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 .
This line of investigation originates in the groundbreaking work of M. Kac @cite_24 and S.O. Rice @cite_15 who studied the distribution of zeros of certain random functions of one real variable. One outcome of their work is the celebrated Kac-Rice formula that gives an explicit description of the expected distribution of zeros of such functions; see @cite_11 .
@cite_24 @cite_15 @cite_11
2109519791 2114509411 1966351146
In this section we use the representations of the noise currents given in section 2.8 to derive some statistical properties of I(t). The first six sections are concerned with the probability distribution of I(t) and of its zeros and maxima. Sections 3.7 and 3.8 are concerned with the statistical properties of the envelope of I(t). Fluctuations of integrals involving I2(t) are discussed in section 3.9. The probability distribution of a sine wave plus a noise current is given in 3.10 and in 3.11 an alternative method of deriving the results of Part III is mentioned. Prof. Uhlenbeck has pointed out that much of the material in this Part is closely connected with the theory of Markoff processes. Also S. Chandrasekhar has written a review of a class of physical problems which is related, in a general way, to the present subject.22 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.