Universality results for zeros of random holomorphic sections

Bayraktar, Turgay; Coman, Dan; Marinescu, George

doi:10.1090/tran/7807

Cited by 17 publications

(25 citation statements)

References 51 publications

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“…Indeed, for a compact Kähler manifold pX, ωq endowed with a Hermitian holomorphic line bundle pL, hq with positive curvature ω " c 1 pL, hq, Shiffman-Zelditch [23] showed that the normalized currents of integration 1 p rDivps p qs over zero divisors of a random sequence of sections s p P H 0 pX, L p q converge almost surely to c 1 pL, hq as p Ñ 8. This result was generalized to the noncompact setting in [14] and to the setting of singular metrics whose curvature is a Kähler current in [9,11,13]. It holds also in our present setting and implies that the number of zeros (counted with multiplicity) of a random section s p in an open set U with negligible boundary is asymptotically equal to p times the area of U in the metric given by c 1 pL, hq.…”

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confidence: 56%

“…On the probabilistic side we allow for probability measures which are no longer Gaussian anymore; instead, these probability measures will be assumed to fulfill some rather general conditions which entail a certain universality of the results we obtain. In [9] (see also [8] for a survey) it was shown that the equidistribution of zeros takes place for a large class of probability measures satisfying a certain moment condition (e. g. measures with heavy tail probability and small ball probability, or measures with support contained in totally real subsets of the complex probability space). Analogous equidistribution results for non-Gaussian ensembles are proved in [6,7,10,12].…”

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confidence: 99%

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Large deviations for zeros of holomorphic sections on punctured Riemann surfaces

Drewitz,

Liu,

Marinescu

2021

Preprint

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In this article we obtain large deviation estimates for zeros of random holomorphic sections on punctured Riemann surfaces. These estimates are then employed to yield estimates for the respective hole probabilities. A particular case of relevance that is covered by our setting is that of cusp forms on arithmetic surfaces. Most of the results we obtain also allow for reasonably general probability distributions on holomorphic sections, which shows the universal character of these estimates. Finally, we also extend our results to the case of certain higher dimensional complete Hermitian manifolds, which are not necessarily assumed to be compact.

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confidence: 56%

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confidence: 99%

Large deviations for zeros of holomorphic sections on punctured Riemann surfaces

Drewitz,

Liu,

Marinescu

2021

Preprint

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“…For random polynomials using a basis other than orthogonal polynomials, see [20]. For random holomorphic sections of a line bundle see [3], [25].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic zero distribution of random orthogonal polynomials

Bloom

Dauvergne²

2019

Ann. Probab.

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We consider random polynomials of the form H n (z) = n j=0 ξ j q j (z) where the {ξ j } are i.i.d non-degenerate complex random variables, and the {q j (z)} are orthonormal polynomials with respect to a compactly supported measure τ satisfying the Bernstein-Markov property on a regular compact set K ⊂ C. We show that if P(|ξ 0 | > e |z| ) = o(|z| −1 ), then the normalized counting measure of the zeros of H n converges weakly in probability to the equilibrium measure of K. This is the best possible result, in the sense that the roots of G n (z) = n j=0 ξ j z j fail to converge in probability to the appropriate equilibrium measure when the above condition on the ξ j is not satisfied. In addition, we give a multivariable version of this result.We also consider random polynomials of the form n k=0 ξ k f n,k z k , where the coefficients f n,k are complex constants satisfying certain conditions, and the random variables {ξ k } satisfy E log(1 + |ξ 0 |) < ∞. In this case, we establish almost sure convergence of the normalized counting measure of the zeros to an appropriate limiting measure. Again, this is the best possible result in the same sense as above.

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“…Theorem 4.1 remains true in certain cases for independent random variables which are not necessarily identically distributed. Some natural tail probability and small ball assumptions suffice; we refer the reader to [4] and the conditions (B1) and (B2) therein.…”

Section: Distribution Of Zeros In Higher Codimensionsmentioning

confidence: 99%