Convex hulls of random walks: expected number of faces and face probabilities

Consider a sequence of partial sums Si=ξ1+…+ξi, 1≤i≤n, starting at S0=0, whose increments ξ1,…,ξn are random vectors in Rd, d≤n. We are interested in the properties of the convex hull Cn:=Conv(S0,S1,…,Sn). Assuming that the tuple (ξ1,…,ξn) is exchangeable and a certain general position condition holds, we prove that the expected number of k-dimensional faces of Cn is given by the formula E[fk(Cn)]=2⋅k!n!∑l=0∞[n+1d−2l]{d−2lk+1}, for all 0≤k≤d−1, where [nm] and {nm} are Stirling numbers of the first and second kind, respectively. Further, we compute explicitly the probability that for given indices 0≤i1<…<ik+1≤n, the points Si1,…,Sik+1 form a k-dimensional face of Conv(S0,S1,…,Sn). This is done in two different settings: for random walks with symmetrically exchangeable increments and for random bridges with exchangeable increments. These results generalize the classical one-dimensional discrete arcsine law for the position of the maximum due to E. Sparre Andersen. All our formulae are distribution-free, that is do not depend on the distribution of the increments ξk's. The main ingredient in the proof is the computation of the probability that the origin is absorbed by a joint convex hull of several random walks and bridges whose increments are invariant with respect to the action of direct product of finitely many reflection groups of types An−1 and Bn. This probability, in turn, is related to the number of Weyl chambers of a product-type reflection group that are intersected by a linear subspace in general position

Discrete structure of ultrathin dielectric films and their surface optical properties

The boundary problem of linear classical optics about the interaction of electromagnetic radiation with a thin dielectric film has been solved under explicit consideration of its discrete structure. The main attention has been paid to the investigation of the near-zone optical response of dielectrics. The laws of reflection and refraction for discrete structures in the case of a regular atomic distribution are studied and the structure of evanescent harmonics induced by an external plane wave near the surface is investigated in details. It is shown by means of analytical and numerical calculations that due to the existence of the evanescent harmonics the laws of reflection and refraction at the distances from the surface less than two interatomic distances are principally different from the Fresnel laws. From the practical point of view the results of this work might be useful for the near-field optical microscopy of ultrahigh resolution.Comment: 25 pages, 16 figures, LaTeX2.09, to be published in Phys.Rev.

Shintani Cocycles on \GL_{n}

The aim of this paper is to define an n-1-cocycle $\sigma$ on \GL_{n}(\Q) with values in a certain space \hD of distributions on \A_f^{n}\setminus\{0\}. Here \A_f denotes the ring of finite ad\`{e}les of \Q, and the distributions take values in the Laurent series \C((z_{1},...,z_{n})). This cocycle can be used to evaluate special values of Artin L-functions on number fields at negative integers. The construction generalizes that of Solomon in the case n=2.Comment: 14 pages. To appear in LMS Bulleti

A multivariate CLT for bounded decomposable random vectors with the best known rate

We prove a multivariate central limit theorem with explicit error bound on a non-smooth function distance for sums of bounded decomposable $d$-dimensional random vectors. The decomposition structure is similar to that of Barbour, Karo\'nski and Ruci\'nski (1989) and is more general than the local dependence structure considered in Chen and Shao (2004). The error bound is of the order $d^{\frac{1}{4}} n^{-\frac{1}{2}}$, where $d$ is the dimension and $n$ is the number of summands. The dependence on $d$, namely $d^{\frac{1}{4}}$, is the best known dependence even for sums of independent and identically distributed random vectors, and the dependence on $n$, namely $n^{-\frac{1}{2}}$, is optimal. We apply our main result to a random graph example.Comment: 12 page

An analysis of the far-field response to external forcing of a suspension in Stokes flow in a parallel-wall channel

The leading-order far-field scattered flow produced by a particle in a parallel-wall channel under creeping flow conditions has a form of the parabolic velocity field driven by a 2D dipolar pressure distribution. We show that in a system of hydrodynamically interacting particles, the pressure dipoles contribute to the macroscopic suspension flow in a similar way as the induced electric dipoles contribute to the electrostatic displacement field. Using this result we derive macroscopic equations governing suspension transport under the action of a lateral force, a lateral torque or a macroscopic pressure gradient in the channel. The matrix of linear transport coefficients in the constitutive relations linking the external forcing to the particle and fluid fluxes satisfies the Onsager reciprocal relation. The transport coefficients are evaluated for square and hexagonal periodic arrays of fixed and freely suspended particles, and a simple approximation in a Clausius-Mossotti form is proposed for the channel permeability coefficient. We also find explicit expressions for evaluating the periodic Green's functions for Stokes flow between two parallel walls.Comment: 23 pages, 12 figure