Zeros of random polynomials and its higher derivatives
Sung-Soo Byun Jaehun Lee Tulasi Ram Reddy
math.PR 60G99, 30C15

Zeros of random polynomials and its higher derivatives]Zeros of random polynomials and its higher derivatives


 

Department of Mathematical Sciences, Seoul National University,Seoul, 151-747, Republic of Korea
sungsoobyun@snu.ac.kr  



Department of Mathematical Sciences, Seoul National University,Seoul, 151-747, Republic of Korea
hun618@snu.ac.kr  


Division of Sciences, New York University Abu Dhabi, Saadiyat Island, Abu Dhabi, United Arab Emirates
tulasi@nyu.edu


S.-S. Byun was partially supported by Samsung Science and Technology Foundation (SSTF-BA1401-01). J.-H. Lee was partially supported by the National Research Foundation of Korea (NRF-2016K2A9A2A13003815). Financial support by German Research Foundation (DFG) for S.-S. Byun and J.-H. Lee through the IRTG 2235 is gratefully acknowledged. 



 In this article we study the limiting empirical measure of zeros of higher derivatives for sequences of random polynomials. We show that these measures agree with the limiting empirical measure of zeros of corresponding random polynomials. Various models of random polynomials are considered by introducing randomness through multiplying a factor with a random zero or removing a zero at random for a given sequence of deterministic polynomials. We also obtain similar results for random polynomials whose zeros are given by i.i.d. random variables. As an application, we show that these phenomenon appear for random polynomials whose zeros are given by the 2D Coulomb gas density. 



[
    Tulasi Ram Reddy
    December 30, 2023
=====================




2010 Mathematics Subject Classification: Primary 60G99; Secondary 30C15.


Key words and phrases: random polynomials, zeros of derivatives, 2D Coulomb gas, zeros of characteristic polynomials, logarithmic potential theory.


§ INTRODUCTION

The study of zeros of polynomials and its derivatives have been of interest for long. A well known result relating these refer to Gauss and Lucas' theorem which states that the critical points of a polynomial  lie in the convex hull formed by the zeros of the polynomial. There have been several extensions and refinements of this result. For instance, Pereira <cit.> and Malamud <cit.> independently extended this result to relating the zeros and critical points with a doubly stochastic matrix. For a deeper discussion on this topic, we refer the reader to classical texts <cit.> and <cit.>.

In the theory of random polynomials, one natural way of constructing model is imposing randomness to the coefficients of the polynomials. Then it is of interest to understand the limiting behavior of zero sets when the degree of polynomials goes to infinity. This question was answered by Kabluchko and Zaparozhets in the case that coefficients are taken to be scaled i.i.d. random variables, see <cit.>. They showed that the limiting empirical measure of zeros is radially symmetric and depends only on the variance profile of the coefficients. Following the result of Kabluchko and Zaparozhets, one can observe that the empirical measure of zeros of higher derivatives of these polynomials converge to the same measure as that of the zeros of these polynomials.

It is to be noted that when the zeros are all taken to be on the real line, this phenomenon holds for any suitable deterministic sequence of polynomials. This follows from the elementary property that the zeros and critical points of the polynomial interlace. However this phenomenon cease to hold at this generality when the zeros are allowed to be complex numbers. A simple example can be constructed by choosing P_n(z)=z^n-1, where the zeros are uniformly distributed on the unit circle, whereas the critical points are accumulated at the origin. For more examples, where this phenomenon does not hold, see the discussion in <cit.>.

It was first conjectured by Pemantle and Rivin in <cit.> that the limiting empirical measures of zero set and critical point coincide for general random polynomials. Moreover in <cit.>, they studied a model having zeros chosen by i.i.d. random variables and established this phenomenon when the measure μ from which the zeros are chosen has finite 1-energy. In <cit.>, Subramanian showed a similar result in the case of general probability measure μ supported on unit circle. Kabluchko extended this phenomenon for general probability measure μ, see <cit.>.  

In addition, the case that random zeros interact with each other has also been studied. One of the important examples is the characteristic polynomial of some random matrix model. For instance, in <cit.>,  O'Rourke showed this phenomenon in the case of characteristic polynomials of circular ensembles. Dennis and Hannay studied the association of critical points of the characteristic polynomial of Ginibre random matrix to its zeros, see <cit.>. Hanin in <cit.>, studied the correlation functions of critical points and zeros of spherical polynomial ensembles. It is pertinent to note that whenever the support of the limiting measure of the zeros does not divide the complex plane into disconnected components this phenomenon holds without invoking any randomness, see <cit.>. As a consequence, this phenomenon holds for the characteristic polynomial of Ginibre random matrix.


It is  natural to study this phenomenon for the higher derivatives too.  We remark that in the case that the limiting empirical measure of zero set is purely atomic, this phenomenon trivially holds. For non-atomic case, Hu and Chang in <cit.> studied this problem when the zeros of polynomials are in a bounded strip. 

Our first purpose of this paper is to extend some known results of this phenomena from critical points to zeros of higher derivatives. See Theorem <ref> for the generalization of result in <cit.> and Theorem <ref>, <ref> for those of theorems in <cit.>.
Our second aim is to extend the class of random polynomials in which this phenomenon occurs by virtue of introducing randomness to a given deterministic polynomials. More precisely, we will impose randomness in such a way as to exclude one zero at random (Theorem <ref>) or to include finite random zeros (Theorem <ref>). We remark that Theorem <ref> affirmatively settles the conjecture posed in <cit.>, when measure is non-atomic. In the last section, we will utilize methods used above to show that this phenomenon also occurs in the case when zeros of the polynomials are given by the 2D Coulomb gas density with general external potential, see Theorem <ref>. 




 §.§ Symbols and notation



Throughout this paper, we will use the following symbols and notation. Let _r be the disk of radius r, centered at origin. For any polynomial
P, we denote by 𝒵(P) the multi-set of zeros of P and ℳ(P) the uniform probability measure supported on 𝒵(P). We write δ_a the Dirac measure supported at a. We recall preliminary definitions introduced in <cit.>.


	Let {a_n}_n≥ 1 (resp, {a_n,i}_n≥1;1≤ i≤ n) be a sequence (resp., triangular array) of complex numbers. If the measure 1/n∑_i=1^n δ_a_i (resp., 1/n∑_i=1^n δ_a_n,i) converges weakly to a probability measure μ, we call such a sequence (resp, triangular array) to be μ-distributed. 



	We say a sequence (resp, triangular array) of complex numbers {a_n}_n≥ 1 (resp, {a_n,i}_n≥1;1≤ i≤ n) is log-Cesáro-bounded if the Cesáro means of the positive part of their logarithms are bounded, i.e., the sequence {1/n∑_i=1^n log_+|a_i| }_n ≥ 1, (resp, {1/n∑_i=1^n log_+|a_n,i|}_n ≥ 1) is bounded. 







 §.§ Results



Our first result deals with the random polynomial whose zeros are chosen to be i.i.d. random variables. Note that the empirical measure of these zeros converge to the probability measure from which the random variables are drawn. We show that the same phenomenon appear for the zeros of any derivative of these random polynomials. This result generalizes the result of Kabluchko <cit.> which was established for zeros of first derivative (critical points).

 
	Let  { z_i }_i≥1 be i.i.d. random variables distributed according to μ, where μ is an arbitrary probability measure on . For each n∈ℕ, let 
	
    P_n(z):=(z-z_1)⋯(z-z_n).

	Then for any k ∈ℕ, ℳ(P^(k)_n) →μ in probability. 


We show that the same result holds when the zeros of polynomials are independently sampled, from two deterministic sequences (triangular arrays), see Theorem <ref>  (Theorem <ref>) below. We remark that Theorem <ref> can be utilized to verify this phenomenon in the case that the deterministic zeros are perturbed independently at random. For example, choose a_i=z_i+σ_iX_i and b_i=z_i-σ_iX_i, where {z_i}_i≥1 is a μ-distributed deterministic sequence, X_i's are  i.i.d. symmetric random variables and {σ_i}_i≥1 is a sequence of positive numbers converging to 0. This is stated as Corollary <ref>. 
These results in the case of zeros of the first derivative were shown in <cit.>.


 
	Let { a_i }_i ≥ 1 and { b_i }_i ≥ 1 be two μ-distributed, log-Cesáro bounded sequences of complex numbers. Suppose that a_i ≠ b_i for infinitely many i. Let {ξ_i}_i ≥ 1 be a sequence of independent random variables such that ξ_i=a_i or ξ_i=b_i with equal probability. For each n ∈ℕ, let 	
	
    P_n(z):=(z-ξ_1) ⋯(z-ξ_n).

	Then ℳ(P_n) →μ almost surely and ℳ(P^(k)_n) →μ in probability for any k ∈ℕ.







 
	Let { z_i }_ı≥ 1 be a μ-distributed log-Cesáro bounded sequences of complex numbers. For a non-zero i.i.d. sequence of symmetric random variables { X_i } satisfying [|X_1|] < ∞, let 
	
    P_n(z):= ∏_i=1^n (z-z_i+σ_i X_i),

	where {σ_i }_i≥1 is a decreasing sequence of real number satisfying lim_i σ_i=0. Then ℳ(P_n) →μ almost surely and ℳ(P^(k)_n) →μ in probability for any k ∈ℕ.  









	Let { a_i,j}_i ≥ 1 ; 1 ≤ j ≤ i and { b_i,j}_i ≥ 1 ; 1 ≤ j ≤ i be two μ-distributed and log-Cesáro bounded triangular arrays of complex numbers satisfying ∑_i=1^n log_+1/|a_n,i-b_n,i|=o(n^2). For each i ≥ 1, let {ξ_i,j}_j ≤ i be a sequence of independent random variables such that ξ_i,j=a_i,j or ξ_i,j=b_i,j with equal probability. For each n ∈ℕ, let 
	
    P_n(z):=(z-ξ_n,1) ⋯(z-ξ_n,n).

	Then ℳ(P_n) →μ almost surely and ℳ(P^(k)_n) →μ in probability for any k ∈ℕ.



Our next result deals with a question appeared in <cit.> which states that the same phenomenon will happen when a zero is removed uniformly at random from a deterministic sequence of polynomials. We resolve this conjecture positively when the empirical measure of zeros of the polynomials converge to a non-atomic probability measure. 





 
	Suppose {z_i}_i ≥ 0 is a μ-distributed, log-Cesáro bounded sequence of complex number, where μ is a non-atomic probability measure on . For each n ∈ℕ, let
	
    P_n(z)=(z-z_0)(z-z_1)… (z-z_n)/z-z_s_n,
 
	where s_n is a random number distributed uniformly on the set {0,1,⋯,n}. Then ℳ(P_n) →μ almost surely and ℳ(P'_n) →μ in probability.




We now consider sequence of polynomials whose zeros are deterministic except for finite ones. Further we assume that the zero set {z_n,i}_n ≥ 1, i ≤ n is μ-distributed triangular array of complex numbers. For such polynomials, we show that the empirical measure of zeros of higher derivatives (up-to the number of random zeros) converge to the same limiting measure as that of the zeros of these polynomials. We remark that it can be interpreted as a random perturbation of polynomials where the perturbed polynomial is obtained by multiplying with a random factor, which strengthens Theorem 2.1 in <cit.>. 

Before stating our results we introduce the assumptions on the random zeros of our polynomials. Consider a random vector (X_n,1, …, X_n,k), where X_n,j's are complex-valued random variables distributed according to the joint probability density function ν_n(w_1, …, w_k). Here, we assume that for any k, there exist positive constants C_1,C_2>0 and a ∈ [0,1), which does not depend on n,i such that ν_n satisfies the following conditions. 

    ∫_^k∑_i=1^klog_+|w_i| ν_n(w_1,w_2,… w_k)dw_1 dw_2 … dw_k ≤ C_1 <∞;
 

    sup_w_i ∈ν_n(w_1, …, w_k)/∫_ν_n(w_1, …, w_k) dw_i ≤ C_2 exp( n^a )      w_1,…, w_i-1,w_i+1, …, w_k ∈;
 

    lim_ r →∞lim sup_n →∞(  max_1 ≤ i ≤ k|X_n,i| ≥ r ) = 0.

Notice that (<ref>) ensures that X_n,1, …, X_n,k have finite log_+-moments. Note also that (<ref>) has the following probabilistic interpretation: for any given complex numbers 
    w_1, …, w_i-1, w_i+1, … w_k,
 the conditional density of w_i is sub-exponentially bounded. 








 
	For fixed k ∈ and each n∈ℕ, suppose that sequence of complex-valued random vector (X_n,1, …, X_n,k) with joint probability density ν_n(w_1, …, w_k) satisfies (<ref>), (<ref>) and (<ref>). Let
	
    P_n(z):=(z-z_n,1)⋯(z-z_n,n)(z-X_n,1) ⋯ (z-X_n,k).

	Then  ℳ(P_n) →μ and ℳ(P^(ℓ)_n) →μ in probability for any 1≤ℓ≤ k.


We remark that one of simple examples of probability distributions satisfying above conditions (<ref>), (<ref>) and (<ref>) is the mutually independent random variables with bounded densities. We state this specific case as the following corollary.  
 
	Suppose that {z_i}_i ≥1 is a log-Cesáro bounded μ-distributed sequence of complex numbers, where μ is any probability measure on . Let k ∈ and X_1, … X_k be independent complex-valued random variables according to bounded density ν_1, …, ν_k on , respectively. For each n∈ℕ, let 
	
    P_n(z):=(z-z_1)⋯(z-z_n)(z-X_1)⋯(z-X_k).

	Then ℳ(P_n) →μ and ℳ(P^(ℓ)_n) →μ in probability for any 1≤ℓ≤ k.


As a consequence of Corollary <ref>, Theorem <ref> can be obtained in a special case, when the measure μ has bounded density and satisfy ∫_log_+|z|dμ(z) < ∞. This is obtained by conditioning on all the zeros except for the first k of them.


We further extend this phenomenon in the case where the zeros of the random polynomials follow 2D Coulomb gas density. For the benefit of the reader we recall some definitions and existing results concerning 2D Coulomb gases. For a fixed positive value β and given external field Q: →, let _n^β be the point process distributed as

    d𝐏_n^β(ζ_1, ⋯, ζ_n)=  
    1/ Z_n^β∏_j,k:j<k|ζ_j-ζ_k|^2βe^-β n ∑_i Q(ζ_i) d_2n,

where Z_n^β stands for the partition function and d_2n is the Lebesgue measure in ^2n.

As the number of particles goes to infinity, the system {ζ_i }_1 ≤ i ≤ n tends to be concentrated in a certain compact set S called the droplet. One of the well-known examples is the complex Ginibre ensemble, in which β=1 and Q(z)=|z|^2. In this case the droplet is given as S= { z : |z| ≤1 }. 
In general, Hedenmalm and Makarov showed that under the mild assumptions on Q, the empirical measure of the system {ζ_i }_1 ≤ i ≤ n converges weakly to the equilibrium measure given by weighted (logarithmic) potential. See <cit.> for more details. Also when Q satisfies some regularity conditions in a neighborhood of S, the limiting equilibrium measure σ_Q is absolutely continuous with respect to Lebesgue measure dm, and takes the following explicit form:

    dσ_Q(z)=1/4π χ_S ·Δ Q(z) dm(z).


Before we state our theorem below, we introduce the assumptions on the external potential Q. One of the main ingredients in proving Theorem <ref> is a certain type of concentration inequality for 2D Coulomb gas due to Chafaï, Hardy and Maïda, see <cit.>. Therefore, we also consider the same assumptions on Q as follows. 

∙ Assumptions (A0). 

	
  * Q is finite on a set of positive capacity and C^2-differentiable;
	
  * lim inf_|z|→∞Q(z)/|z|^2>0;
	
  * lim sup_|z|→∞1/Q(z)sup_|w-z|<1Δ Q(w) <4.

For basic notions in logarithmic potential theory, we refer the reader to <cit.>. For instance, the potentials Q(z)=|z|^2α (α≥ 1) satisfies (A0). Note that (1) implies that Q is admissible and the Boltzmann-Shannon entropy 
S(σ_Q)= -∫dσ_Q/dxlogdσ_Q/dx dx
of equilibrium measure σ_Q is finite.
We remark that as the authors pointed out, the assumptions (A0) can be weakened as follows, see <cit.>.

∙ Assumptions (A1). 
 
	
  * Q is finite on a set of positive Lebesgue measure and σ_Q is of the form (<ref>);
	
  * lim inf_|z|→∞Q(z)/|z|^2>0;
	
  * Q can be decomposed as Q=Q̃+h, where Q̃ is twice differentiable function satisfying condition (A0)-(3), and h is a super-harmonic function.   

Note that by (i) and (iii), σ_Q has bounded density inside the support, which implies S(σ_Q) is finite. We remark that if Q has a Lipschitz continuous derivative, then σ_Q is of the form (<ref>), see e.g., <cit.>. In the case of radially symmetric potentials given as Q(z)=g(|z|) for some g: _+ →, the following condition implies (i): Q is finite on a set of positive Lebesgue measure and r g'(r) is increasing on _+ (or g is convex on _+), see e.g., <cit.>


We emphasize that under our assumption, the “shape” of droplets is not restricted to a simply connected domain. For example our theorem cover the case that the external potential is given by “Mittag-Leffler” potential
Q(z)=|z|^2α-2νlog|z|, (α≥ 1,ν > 0). In this case, the droplet is given by annulus where its modulus depends on α, ν. We remark that these cases are not covered by Eremenko's result, see <cit.>.
 
	For any β>0 and any external potential Q satisfying (A1), let {ζ_i }_1 ≤ i ≤ n be the corresponding 2D Coulomb gas ensemble, and define 
	
    P_n(z):=(z-ζ_1)⋯(z-ζ_n).

	Then for any k ∈,
	
    ℳ( P^(k)_n ) →1/4π χ_S ·Δ Q(z) dm(z)   in probability.




















§ OUTLINE OF METHOD


To prove our results, we follow the potential theoretic approach introduced by Kabluchko in <cit.>. For given polynomial P_n(z) with 𝒵(P)={ w_1, ⋯, w_n  } and 1 ≤ k ≤ n, let us define 

    L^k_n(z):=1/k!P^(k)_n(z)/P_n(z) = ∑_1≤ i_1 < i_2<…<i_k ≤ n1/z-w_i_1…1/z-w_i_k.
 
Note that L_n^k is the product of logarithmic derivatives of P_n,P_n^(1),…,P_n^(k-1), with a scaling of 1/k!. For fixed k ∈ and any f ∈ C_c^∞() whose support is contained in _r, we define f_n(z):=1/n(log|L_n^k(z)|)Δ f(z). Then by applying Green's theorem twice, we have

    1/2π∫__r f(z) Δ1/nlog |L_n^k(z)|  =∫__r f_n(z)dm(z).

Here Δ1/nlog |L_n^k(z)| is interpreted in the sense of distributions. Notice that,

    1/2π∫__r f(z) Δ1/nlog |L_n^k(z)|=1/n∑_j=1^n f(w_j)-1/n∑_j=1^n-k f(ξ^k_j),

where {ξ_j^k : 1 ≤ j ≤ n-k }= 𝒵(P_n^(k)).
Therefore to verify the concentration of empirical measures ℳ(P_n) and ℳ(P_n^(k)), it is enough to show that ∫__r f_n(z)dm(z) converges to 0. To ensure the tightness, we recall a lemma of Tao and Vu.

 <cit.>.  
	Let (X,𝒜,ν) be a finite measure space and f_n: X →, n ≥ 1 be  random functions which are defined over a probability space (Ω, ℬ,) and jointly measurable with respect to 𝒜⊗ℬ. Assume that : 
		
		
  * For ν-a.e. x∈ X we have f_n(x) → 0 in probability, as n →∞; 
		
  * For some δ>0, the sequence ∫_X |f_n(x)|^1+δdν(x) is tight.
	
	Then, ∫_X f_n(x)dν(x) → 0 in probability, as n → 0.


By Lemma <ref>, it is enough to show that the following two statements hold.

    1/nlog|L^k_n(z)| → 0   z ∈;


    the sequence  {∫_^r1/n^2log^2 |L^k_n(z)|dm(z) }_n≥ 1  is tight.











































§ CONTROLLING LARGE VALUES OF L_N^K AND TIGHTNESS


As we explained above in Section 2, all we need to show is the upper and lower  estimate of L_n^k and (<ref>) for given sequence of random polynomials P_n. In this section, we prove the following two lemmas which provide the upper estimate and tightness which can be applied for every cases in our theorems. The following lemma is counterpart of <cit.>.
 
	Let { a_i,j}_i ≥1; 1 ≤ j ≤ i be any triangular array of numbers. Define  
    L_n^k(z)=∑_1 ≤ j_1 < ⋯ < j_k ≤ n 1/|z-a_n,j_1|1/|z-a_n,j_2|⋯1/|z-a_n,j_k|.
 
	Then for Lebesgue a.e. z ∈,
	
    lim sup_n →∞1/nlogL_n^k(z) ≤ 0.
		
 
 First, notice that logL_n^k(z) ≤ klogL_n^1(z). 
	Now it is enough to bound logL_n^1(z)  which follows from the proof of Lemma 4.2 in <cit.>.



	Let { a_i,j}_i≥1; 1≤ j ≤ n be a log-Cesáro bounded triangular array of numbers. Define P_n(z)= ∏_j=1^n (z-a_n,j). Then, for any r>0 and k ∈, the sequence 
	
    {1/n^2∫__rlog^2 | P_n^(k)(z)/P_n(z)| dm(z)}_n≥ k
 
	is bounded.


	Notice that
	
    ∫__rlog^2 | P_n^(k)(z)/P_n(z)| dm(z)=∫__rlog_+^2 | P_n^(k)(z)/P_n(z)| dm(z)+∫__rlog_-^2 | P_n^(k)(z)/P_n(z)| dm(z).

	We now analyse above positive and negative parts of the logarithm separately. For the positive part, observe that by Cauchy-Schwarz inequality, we obtain
	
    ∫__rlog_+^2 | P_n^(k)(z)/P_n(z)| dm(z) 
    	   = ∫__rlog_+^2 | P_n^(k)(z)/P_n^(k-1)(z)…P^(1)_n(z)/P_n(z)| dm(z),
       ≤∫__r( log_+ | P_n^(k)(z)/P_n^(k-1)(z)|+ ⋯ + log_+ | P^(1)_n(z)/P_n(z)| )^2 dm(z),
    	
       ≤ k ∫__rlog_+^2 | P_n^(k)(z)/P_n^(k-1)(z)|+ ⋯ + log_+^2 | P_n^(1)(z)/P_n(z)| dm(z).

	Note that for any 0 ≤ j ≤ k-1, each term in the above expression is bounded as 
	
    ∫__rlog_+^2 | P_n^(j+1)(z)/P^(j)_n(z)| dm(z) 
    	= ∫__rlog_+^2 | ∑_ω: P^(j)_n(ω)=01/z-ω| dm(z),
    	
    ≤   ∫__r(log(n-j)+∑_ω: P^(j)_n(ω)=0log_+ | 1/z-ω| )^2 dm(z),
    	
    ≤    (n-j+1)∫__r( log^2(n-j)+∑_ω: P^(j)_n(ω)=0log_+^2 | 1/z-ω| ) dm(z),
    	
    
    	=     (n-j+1)( π r^2 log^2(n-j)+∫__r∑_ω: P^(j)_n(ω)=0log_-^2 | z-ω| dm(z) ),

	where in the first inequality we have used log_+( ∑_k=1^n a_k) ≤log n + ∑_k=1^n log_+ a_k for any n ∈ and a_1,…,a_n ∈, see <cit.> for instance. By the translation invariance of Lebesgue measure, 

    ∫__rlog_-^2|z-ξ|dm(z)=∫__r+ξlog_-^2|z|dm(z) 
       ≤sup_K ⊂∫_Klog_-^2|z|dm(z)=  ∫__1log_-^2|z|dm(z)< ∞.

 Therefore, we conclude that {1/n^2∫__rlog_+^2 | P_n^(k)(z)/P_n(z)| dm(z)}_n≥ k is bounded. 
	
	Now we show that the negative part is also bounded. Note that using log_-|ab| ≤log_-|a| + log_-|b| for any a,b ∈ we have
	
    1/n^2∫__rlog^2_-| P_n^(k)(z)/P_n(z)| dm(z) ≤1/n^2∫__rlog^2_-| P_n^(k)(z) | + log^2_-| 1/P_n(z)| dm(z).

	As in the same way above, we get the following inequality for the first term: 
	
    1/n^2∫__rlog^2_-| P_n^(k)(z) | dm(z) ≤n-k/n^2∫__r∑_w∈𝒵(P_n^(k))log_-^2|z-w|dm(z).

	which is uniformly bounded in n by (<ref>). Also, we have
	
    1/n^2∫__rlog^2_-| 1/P_n(z)| dm(z)=   1/n^2∫__rlog^2_+| P_n(z) | dm(z),
    	
    ≤   1/n^2∫__r( ∑_j=1^n  log_+| z-a_n,j| )^2 dm(z),
    	
    ≤   ∫__r( log 2 +log_+|z|+1/n∑_j=1^n  log_+|a_n,j| )^2 dm(z).
 
	Now the lemma follows from the fact that { a_n,j} is log-Cesáro bounded.























§ CONTROLLING SMALL VALUES OF L_N^K.





 §.§ Proof of Theorem <ref>,  <ref> and  <ref>.



In this subsection, we present the proof of Theorem <ref>,  <ref> and  <ref>. 

Controlling the small values of L_n^1 were given in  <cit.> and <cit.>. We will use these lemmas and the induction argument to control the small values of L_n(z) in the corresponding  theorems.




  §.§.§ Proof of Theorem <ref>
 


Recall that under the conditions of Theorem <ref>, zeros of random polynomials P_n(z) are given by i.i.d. random variables { z_i }_i≥1 distributed according to μ. First we introduce the following lemma due to Kabluchko.




















<cit.>
	For Lebesgue a.e. z ∈,
	
    lim_n→∞(|L^1_n(z)| < e^-n ) =0
 
	for any >0, where 
	L^1_n(z):=P'_n(z)/P_n(z).
	Here P_n(z)'s are random polynomials in Theorem <ref>.

Recall that L_n^k is given as (<ref>) for each k ≤ n. Now, to complete the proof of Theorem <ref>, all we need to show is the following lemma. 

	For Lebesgue a.e. z ∈,
	
    lim_n→∞(|L^k_n(z)| < e^-n ) =0
 
	for every k ∈ and >0, where L^k_n(z):=1/k!P^(k)_n(z)/P_n(z).
	Here P_n(z)'s are random polynomials in Theorem <ref>.



	When μ is purely atomic measure, the proof of (<ref>) is straightforward so without loss of generality we can assume that μ is not purely atomic. Note that in the case k=1, (<ref>) follows from <cit.>. Suppose that (<ref>) holds for some k ≥ 1. First, we decompose μ by μ_1+μ_2 such that μ_1 is purely atomic, and μ_2 is non-atomic. Then 0 ≤ x:=μ_1()<1 since μ is not purely atomic. Let E:=supp(μ_1).
	For j,m<n, define a random variable L^k,j_n(z) by 
	
    L^m,j_n(z):=∑_1≤ i_1 < … < i_m ≤ n, i_ℓ≠ j ∀ℓ1/z-z_i_1…1/z-z_i_m.

	Note that L^m,j_n(z) is independent of z_j. 
	Fix z ∈ which satisfies (<ref>) for every k ≤ k_0 and >0. First, for any ℓ∈, by definition of L_n^m,j(z) we have
	
    L^k_0+1_n+ℓ(z)=1/z-z_j L^k_0,j_n+ℓ(z) + L^k_0+1,j_n+ℓ(z)      1≤ j ≤ℓ.

	For 1≤ j ≤ℓ, let Ω be the sample space and denote
	
    A_j   := {ω∈Ω : |L^k_0,j_n+ℓ(z)| < e^-n}; 
    
    	B_j   := {ω∈Ω : z_j ∉ E }; 
    
    	C   := {ω∈Ω : |L^k_0+1_n+ℓ(z)| < e^-2n},

	for fixed >0. Note that 	
	
    Ω =  (∪_j=1^ℓ A_j ) ∪(∪_j=1^ℓ(A^c_j ∩ B_j)) ∪(∩_j=1^ℓ B_j^c),

	which implies
	
    C ⊂(∪_j=1^ℓ A_j ) ∪(∪_j=1^ℓ(A^c_j ∩ B_j ∩ C)) ∪(∩_j=1^ℓ B_j^c).

	Therefore, we immediately obtain
	
    (C)≤∑_j=1^ℓ(A_j)+∑_j=1^ℓ(A_j^c ∩ B_j ∩ C) + (∩_j=1^ℓ B_j^c).

	Note that, since L^k_0,j_n+ℓ(z) and L^k_0_n+ℓ-1(z) are identically distributed, we have 
	
    (A_j) = (|L^k_0_n+ℓ-1(z)|<e^-n)      1 ≤ j ≤ℓ.

	which converge to 0 as n goes to ∞, for all 1 ≤ j ≤ℓ by the induction hypothesis. For the second term, using (<ref>), we get 
	
    (A_j^c ∩ B_j ∩ C) 
    	
    
    	=   (z_j ∉ E, |L^k_0,j_n+ℓ(z)| ≥ e^-n, |1/z-z_j L^k_0,j_n+ℓ(z) +L^k_0+1,j_n+ℓ(z)|<e^-2n),
    	
     
    	=    (z_j ∉ E, |L^k_0,j_n+ℓ(z) | ≥ e^-n, |1/z-z_j +L^k_0+1,j_n+ℓ(z)/ L^k_0,j_n+ℓ(z)|<e^-2n/ L^k_0,j_n+ℓ(z)) ,
    	
    ≤   (z_j ∉ E, |1/z-z_j +L^k_0+1,j_n+ℓ(z)/ L^k_0,j_n+ℓ(z)| < e^-n).

	Let ν_z(dw) be a distribution of 1/z-z_1 1_{z_1 ∉ E}. Then since ν_z(dw) is non-atomic measure, we have 
	
    lim_r ↓ 0sup_z_0 ∈ν_z(B(z_0,r)) =0.

	Using (<ref>) to (<ref>), we obtain 
	
    (A_j^c ∩ B_j ∩ C)    ≤(z_j ∉ E, |1/z-z_j +L^k_0+1,j_n+ℓ(z)/ L^k_0,j_n+ℓ(z)| < e^-n), 
    	
       = [ν_z(B (L^k_0+1,j_n+ℓ(z)/ L^k_0,j_n+ℓ(z),e^-n) ) ], 
    	
       ≤sup_z_0 ∈ν_z(B(z_0,e^-n)).

	Also, since z_i are i.i.d. we have
	
    (∩_j=1^ℓ B_j^c) = ∏_j=1^ℓ(z_j ∈ E) = x^ℓ.

	Combining all (<ref>), (<ref>), (<ref>) and (<ref>), we obtain
	
    lim sup_n→∞(|L^k_0+1_n(z)|<e^-2n)= lim sup_n →∞(C) ≤ x^ℓ.

	for arbitrarily ℓ∈. Now, (<ref>) for k=k_0+1 follows from the fact that x<1. This finishes the proof.




  §.§.§ Proof of Theorem <ref> and Theorem <ref>
 


In this section we will prove Theorem <ref> and Theorem <ref> at once. Let {a_n,i} and {b_n,i} be two μ-distributed log-Cesáro bounded triangular arrays of complex numbers. Here we assumed in Theorem <ref> that ∑_i=1^n log_+ 1/|a_n,i-b_n,i| = o(n^2). Note that this condition is used only for the case k=1. Therefore it suffices to prove Theorem <ref> and the proof of Theorem <ref> follows from similar argument. Recall that zeros of random polynomials P_n(z) are given by the random sequence ξ_n,i where ξ_n,i=a_n,i or b_n,i with equal probability. 


Now following lemma completes the proof of Theorem <ref>. 

	For a.e. z ∈,		
	
    lim_n→∞( |L^k_n(z)| < e^-n ) =0
 
	for all k ∈ℕ and >0, where L^k_n(z):=1/k!P^(k)_n(z)/P_n(z).
	Here P_n(z)'s are random polynomials in Theorem <ref>.



	Note that the case k=1 is given by <cit.>. In particular, for the proof of <cit.> the author has loosened the conditions ∑_i=1^n log_+ 1/|a_n,i-b_n,i| = o(n^2) to the following condition : 
	
    lim_↓ 0lim inf_n →∞ |{ i: log_+ 1/|a_n,i-b_n,i|≤ n  }|/n≥3/4.

	Also, (<ref>) holds for any z ∈ that does not agree with any a_n,i or b_n,i in the triangular array. 
	
	We will prove Lemma <ref> by using induction argument on k. 
	
	Let 
    Ω= {ξ_n={ξ_n,i}_i≤ n  |  ξ_n,i = a_n,i b_n,i,   n ∈}
 be the sample space of all possible finite sequences {ξ_n,i}_i≤ n equipped with uniform probability measure . By (<ref>), we have
	
    |{ i: a_n,i≠ b_n,i, i ≤ n }| →∞     n →∞.

Thus, for any ℓ∈, we may assume that a_n,i≠ b_n,i for 1 ≤ i ≤ℓ for large n, since finite permutations on sequences does not affect (<ref>).
	
	Fix z ∈, ℓ,k ∈, and let 𝒩_n=𝒩_n(z) be a set of finite sequence ξ_n={ξ_n,i}_i ≤ n such that
	
    𝒩_n:= {ξ_n ∈Ω  |   |L^k_n(z) | < e^-n}.

	Note that for any n ∈ and z ∈, the value of L_n^k(z) depends only on ξ_n. Set
	
    𝒩_n^ℓ := {ξ_n ∈𝒩_n  | ∃η_n ∈𝒩_n η_n ≠ξ_n, η_n,i = ξ_n,i l < i ≤ℓ}.

	Note that if there exist two sequences ξ_n, ξ_n in 𝒩_n \𝒩^ℓ_n with ξ_n,i=ξ_n,i for 1 ≤ i ≤ℓ, then ξ_n=ξ_n by the construction. Therefore for each given sequence of tails {ξ_n,i}_ℓ <i ≤ n, there is at most one sample contained in 𝒩_n \𝒩^ℓ_n, which implies 
	
    (𝒩_n \𝒩^ℓ_n) ≤ 2^-ℓ.
 
	For any n ∈ and ξ_n ∈𝒩^ℓ_n, using the definition we can find a sequence η_n ∈𝒩^ℓ_n such that η_n ≠ξ_n and η_n,i = ξ_n,i for all l<i ≤ℓ. Note that since ξ_n,  η_n ∈𝒩_n, 
	
    |L^k_n(z) (ξ_n ) |<e^-n,     |L^k_n(z)(η_n ) |<e^-n.

	Let F={ i_j }_1 ≤ j ≤ m be the set of index such that ξ_n,i_j≠η_n,i_j. Without loss of generality, we may assume that 1 ≤ i_1<i_2<...<i_m ≤ℓ. 
	For 1≤ p ≤ m, let us denote  
	
    α_p:= ∑_1 ≤ j_1<…<j_p ≤ m1/z-ξ_n,i_j_1…1/z-ξ_n,i_j_p,   β_p:=∑_1 ≤ j_1<… <j_p ≤ m1/z-η_n,i_j_1…1/z-η_n,i_j_p,
 
	and α_0=β_0=1. 
	Note that, by construction, 
	
    ∑_p=0^m α_p w^m-p=∏_j=1^m (w+ 1/z-ξ_n,i_j),   ∑_p=0^m β_p w^m-p=∏_j=1^m (w+ 1/z-η_n,i_j),

	which implies that two polynomials  ∑_p=0^m α_p w^m-p and  ∑_p=0^m β_p w^m-p cannot be the same as polynomials in `w'. Therefore, α_p ≠β_p for at least one of 1 ≤ p ≤ m. 
	
	For each k_1 ∈ and E ⊂{1,2,…,n}, let us define L^k_1,E_n(z) by 
	
    L^k_1,E_n(z)=L^k_1,E_n(z)(ξ_n) :=∑_1≤ i_1 < … < i_k_1≤ n, i_j ∉ E ∀ j1/z-ξ_n,i_1…1/z-ξ_n,i_k.

	Note that L^k_1,E_n(z) is independent of {ξ_n,i}_i ∈ E by the independence of ξ_n,i. Also by definition of α_p, β_p and F, we have the following decomposition of L^k_n(z):
	
    L^k_n(z)(ξ_n )=∑_p=0^m α_p L^k-p,F_n(z)(ξ_n ),    L^k_n(z)(η_n )=∑_p=0^m β_p L^k-p,F_n(z)(η_n ).

	Notice that since ξ_n,i = η_n,i for all i ∉ F, L^k-p,F_n(z)(ξ_n)= L^k-p,F_n(z)(η_n) for all 0 ≤ p ≤ m. Therefore by (<ref>) and (<ref>), we get
	
    2e^-n     ≥   |L_n^k(z)(ξ_n)-L_n^k(z)(η_n) |, 
          =    |∑_p=0^m α_p L^k-p,F_n(z)(ξ_n) - ∑_p=0^m β_p L^k-p,F_n(z)(η_n)|,  
          =    |∑_p=1^m (α_p- β_p) L^k-p,F_n(z)(ξ_n) |,

	for sufficiently large n.
	
	Let m_0:= max{1 ≤ p ≤ m: α_p ≠β_p }, m_1:= min{1 ≤ p ≤ m: α_p ≠β_p } and κ_1,…,κ_q be zeros of polynomial 
	
    f(w):=∑_p=m_1^m_0 (α_p - β_p)w^m_0 - p.
 
	Note that q ≤ m-1 since q= f = m_0 - m_1 ≤ m-1. Also the fact that f(0)= α_m_0 - β_m_0≠ 0 implies κ_i ≠ 0 for every 1 ≤ i ≤ q. Let γ_i= z+ 1/κ_i for each 1 ≤ i ≤ q. Then we obtain
	
    f(w)= ∏_i=1^q (w+1/z-γ_i),   α_p-β_p= L_q^p({γ_i}_i ≤ q).

	Now let us define a random sequence {ν_n,i}_i ≤ n-m+q by 
	
    ν_n,i =γ_i       1 ≤ i ≤ q 
    ξ'_i-q      q < i ≤ n-m+q,

	where the random sequence {ξ'_i}_1 ≤ i ≤ n-m is constructed as follows: 
	
		
  * For 1 ≤ i ≤ℓ-m, by shortening the sequence {ξ_i}_i ∈{1,…,ℓ} to {ξ_i}_i ∈{1,…,ℓ}\ F; 
		
  * For i > ℓ-m, pick a_n,i+m or b_n,i+m with equal probability.  
	 
	Combining all above, we obtain
	
    |∑_p=1^m (α_p-β_p) L^k_0-p,F_n(z) (ξ_n ) |
    	      =  |∑_p=m_1^m_0 (α_p-β_p) L^k_0-p,F_n(z) ({ξ_n,i}_i ≤ n) | ,
    	
          =  |∑_p=0^q L_q^p (z)({γ_i}_i ≤ q) L_n^k-m_1-p,F(z)({ξ_n,i}_i ≤ n)| ,
     
          =  |∑_p=0^q L_q^p (z)({γ_i}_i ≤ q) L_n-m^k-m_1-p(z)({ξ'_i}_i ≤ n-m) |,
     
          =  |L^k-m_1 _n-m+q(z)({ν_n,i}) |.

	Note that since ℓ is fixed number, there are only finite cases of {ν_n,i}_i ≤ n-m+q. More precisely, since the deterministic part of {ν_n,i}_i ≤ n-m+q is determined by the values of ξ_n,1,…,ξ_n,ℓ and η_n,1,…,η_n,ℓ, the number of cases is less than 2^2ℓ for each n. Now we inductively define 𝒵_k,n,ℓ and C_k,n,ℓ as follows.
	
		
  * First we set 
		
    C_1,n,ℓ:= {{(a_n,i,b_n,i) }_i ≤ n}    𝒵_1,n,ℓ:= {a_n,i,b_n,i : {(a_n,i,b_n,i) }_i ≤ n∈ C_1,n,ℓ}.
 
		
  * We denote by C_2,n,ℓ the collection of {ν_n,i}, where ν is a sequence of 2-vector defined by (<ref>) with the sequence of 2-vector ξ∈ C_1,n,ℓ and set 
		
    𝒵_2,n,ℓ:= 𝒵_1,n,ℓ∪{ν^1_n,i, ν^2_n,i : {ν_n,i} = { (ν^1_n,i, ν^2_n,i)}∈ C_1,n,ℓ}.

		
  * Finally, for any k ≥ 1, we define C_k+1,n,ℓ as the collection of {ν_n,i}, where ν is a sequence of 2-vector defined by (<ref>) with the sequence of 2-vector ξ∈ C_k,n,ℓ and set 
		
    𝒵_k+1,n,ℓ:=𝒵_k,n,ℓ∪{ν^1_n,i, ν^2_n,i : {ν_n,i} = { (ν^1_n,i, ν^2_n,i)}∈ C_k+1,n,ℓ}.

	 
	Note that for each k,n,ℓ∈, |C_k,n,ℓ| ≤ 2^2kℓ since there is at most 2^2 ℓ choice of {ν_n,i} for each sequence of 2-vector. Thus, Z_k,n,ℓ is also finite set. Let
	
    𝒵_k := ⋃_n=1^∞⋃_ℓ=1^∞𝒵_k,n,ℓ,

	which will be exceptional set of (<ref>) for k. Note that 𝒵_k⊂ is countable set since each 𝒵_k,n,ℓ is finite. Also,  𝒵_k is increasing in k since 𝒵_k,n,ℓ is increasing in k for each n,  ℓ. Now we are ready to state our induction hypothesis. We claim that for any {a_n,i} and {b_n,i} satisfying (<ref>), (<ref>) holds for any k ∈ and z ∈𝒵_k^c. Let us emphasize that we already proved the case k=1 at the beginning of this proof. Assume that our claim is valid for k=1,2,…,k_0 - 1. Fix w ∈𝒵^c_k_0 and ℓ∈, and define 𝒩_n and 𝒩_n^ℓ as above. Note that it suffices to show that
	
    lim_n →∞(𝒩_n  ) = lim_n →∞(𝒩_n(w)  )=0

	By (<ref>), (<ref>) and (<ref>) we obtain
	
    lim sup_n →∞(𝒩_n) ≤lim sup_n →∞(𝒩_n \𝒩_n^ℓ ) + lim sup_n →∞( 𝒩_n^ℓ ) ≤ 2^-ℓ + lim sup_n →∞( 𝒩_n^ℓ ) ,
       ≤ 2^-ℓ + lim sup_n →∞(|L_n^k_0(w)(ξ_n)|<e^-n ),
       ≤ 2^-ℓ +	lim sup_n →∞(|L^k_0-m_1_n-m+q(w)(ν_n )| < 2e^-nν_n={ν_n,i}∈ C_2,n,ℓ), 
       ≤ 2^-ℓ + ∑_ν_n ∈ C_2,n,ℓlim sup_n →∞(|L^k_0-m_1_n-m+q(w)(ν_n )| < 2e^-n).

	By the definition of C_k_0,n,ℓ and induction hypothesis, we first  have |C_2,n,l| ≤ 2^2 ℓ. 
	
	For the sequence of 2-vector ξ= {ξ_n }_n ≥ 1 = {  (ξ^1_n,i, ξ^2_n,i    )   }_i ≤ n, n ≥ 1, let us denote by 𝒵_k,n,ℓ(ξ_n) the set defined same as 𝒵_k,n,ℓ, except that C_1,n,ℓ= {ξ_n  }. Then we observe that for any m ≥ 1 and ν_n ∈ C_2,n,ℓ,
 
    𝒵_k_0-m,n,ℓ(ν_n) ⊂𝒵_k_0-m+1,n,ℓ⊂𝒵_k_0,n,ℓ.

	 
Thus, we conclude that since w ∉𝒵_k_0 and m_1 ≥ 1,
	
    lim_n →∞(|L^k_0-m_1_n-m+q(w)(ν_n )| < 2e^-n)=0     ν_n ∈ C_2,n,ℓ.

  Therefore, 
  
    lim sup_n →∞(𝒩_n^ℓ)=0.

	Now the claim follows from the fact that ℓ is arbitrary, which completes the proof.















 §.§ Proof of Theorem <ref> and  <ref>
 


In this subsection, we prove Theorem <ref> and  <ref>. Recall that in these cases, considering random polynomials are constructed as giving randomness to deterministic ones. 




  §.§.§ Proof of Theorem <ref>



Let {a_n,i}_n≥ 1, 0≤ i ≤ n be a triangular array defined by
a_n,i:=1-δ_s_n(i), where s_n is a random number distributed uniformly on the set {0,1,⋯,n}. Assume that μ is non-atomic probability measure. Let {z_n}_n ≥ 0 be a μ-distributed and log-Cesaro bounded sequence.

	For Lebesgue a.e. z ∈,
	
    lim_n→∞(|L^1_n(z)| < e^-n ) =0
 
	for every >0, where L^1_n is given as 
	
    L^1_n(z):= a_n,0/z-z_0 + a_n,1/z-z_1 + … + a_n,n/z-z_n.
 

 Since μ is non-atomic, it suffices to show that if 
	lim sup_n→∞(|L_n(z)|<e^-n)>0 for some >0, z∈, there exist x∈ with μ(x)>0. Suppose that 
	
    lim sup_n→∞(|L_n(z)|<e^-n)=3δ>0
 
	for some fixed >0, z∈.
	Then, there exists a subsequence {n_k}_k≥1 satisfying 	
	
    (|L_n_k(z)|<e^-n_k)>2δ

	Since the value L_n(z) is just determined by s_n ∈{0,1,…,n}, (<ref>) implies that 
	
    | 0≤ i≤ n_k : |∑_j=0^n_k1/z-z_j - 1/z-z_i|<e^-n_k| > 2δ (n_k+1).

	Let 
	
    N_k:=|0≤ i≤ n_k : |∑_j=0^n_k1/z-z_j - 1/z-z_i|<e^-n_k|,

	and w_k,1,w_k_2,…,w_k,N_k are those values of z_i's, i.e.,
	
    |∑_j=0^n_k1/z-z_j - 1/z-w_k,j|<e^-n_k,    1 ≤ j ≤ N_k.
 
	Note that w_k,i may have same values. Without loss of generality, we may assume |w_k,1| ≤ |w_k,2| ≤…≤ |w_k,N_k|. Then by (<ref>), for all 1≤ i_1,i_2 ≤ N_k,
	
    |1/z-w_k,i_1-1/z-w_k,i_2| < 2e^-n_k .

	
	We will use the following inequality: 
	
    1/n_k∑_i=1^n_klog_+ |z_i| ≥1/n_k∑_i=1^N_klog_+ |w_k,i| ≥N_k/n_klog_+ |w_n,1| ≥ 2δlog_+|w_n,1|.

	Notice that since {z_k} is log Cesáro bounded, log_+|w_n,1| is also bounded.
	
	Let B_k:=B̅(w_k,1,r_k) be closed ball centered at w_k,1 with radius 
	
    r_k:=2exp(-n_k)R_k^2/1-2R_k exp(-n_k),     R_k:=|z-w_k,1|.

	By (<ref>), R_k is bounded, which implies 1>r_k>0 for large k. Therefore, we may assume that 1>r_k>0 for all k≥ 1 without loss of generality.
	Note that if w ∉ B_k,  
	
    [                                 |1/z-w_k,1-1/z-w|                                                 =                      |w-w_k,1/(z-w_k,1)(z-w_k)| ,;                                                  ;                                                                                                   ≥ |r_k/(z-w_k,1)(z-w)| ≥r_k/R_k(R_k+r_k)≥ 2e^-n_k . ]

	

	Then by (<ref>) and (<ref>), w_k,i∈ B_k for all 1 ≤ i ≤ N_k.
	
	Let μ_k:= 1/n_k+1∑_i=0^n_kδ_z_i. Then, μ_k(B_k)=N_k/n_k+1≥ 2δ and μ_k →μ weakly. Note that if μ_k →μ weakly, lim sup_k→∞μ_k(C) ≤μ(C) for all closed set C and lim inf_k→∞μ_k(U) ≥μ(U) for all open set U.  
	
	We can find R̃>0 satisfying μ(B(0,R̃-2)) > 1-δ/3. Since lim inf_k→∞μ_k(B(0,R̃-2)) ≥μ(B(0,R̃-2)), μ_k(B(0,R̃-2)) > 1-δ/2. for large k. So B_k ∩ B(0,R̃-2) ≠∅. Since r_k<1, we can say that B_k ⊂B̅(0,R̃):=K for large k. 
	
	Suppose μ has no point measure. Then, for all x∈ K, there exist =(x)>0 such that μ(B̅(x,(x)))<δ. {B(x,(x)/2)}_x∈ K makes open cover of compact set K and there exist finite open cover {B(x_i,(x_i)/2)}_i≤ N.
	
	Let =min{(x_i)/2 : i ≤ N}. r_k< for large k since r_k ↓ 0. For each k with r_k< , there exists i≤ N satisfying B_k ∩ B(x_i, (x_i)/2) ≠∅. So, B_k ⊂B̅(x_i,(x_i)) for some i. Therefore, there exists i≤ N such that 	
	
    B_k ⊂B̅(x_i,(x_i))=:Ck.

	Therefore, 2δ≤lim sup_k μ_k(C) ≤μ(C) < δ is contradiction, where we used (<ref>) for the first inequality, and the fact C is closed and μ_k →μ for the second inequality. Therefore, μ has point mass and the conclusion of the lemma follows.
	







  §.§.§ Proof of  Theorem <ref>




Now we prove Theorem <ref>. Recall that sequence of complex-valued random vector (X_n,1, …, X_n,k) with joint probability density ν_n(w_1, …, w_k) satisfies (<ref>), (<ref>) and (<ref>). The following lemma shows that (<ref>) holds for L_n^1(z).

	For Lebesgue a.e. z ∈,
	
    lim_n→∞(|L^1_n(z)| < e^-n ) =0
 
	for every >0, where 
	L^1_n(z):=P'_n(z)/P_n(z). Here P_n(z)'s are random polynomials given as 
	
    P_n(z):=(z-z_n,1)⋯(z-z_n,n)(z-X_n,1) ⋯ (z-X_n,k).
.


	First we claim that it suffices to show the case k=1. Assume that (<ref>) holds for k=1. Now for any fixed X_n,1, …, X_n,k-1 with general k ≥ 2, we can define w_n+k-1,i= z_n,i for 1 ≤ i ≤ n, w_n+k-1,n+j = X_n,j for 1 ≤ j ≤ k-1 and Y_n,1=X_n+k,1 so that {w_n,i}_n ∈, 1 ≤ i ≤ n is μ-distributed and Y_n,1 satisfies (<ref>) and (<ref>). Thus we have (<ref>) for any fixed X_n,1, …, X_n,k-1, which implies (<ref>). So we can assume k=1 without loss of generality. 
	
	Fix δ >0 and z ∈ satisfying z ≠ z_n,i for any n ∈ and 1 ≤ i ≤ n. By (<ref>), we have constants r_δ>0 and N_δ∈ such that
	
    (|X_n,1| ≥ r_δ - |z|) ≤δ/2     n ≥ N_δ.

	Let θ^(n)_z(w)dw be the conditional density of 1/z-X_n,1_{ |X_n,1| ≤ r_δ - |z|  } given the others. Note that by (<ref>) we have
	
    θ^(n)_z(w) ≤|z-X_n,1|^2 sup_w_i ∈ν_n(w)/∫_ν_n(w) dw ≤  C_2  e^cn^a|z-X_n,1|^2 ≤ C_2 r_δ^2 e^cn^a.
 
	By definition of L_n^1, we may write 
    L_n^1(z)=1/z-X_n,1 +  ∑_i=1^n 1/z-z_i .

	Let us denote
	
    Y_n := 1/z-X_n,1     x_n:= ∑_i=1^n 1/z-z_i,

	i.e., L_n^1(z) = Y_n + x_n. Then for large n satisfying n ≥ N_δ and π C_2 r_δ^2  e^-2n+cn^a≤δ/2, we have
	
    (L_n^1(z) ≤ e^-n)    ≤(|X_n,1| ≥ r_δ - |z|) + ( |Y_n+x_n| ≤ e^-n ; |X_n,1| ≤ r_δ - |z|) ,
       ≤δ/2 + sup_x ∈( |Y_n| ≤ B(-x,e^-n);|X_n,1| ≤ r_δ - |z|), 
       ≤δ/2 + C_3 e^-2n+cn^a≤δ.

	for some constant C_3. Since δ>0 is arbitrary, we obtain (<ref>). 
 

Now we are ready to use induction on k to obatin (<ref>). The following lemma completes the proof of Theorem <ref>. 


	Let k ∈ be the number of X_n,j's. For Lebesgue a.e. z ∈, we have
	
    lim_n→∞(|L^ℓ_n(z)| < e^-n ) =0
 
	for every >0 and ℓ≤ k, where L^ℓ_n(z):=1/ℓ!P^(ℓ)_n(z)/P_n(z).
	Here P_n(z)'s are random polynomials defined by (<ref>).

 Repeating the argument of Lemma <ref>, we can obtain that it suffices to prove Lemma <ref> for the case ℓ=k.
	Define 
	
    Q_n(z):= (z-z_n,1)⋯(z-z_n,n)(z-X_n,1)⋯(z-X_n,k-1)= P_n(z)/z-X_n,k.
  Then we observe that
	
    L_n^k(z) = 1/z-X_n,k M^k-1_n(z) + M^k_n(z),

	where M^l_n(z):= 1/l!Q_n^(l)(z)/Q_n(z) for l ∈. Thus, for any ,δ>0 and any z ∈ satisfying z ≠ z_n,i for all n,i, we have	
	
    (|L^k_n(z)| < e^-2n )    =( | M^k-1_n(z)/z-X_n,k+M^k_n(z) | < e^-2n ) ,
       ≤( | 1/z-X_n,k+M^k_n(z)/M^k-1_n(z)| < e^-n  ; |X_n,k| ≤ r_δ - |z| ) ,
    	
       +(|M_n^k-1(z)| < e^-n) + (|X_n,k| ≥ r_δ - |z|),

	where r_δ>0 is a constant in the proof of Lemma <ref>. Note that by induction hypothesis and definition of r_δ we have
	
    lim_n →∞(|M_n^k-1(z)| < e^-n) = 0     lim sup_n →∞(|X_n,k| ≥ r_δ - |z|) ≤δ/2.

	Set 
	
    Y_n:= 1/z-X_n,k_{|X_n,k| ≤ r_δ -z}.

	Then by (<ref>), we obtain
	
    θ_z(w) ≤ C_3 r_δ^2 n^a

	for density θ_z(w) of Y_n, and
	
    ( | 1/z-X_n,k+M^k_n(z)/M^k-1_n(z)| < e^-n  ; |X_n,k| ≤ r_δ - |z| )  ≤sup_x ∈(|Y_n+x| < e^-n),

	which goes to 0 as n →∞. Therefore, we have
	
    lim sup_n →∞(|L^k_n(z)| < e^-2n ) ≤δ.

	Now lemma follows from the fact that δ is an arbitrary constant.










§ APPLICATION TO 2D COULOMB GAS ENSEMBLES


In this section, we prove Theorem <ref>. Recall that the joint probability density of 2D Coulomb gas ensemble is given as

    d𝐏_n^β(ζ_1, ⋯, ζ_n)=  
    1/ Z_n^β∏_j,k:j<k|ζ_j-ζ_k|^2βe^-β n ∑_j Q(ζ_j) d_2n,

where β>0 is inverse temperature and Q: → is external potential satisfying the assumptions (A1) in Section 1.

First we recall some definitions and properties of equilibrium measure which we will use in this section.
For any probability measure μ, logarithmic potential U^μ : ℂ→ (-∞, ∞]  is defined by 
U^μ(ζ):=∫_ℂlog1/|ζ-η|^2dμ(η)  
and logarithmic energy I[μ] is given as
I[μ]:=∫_ℂ^2log1/|ζ-η|^2dμ(ζ)dμ(η)= 
∫_ℂ U^μ(ζ) dμ(ζ). 
A subset 𝒩 of  is said to be polar if I[μ]=∞ for all compactly supported probability measures with (μ) ∈𝒩. 
We say that some property holds quasi-everywhere (q.e) on  E ⊂ℂ if it holds everywhere on E except some Borel polar set. Note that every Borel probability measure with finite logarithmic energy assigns zero Lebesgue measure
to Borel polar sets.  

For given admissible potential Q, the weighted logarithmic energy I_Q[μ] for each probability measure μ is defined as 

    I_Q[μ]:=∬_ℂ^2log1/|ζ-η|^2dμ(ζ)dμ(η)+2∫_ℂ Q dμ.

The following theorem is a collection of properties of equilibrium measure.
  <cit.>
	
	Suppose that the potential Q is admissible and let 
    I_Q := inf{ I_Q[μ] },
 where infimum is over all probability measures on . 
	Then the following properties hold. 
	
		
  * There exists the unique probability measure σ_Q such that 
    I_Q [σ_Q]=I_Q.

		
  * I_Q[σ_Q]=I_Q is finite. 
		
  * I[σ_Q]=I_Q-2 ∫_Qdσ_Q  is finite. 
		
  * S_Q:=  (σ_Q) is compact.
		
  * Let 
		
    F_Q:=I_Q-∫ Q dσ_Q.
 
		Then 
		
    U^σ_Q(ζ) +Q(ζ) ≥ F_Q  holds for q.e.  ζ∈.

		and 
		
    U^σ_Q(ζ)+Q(ζ) = F_Q  holds for q.e.  ζ∈ S_Q.
 
	

The measure σ_Q is called the equilibrium measure associated with Q. The constant F_Q is called the modified Robin constant for Q.	

Using the notion of equilibrium measure, we immediately obtain the following lemma. 
 For Lebesgue a.e. ζ∈, there exists a positive constant C>0 satisfying
	
    e^-β n ( Q(ζ)-2∫_log|ζ-z| dσ_Q(z)  ) /∫_ e^-β n ( Q(ζ)-2∫_log|ζ-z| dσ_Q(z)  )  dζ < C.



	Note that by Theorem <ref>(3), the Borel polar set has Lebesgue measure zero. Therefore by  Theorem <ref>(5),  we have 
	
    Q(ζ)-2∫_log|ζ-z| dσ_Q(z)=	U^σ_Q(ζ)+Q(ζ) ≥ F_Q,
   
	for a.e. ζ∈ and
	
    Q(ζ)-2∫_log|ζ-z| dσ_Q(z)=	U^σ_Q(ζ)+Q(ζ) = F_Q,
  
	for a.e. ζ∈ S_Q, which implies 
	
    e^-β n ( Q(ζ)-2∫_log|ζ-z| dσ_Q(z)  ) /∫_ e^-β n ( Q(ζ)-2∫_log|ζ-z| dσ_Q(z)  )  dζ   ≤ e^-β n F_Q /∫_S_Q e^-β n F_Q  dζ≤1/m(S_Q),

	where m denotes the Lebesgue measure in . Therefore, Theorem <ref>(4) concludes the lemma.


We denote by W_p the Wasserstein distance of order p.  
In particular, if p=1, by Kantorovich-Rubinstein dual representation, we have

    W_1(μ, ν)= sup_||f||_ Lip≤ 1∫ f(x) (μ-ν) (dx),    ||f||_ Lip := sup_x ≠ y|f(x)-f(y)|/|x-y|.
 
Let μ_n be the empirical measure of Coulomb gas ensemble, i.e., 
μ_n = 1/n∑_j=1^nδ_ζ_j. The following concentration inequality is due to Chafaï, Hardy and Maïda.  
 <cit.>
	There exists a constant a'>0 such that for any n ≥ 2, and r >0,
	
    _n^β( W_1(μ_n, σ_Q)≥ r ) ≤ e^-a' n^2 r^2  .



Using this concentration inequality, we prove following lemma. 
 
	There exists >0 such that for Lebesgue a.e. ζ∈,
	
    lim_n →∞_n^β( e^-β n Q(ζ)∏_j=1^n-1| ζ-ζ_j |^2β/∫_ e^-β n Q(ζ)∏_j=1^n-1| ζ-ζ_j |^2βdζ≤exp( n^1-/2) )= 1.



	For some small >0, set 
	
    f_n,ζ(z)   :=  log (n^1/2-2|z-ζ|)_+ - (1/2 -2) log n; 
       = log |z-ζ|,    if   |z-ζ| ≥ n^-1/2+2
     (-1/2+2)log n,    if   |z-ζ| < n^-1/2+2.

	Using the concentration inequality (<ref>) with the choice r = 1/2n^-1/2+, we obtain
	
    _n^β(W_1(μ_n, σ_Q) ≥1/2n^-1/2+) ≤ e^-2a n^1+2,

	for some positive constant a>0. Set μ'_n:= 1/n-1∑_j=1^n-1δ_ζ_j. Then we have
	
    _n^β(W_1(μ'_n, σ_Q) ≥ n^-1/2+) ≤ e^-a n^1+2,

	for large n. Indeed, we have W_1(μ_n,μ'_n) ≤1/n(n-1)∑_j=1^n-1 |ζ_j - ζ_n| → 0 as n →∞ in probability, which follows from the tightness property of ζ_j, see <cit.>. Thus, using f_n,ζ_ Lip = n^1/2-2 for every ζ∈, we obtain
	
    _n^β( sup_ζ∈| ∫ f_n,ζ(z) μ'_n(dz) - ∫ f_n,ζ(z) σ_Q(dz) |  ≥ n^-) ≤ e^-a n^1+2.

	Also recall that conditions (A1)-(i),(iii) deduce the boundedness of the density of σ_Q. Thus, using the definition of f_n,ζ we have 
	
    ∫( f_n,ζ(z)- log|z-ζ| ) σ_Q(dz) ≤∫_B(ζ,n^-1/2+2)  -log|z-ζ| σ_Q(dz) ≤ c   n^-1+5.

	Combining these estimates with the fact that f_n,ζ(z) ≥log|z-ζ|, we conclude that for large n,
	
    _n^β( sup_ζ∈∫log|z-ζ| μ'_n(dz) - ∫log|z-ζ| σ_Q(dz) ≥ 2n^-)  	
    ≤   _n^β( sup_ζ∈∫log|z-ζ| μ'_n(dz) - ∫ f_n,ζ(z) σ_Q(dz)   ≥  2n^- - c   n^-1+5) , 
    ≤   _n^β(sup_ζ∈∫ f_n,ζ(z) μ'_n(dz) - ∫ f_n,ζ(z) σ_Q(dz) ≥ n^-) ≤ e^-an^1+2.

	Then by Theorem <ref>(5) and the fact that 
	
    e^-β nQ(ζ)∏_j=1^n |ζ - ζ_j|^2β = exp[ -β n {Q(ζ) + ∫log1/|z-ζ|^2μ_n(dz) }],

	we have
	
    _n^β( sup_ζ∈ e^-β nQ(ζ)∏_j=1^n |ζ - ζ_j|^2β≥exp(-n β F_Q + β n^1-) ) ≤ e^-an^1+2.

	On the other hand, since  ∫_ f_n,ζ(z) - log|z-ζ|  σ_Q(dζ) ≤  c  n^-1+5, we have
	
    ∫_S_Q∫_ f_n,ζ(z) - log|z-ζ| μ'_n(dz) σ_Q(dζ) 
       = ∫_∫_S_Q f_n,ζ(z) - log|z-ζ|   σ_Q(dζ) μ'_n(dz),  
       ≤∫_ cn^-1+5μ'_n(dz) = c   n^-1+5.

	Set 
	
    A:={ζ∈ S_Q : ∫_| log|z-ζ| - f_n,ζ(z)| μ'_n(dz) ≥ n^-1+6}.

	Then by Chebyshev inequality and (<ref>), we have
	
    lim_n →∞σ_Q(A) ≤lim_n →∞c   n^-1+5/n^-1+6=0.

  Note that	we may assume that m(A) ≤ m(S_Q)/2 since σ_Q is absolutely continuous with respect to Lebesgue measure.  
	Therefore, we obtain that for large n,
	
    _n^β( ∫_ e^-β n Q(ζ)∏_j=1^n-1| ζ-ζ_j |^2βdζ≤∫_S_Q A e^-β n F_Q - β n ^1- - cβ n^5 dζ) 
    ≤   _n^β( ∫_S_Q  A e^-β n Q(ζ)∏_j=1^n-1| ζ-ζ_j |^2βdζ≤∫_S_Q  A e^-β n F_Q - β n ^1-- cβ n^5 dζ),
    	
    ≤   _n^β(  inf_ζ∈ e^-β n Q(ζ)∏_j=1^n-1| ζ-ζ_j |^2β≤ e^-β n F_Q - β n^1- -cβ n^5),
    	
    ≤   _n^β(sup_ζ∈|  ∫ f_n,ζ(z) μ'_n(dz) - ∫log|z-ζ|σ_Q(dz)| ≥ n^- + cn^-1+5), 
    ≤   _n^β(sup_ζ∈|  ∫ f_n,ζ(z) μ'_n(dz) - ∫ f_n,ζ(z) σ_Q(dz) | ≥ n^-)
    	≤ e^-an^1+2.
   
	This and (<ref>) proves the lemma.         
	




	Fix k ∈. For n ∈, let us denote by ζ_n, …, ζ_n the n-th Coulomb gas ensembles. Set z_n,i= ζ_i for 1 ≤ i ≤ n-k and Y_n,i=ζ_n-k+i for 1 ≤ i ≤ k. Recall that all we need to show is 

    lim_n →∞(|L_n^k(z)| < e^-n)= 0,     L^k_n(z):=1/k!P^(k)_n(z)/P_n(z),

where P_n(z):=(z-ζ_1)⋯(z-ζ_n).

For each 0 ≤ i ≤ k-1, let 𝒩_n,i be the subset of sample space, which satisfies 

    sup_ζ∈[ e^-β n Q(ζ)∏_j=1, j ≠ n-i^n| ζ-ζ_j |^2β/∫_ e^-β n Q(z)∏_j=1^n-1| z-ζ_j |^2βdz  ] > exp( n^1-ϵ/2)

and set 𝒩_n:= ∪_i=0^k-1𝒩_n,i. Then by Lemma <ref>, 

    _n^β(𝒩_n) ≤∑_i=0^k-1_n^β(𝒩_n,i) ≤ k e^-an^1+2,
 
which implies lim_n →∞_n^β(𝒩_n)=0.

In the case of 𝒩^c_n, we verify (<ref>) as a consequence of Lemma <ref>. Therefore all we need to check is (<ref>), (<ref>), and (<ref>) for Coulomb gas ensemble. First note that (<ref>) is obtained from the construction of 𝒩_n. To establish (<ref>), notice that
	
    log_+ |ζ| e^-β n Q(ζ)≤ e^-β n Q_0(ζ),

	where Q_0(ζ) = Q(ζ) - |ζ|. Then by the assumption (A1)-(ii), the partition function of the Coulomb gas with potential Q_0 is also finite, which implies (<ref>).
 Finally, (<ref>) follows from the well-known tightness of Coulomb  gas ensemble, see e.g., <cit.>.
Combining these, we obtain that 
	
    lim_n →∞(|L_n^k(z)| < e^-n ; 𝒩_n^c) = 0.

	Therefore we conclude
	
    lim_n →∞(|L_n^k(z)| < e^-n) ≤lim_n →∞ [(|L_n^k(z)| < e^-n ; 𝒩_n^c) + (𝒩_n)] = 0,

	which completes the proof.






§ QUESTIONS



	
  * It is expected that the Theorem <ref> hold for any probability measure μ. Does the proof of Theorem <ref> extend to the case when the limiting measure μ has atoms as well? 
	
  * In the spirit of <cit.>, one may ask if it is possible to show that most of the zeros have a critical point with in a distance of O(1/n). Establishing this would imply a natural pairing between zeros and critical points. Once the pairing is established, it will be of interest to study how the sum of pairwise distances (matching distance) behaves with n. This was studied in the case when all the zeros are real in <cit.>.
	
  * Extending the previous question, it is pertinent to ask about the density of the distances between zeros and critical points in the scale of 1/n. A particular case being the study of critical points of characteristic polynomial of Haar distributed unitary matrix. In this case one can notice that the law of zeros and hence for the critical points is rotationally invariant. The characteristic polynomial of CUE random matrix is believed to model Riemann zeta function and this problem is of interest in the study of critical points of Riemann zeta function. For more on this problem see <cit.> and references there in.  

	


Acknowledgments: The authors would like to thank the organizers of the Second ZiF Summer School on Randomness in Physics and Mathematics, held at Bielefeld in August 2016, where this work was initiated. 


























 

























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