We prove that the Euler form of a metric connection on real oriented vector bundle @math over a compact oriented manifold @math can be identified, as a current, with the expectation of the random current defined by the zero-locus of a certain random section of the bundle. We also explain how to reconstruct probabilistically the metric and the connection on @math from the statistics of random sections of @math .
As explained e.g. in @cite_11 @cite_38 , the Kac-Rice formula has an extension to complex valued random functions of one complex variable that can be used to study the distribution of complex zeros of various ensembles of random complex polynomials. One such ensemble is obtained by regarding the degree @math polynomials in one complex variable as holomorphic sections of the degree @math holomorphic line bundle over @math . If the ensemble of sections is unitarily equivariant, then the expected distribution of zeros approaches the uniform (invariant) distribution on @math as @math ; see [ 8] EK . The same type of equidistribution phenomenon was observed in the pioneering work of S. Nonnenmaker and A. Voros @cite_37 where, among many other things, the authors studied the distributions of zeros of random holomorphic sections of holomorphic line bundles over elliptic curves.
@cite_38 @cite_37 @cite_11
2000931246 2026924428 1966351146 2012430523
Let f be a random Boolean formula that is an instance of 3-SAT. We consider the problem of computing the least real number k such that if the ratio of the number of clauses over the number of variables of f strictly exceeds k , then f is almost certainly unsatisfiable. By a well-known and more or less straightforward argument, it can be shown that kF5.191. This upper bound was improved by to 4.758 by first providing new improved bounds for the occupancy problem. There is strong experimental evidence that the value of k is around 4.2. In this work, we define, in terms of the random formula f, a decreasing sequence of random variables such that, if the expected value of any one of them converges to zero, then f is almost certainly unsatisfiable. By letting the expected value of the first term of the sequence converge to zero, we obtain, by simple and elementary computations, an upper bound for k equal to 4.667. From the expected value of the second term of the sequence, we get the value 4.601q . In general, by letting the U This work was performed while the first author was visiting the School of Computer Science, Carleton Ž University, and was partially supported by NSERC Natural Sciences and Engineering Research Council . of Canada , and by a grant from the University of Patras for sabbatical leaves. The second and third Ž authors were supported in part by grants from NSERC Natural Sciences and Engineering Research . Council of Canada . During the last stages of this research, the first and last authors were also partially Ž . supported by EU ESPRIT Long-Term Research Project ALCOM-IT Project No. 20244 . †An extended abstract of this paper was published in the Proceedings of the Fourth Annual European Ž Symposium on Algorithms, ESA’96, September 25]27, 1996, Barcelona, Spain Springer-Verlag, LNCS, . pp. 27]38 . That extended abstract was coauthored by the first three authors of the present paper. Correspondence to: L. M. Kirousis Q 1998 John Wiley & Sons, Inc. CCC 1042-9832r98r030253-17 253 where (an)n ENo is some sequence of nonnegative numbers, (Sn),nENo is the sequence of partial sums, S0 = 0, Sn = XflXk, of another sequence (Xk)kEN of i.i.d. random variables, and A c R is a fixed Borel set such as [0,1] or [0, oo). Examples of such convolution series are subordinated distributions (f=0Oan = 1) which arise as distributions of random sums, and harmonic and ordinary renewal measures (a0 = 0, an = 1 n for all n C N in the first, an = 1 for all n C NO in the second case). These examples are in turn essential for the analysis of the large time behaviour of diverse applied models such as branching and queueing processes, they are also of interest in connection with representation theorems such as the Levy representation of infinitely divisible distributions. A traditional approach to such problems is via regular variation: If the underlying random variables are nonnegative we can use Laplace transforms and the related Abelian and Tauberian theorems [see, e.g., Stam (1973) in the context of subordination and Feller (1971, XIV.3) in connection with renewal theory; Embrechts, Maejima, and Omey (1984) is a recent treatment of generalized renewal measures along these lines]. The approach of the present paper is based on the Wiener-Levy-Gel'fand theorem and has occasionally been called the Banach algebra method. In Gruibel (1983) we gave a new variant of this method for the special case of lattice distributions, showing that by using the appropriate Banach algebras of sequences, arbitrarily fine expansions are possible under certain assumptions on the higher-order differences of (P(X1 = n))fnEN. Here we give a corresponding treatment of nonlattice distributions. We restrict ourselves to an analogue of first-order differences and obtain a number of theorems which perhaps are described best as next-term results. To explain this let us consider a special case in more detail. We consider tensor powers L N of a positive Hermitian line bundle (L,h L ) over a non-compact complex manifold X. In the compact case, B. Shiffman and S. Zelditch proved that the zeros of random sections become asymptotically uniformly distributed as N→∞ with respect to the natural measure coming from the curvature of L. Under certain boundedness assumptions on the curvature of the canonical line bundle of X and on the Chern form of L we prove a non-compact version of this result. We give various applications, including the limiting distribution of zeros of cusp forms with respect to the principal congruence subgroups of SL 2(ℤ) and to the hyperbolic measure, the higher dimensional case of arithmetic quotients and the case of orthogonal polynomials with weights at infinity. We also give estimates for the speed of convergence of the currents of integration on the zero-divisors. Let P be a Poisson process of intensity one in a square Sn of area n. For a fixed integer k, join every point of P to its k nearest neighbours, creating an undirected random geometric graph Gn,k. We prove that there exists a critical constant ccrit such that for c ccrit, Gn,⌊clog n⌋ is connected with probability tending to 1 as n → ∞. This answers a question posed by the authors in [1]. Let P be a Poisson process of intensity one in a square Sn of area n. For a fixed integer k, we join every point of P to its k nearest neighbours, creating an undirected random geometric graph GSn,k = Gn,k in which every vertex has degree at least k. The connectivity of these graphs was studied by the present authors in [1]. It is not hard to see that Gn,k becomes connected around k = �(log n), and we proved in [1] that if k(n) ≤ 0.3043log n then the probability that Gn,k(n) is connected tends to zero as n → ∞, while if k(n) ≥ 0.5139log n then the probability that Gn,k(n) is connected tends to one as n → ∞. However, we were unable to prove the natural conjecture that there exists a critical constant ccrit such that for c ccrit, P(Gn,⌊clog n⌋ is connected) → 1 as n → ∞. In this paper we prove this conjecture.