Asymptotic zero distribution of random orthogonal polynomials · arXivDesk
1801.10125Jan 30, 201834 pages. The title has been changed from `Universality for zeros of random polynomials'. Edits have been made throughout the paper, chiefly in the introduction, and in the proofs of Lemma 5.5/Theorem 5.6 and Corollary 6.8. Remark 5.4 and Corollary 6.6 have been added
Asymptotic zero distribution of random orthogonal polynomials
We consider random polynomials of the form Hn(z)=∑j=0nξjqj(z)
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where the
{ξj}
are i.i.d non-degenerate complex random variables, and the
{qj(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(∣ξ0∣>e∣z∣)=o(∣z∣−1)
, then the normalized counting measure of the zeros of
Hn
converges weakly in probability to the equilibrium measure of
K.
This is the best possible result, in the sense that the roots of
Gn(z)=∑j=0nξjzj
fail to converge in probability to the appropriate equilibrium measure when the above condition on the
ξj
is not satisfied. In addition, we give a multivariable version of this result. We also consider random polynomials of the form
∑k=0nξkfn,kzk
, where the coefficients
fn,k
are complex constants satisfying certain conditions, and the random variables
{ξk}
satisfy
Elog(1+∣ξ0∣)<∞
. 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.