RAP), d^2 r/d phi^2 = f(phi), where f is Gaussian noise. On the basis of this relation we derive the asymptotic expansion log p_n = -2n log n + n log(2 pi^2 e^2) - c_0 n^{1/5} + ..., of which the first two terms agree with a rigorous result due to Barany. The nonanalyticity in n of the third term is a new result. The value 1/5 of the exponent follows from recent work on the RAP due to Gyorgyi et al. [Phys. Rev. E 75, 021123 (2007)]. We show that the n-sided polygon is effectively contained in an annulus of width \sim n^{-4/5} along the edge of the disk. The distance delta_n of closest approach to the edge is exponentially distributed with average 1/(2n).Comment: 29 pages, 4 figures; references added and minor change

Semiclassical limit of Liouville Field Theory

Liouville Field Theory (LFT for short) is a two dimensional model of random surfaces, which is for instance involved in $2d$ string theory or in the description of the fluctuations of metrics in $2d$ Liouville quantum gravity. This is a probabilistic model that consists in weighting the classical Free Field action with an interaction term given by the exponential of a Gaussian multiplicative chaos. The main input of our work is the study of the semiclassical limit of the theory, which is a prescribed asymptotic regime of LFT of interest in physics literature (see \cite{witten} and references therein). We derive exact formulas for the Laplace transform of the Liouville field in the case of flat metric on the unit disk with Dirichlet boundary conditions. As a consequence, we prove that the Liouville field concentrates on the solution of the classical Liouville equation with explicit negative scalar curvature. We also characterize the leading fluctuations, which are Gaussian and massive, and establish a large deviation principle. Though considered as an ansatz in the whole physics literature, it seems that it is the first rigorous probabilistic derivation of the semiclassical limit of LFT. On the other hand, we carry out the same analysis when we further weight the Liouville action with heavy matter operators. This procedure appears when computing the $n$-points correlation functions of LFT.Comment: 42 pages; 3 figures; Typos correcte

Aspects of atomic decompositions and Bergman projections

The concept of an atomic decomposition was introduced by Coifman and Rochberg (1980) for weighted Bergman spaces on the unit disk. By the Riemann mapping theorem, functions in every simply connected domain in the complex plane have an atomic decomposition. However, a decomposition resulting from a conformal mapping of the unit disk tends to be very implicit and often lacks a clear connection to the geometry of the domain that it has been mapped into. The lattice of points, where the atoms of the decomposition are evaluated, usually follows the geometry of the original domain, but after mapping the domain into another this connection is easily lost and the layout of points becomes seemingly random. In the first article we construct an atomic decomposition directly on a weighted Bergman space on a class of regulated, simply connected domains. The construction uses the geometric properties of the regulated domain, but does not explicitly involve any conformal Riemann map from the unit disk. It is known that the Bergman projection is not bounded on the space L-infinity of bounded measurable functions. Taskinen (2004) introduced the locally convex spaces LV-infinity consisting of measurable and HV-infinity of analytic functions on the unit disk with the latter being a closed subspace of the former. They have the property that the Bergman projection is continuous from LV-infinity onto HV-infinity and, in some sense, the space HV-infinity is the smallest possible substitute to the space H-infinity of analytic functions. In the second article we extend the above result to a smoothly bounded strictly pseudoconvex domain. Here the related reproducing kernels are usually not known explicitly, and thus the proof of continuity of the Bergman projection is based on generalised Forelli-Rudin estimates instead of integral representations. The minimality of the space LV-infinity is shown by using peaking functions first constructed by Bell (1981). Taskinen (2003) showed that on the unit disk the space HV-infinity admits an atomic decomposition. This result is generalised in the third article by constructing an atomic decomposition for the space HV-infinity on a smoothly bounded strictly pseudoconvex domain. In this case every function can be presented as a linear combination of atoms such that the coefficient sequence belongs to a suitable Köthe co-echelon space.Puhtaan matematiikan alaan kuuluva artikkeliväitöskirja käsittelee sekä Coifmanin ja Rochbergin (1980) kehittämää atomidekompositiota että Bergmanin projektion jatkuvuutta tietyissä funktioavaruuksissa. Atomidekompositiossa tietyn alueen kuvaukset voidaan esittää summana yksinkertaisia rakennuspalikoita, atomeja. Klassisessa tapauksessa tämä alue on kompleksitason yksikkökiekko. Minkä tahansa osajoukon atomidekompositio voidaan melko helposti palauttaa klassisen yksikkökiekon tapaukseen Riemannin kuvauslauseen nojalla, mutta näin saatava atomidekompositio ei ole kovin konkreettinen ja sen yhteys alueen geometriaan jää epäselväksi. Ensimmäisessä artikkelissa rakennetaan atomidekompositio suoraan reguloidulle, yhdesti yhtenäiselle alueelle painotetussa Bergman-avaruudessa. Konstruktiossa käytetään alueen geometrisia piirteitä, mutta siinä ei eksplisiittisesti käytetä hyväksi Riemannin kuvausta yksikkökiekolta. On tunnettua, että Bergmanin projektio ei ole jatkuva rajoitettujen mitallisten funktioiden avaruudesta L-ääretön rajoitettujen analyyttisten funktioiden avaruuteen H-ääretön. Taskinen (2004) julkaisi tuloksen lokaalikonvekseista avaruuksista LV-ääretön ja HV-ääretön, jotka ovat tietyssä mielessä pienimmät sellaiset laajennukset avaruuksiin L-ääretön ja H-ääretön, että Bergmanin projektio avaruuksien välillä on jatkuva. Toisessa artikkelissa yllämainitut tulokset yleistetään useamman kompleksimuuttujan tapaukseen pseudokonveksille alueelle. Tällöin aluetta vastaavaa Bergmanin ydintä ei yleensä tunneta eksplisiittisesti, joten jatkuvuuden todistamiseksi käytetään yleistettyä Forelli-Rudin-approksimaatiota. Taskinen (2003) osoitti, että yksikkökiekolle avaruudessa HV-ääretön voidaan rakentaa atomidekompositio. Kolmannessa artikkelissa sama konstruktio tehdään pseudokonveksille alueelle avaruudessa HV-ääretön. Tässä tapauksessa funktioiden dekomposition kerroinjono kuuluu sopivaan Köthen jonoavaruuteen