B. Tsirelson

Random comples zeroes, III. Decay of the hole probability

Recent works

Mikhail Sodin, Boris Tsirelson,
"Random comples zeroes, III. Decay of the hole probability."
Israel Journal of Mathematics 147, 371-379 (2005).
Available online (free of charge) from e-print archive (USA):
or its Israeli mirror:

A research paper, 9 pages, bibl. 6 refs.

By a hole we mean a disc that contains no flat chaotic analytic zero points (i.e. zeroes of a random entire function whose Taylor coefficients are independent complex-valued Gaussian variables, and the variance of the k-th coefficient is 1/k!). A given disc of radius r has a probability of being a hole, - the hole probability. We show that for large r the hole probability decays as exp(-cr4).

  1. Introduction.
  2. Proof of the lower bound in Theorem 1.
  3. Large deviations.
  4. Mean lower bound.
  5. Proof of Theorem 2.
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