239 research outputs found
Zero Sets for Spaces of Analytic Functions
We show that under mild conditions, a Gaussian analytic function that a.s. does not belong to a given weighted Bergman space or Bargmann-Fock
space has the property that a.s. no non-zero function in that space vanishes
where does. This establishes a conjecture of Shapiro (1979) on
Bergman spaces and allows us to resolve a question of Zhu (1993) on
Bargmann-Fock spaces. We also give a similar result on the union of two (or
more) such zero sets, thereby establishing another conjecture of Shapiro (1979)
on Bergman spaces and allowing us to strengthen a result of Zhu (1993) on
Bargmann-Fock spaces.Comment: 17 p
Surprise probabilities in Markov chains
In a Markov chain started at a state , the hitting time is the
first time that the chain reaches another state . We study the probability
that the first visit to occurs precisely at a
given time . Informally speaking, the event that a new state is visited at a
large time may be considered a "surprise". We prove the following three
bounds:
1) In any Markov chain with states, .
2) In a reversible chain with states, for .
3) For random walk on a simple graph with vertices,
.
We construct examples showing that these bounds are close to optimal. The
main feature of our bounds is that they require very little knowledge of the
structure of the Markov chain.
To prove the bound for random walk on graphs, we establish the following
estimate conjectured by Aldous, Ding and Oveis-Gharan (private communication):
For random walk on an -vertex graph, for every initial vertex ,
\[ \sum_y \left( \sup_{t \ge 0} p^t(x, y) \right) = O(\log n). \
On multiple peaks and moderate deviations for supremum of Gaussian field
We prove two theorems concerning extreme values of general Gaussian fields.
Our first theorem concerns with the concept of multiple peaks. A theorem of
Chatterjee states that when a centered Gaussian field admits the so-called
superconcentration property, it typically attains values near its maximum on
multiple near-orthogonal sites, known as multiple peaks. We improve his theorem
in two aspects: (i) the number of peaks attained by our bound is of the order
(as opposed to Chatterjee's polynomial bound in
), where is the standard deviation of the supremum of the
Gaussian field, which is assumed to have variance at most and (ii) our
bound need not assume that the correlations are non-negative. We also prove a
similar result based on the superconcentration of the free energy. As primary
applications, we infer that for the S-K spin glass model on the -hypercube
and directed polymers on , there are polynomially (in ) many
near-orthogonal sites that achieve values near their respective maxima.
Our second theorem gives an upper bound on moderate deviation for the
supremum of a general Gaussian field. While the Gaussian isoperimetric
inequality implies a sub-Gaussian concentration bound for the supremum, we show
that the exponent in that bound can be improved under the assumption that the
expectation of the supremum is of the same order as that of the independent
case.Comment: 25 pages; The title of the paper is revise
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