THE ZEROS OF RADOM POLYOMIALS CLUSTER UIFORMLY EAR THE UIT CIRCLE C.P. HUGHES AD A. IKEGHBALI Abstract. Given a sequence of rom polynomials, we show that, under some very general conditions, the roots tend to cluster near the unit circle, their angles are uniformly distributed. In particular, we do not assume independence or equidistribution of the coefficients of the polynomial. We apply this result to various problems in both rom deterministic sequences of polynomials, including some problems in rom matrix theory.. Introduction We are interested in the asymptotics of the zeros of the rom polynomial P Z = a,k Z k as. We will denote the zeros as z,..., z rather than z,..., z for simplicity. Let { ν ρ := # z k : ρ z k } ρ denote the number of zeros of P Z lying in the annulus bounded by ρ, where 0 ρ, let ρ ν θ, φ := # {z k : θ arg z k < φ} denote the number of zeros of P Z whose argument lies between θ φ, where 0 θ < φ 2π. Let Ω, F, P be a probability space on which the array a,k 0 k defined. The aim of this paper is to show that under some very general conditions on the distribution of the coefficients a,k, we have that ν ρ = Date: 7 June 2004. 2000 Mathematics Subject Classification. 30C5. Key words phrases. Clustering of zeros, Rom polynomials. is
2 C.P. HUGHES AD A. IKEGHBALI ν θ, φ = φ θ 2π either almost surely or in the p th mean, according to the hypotheses we make. We say that the zeros cluster near the unit circle if remains true when ρ 0 as. In many examples, a natural rescaling turns out to be ρ = α/ so clustering requires α = o. Almost all of our results will follow from the following: Theorem. Let a k be a sequence of complex numbers which satisfy a 0 0 a 0. Denote the zeros of the polynomial P Z = a k Z k by z i for i from to, for 0 ρ let for 0 θ < φ < 2π let ν ρ := # {z k, ρ z k / ρ} ν θ, φ := # {z k, θ arg z k < φ} Then, for 0 α α ν 2 log a k α 2 log a 0 2 log a there exists a constant C such that ν θ, φ φ θ 2 2π C log a k 2 log a 0 2 log a Though the proof of this proposition is very simple, using only Jensen s formula a result of Erdős Turan [7, powerful results follow. ote that the same function F := log a k 2 log a 0 2 log a controls both the clustering of the zeros to the unit circle, the uniformity in the distribution of their arguments. ote further that this result holds for any a 0,..., a subject to a 0 a 0, thus has consequences for non-rom polynomials. It is clear that if there exists a function α = o such that F = oα a.s. 2
ZEROS OF RADOM POLYOMIALS 3 then the zeros of the rom polynomial P Z = a n Z n satisfy, ν n=0 α =, a.s. ν θ, φ = φ θ 2π, a.s. that is, the zeros cluster near the unit circle, their arguments are uniformly distributed. The bulk of this paper is concerned with finding conditions on the coefficients such that we may conclude that either E[F = o or that there exists a deterministic function 0 < α < such that F = oα a.s. For example, in Theorem 8 we show that there exists an α such that 2 holds if the a k satisfy the following three conditions: There exists an s > 0 such that for all k, µ k := E [ a k s <. Furthermore, sup k µ k /k =. For some 0 < δ there exists t > 0 a q > 0, such that for all [ E a t { a δ} = O q. This can be interpreted as a generalization of a theorem of Shmerling Hochberg [8 by removing the following requirements on the coefficients in the rom polynomial: they are independent; they have a finite second moment; they have density functions. Our results also enable us to deal with the general case of sequences of rom polynomials i.e. the coefficients of the polynomials are allowed to change with the degree. For example, consider the sequence of polynomials P Z = a,k Z k If there exists a positive function α = o such that [ α E log a,k 2 log a,0 2 log a, = 0 then by Theorem the zeros of this sequence of rom polynomials cluster uniformly around the unit circle. This more general case is dealt with in view of applications to characteristic polynomials of rom unitary matrices. We recover for example the result of clustering of the zeros of the derivative of the characteristic
4 C.P. HUGHES AD A. IKEGHBALI polynomial of a rom unitary matrix, as found in the work of Mezzadri, [6. We shall return to the study of rom matrix polynomials in a later paper. The structure of this paper is as follows: In section 2 we review some of the relevant history of zeros of rom polynomials, describe which prior results which can be obtained as corollaries of our work. In section 3 we describe the basic estimates we need, then in sections 4 5 we prove the main result, use it to deduce clustering of zeros in many examples. In Section 6 we study the rom empirical measure µ = δ zk, associated with the roots of rom polynomials. We show that they converge in mean or almost-surely weakly to the Haar measure on the unit circle i.e. the uniform measure on the unit circle. k= 2. Review of earlier work on rom polynomials Mark Kac [2 gave an explicit formula for the expectation of the number, ν B, of zeros of a n Z n n=0 in any Borel subset B of the real line, in the case where the variables a n n 0 are real independent stard Gaussian. His results were exped in various directions for example, to the non-gaussian case, but most of the work has focused on the real zeros see [8, [6 [7 for more details references. Almost fifty years later, L.A. Shepp J. Verbei [7 extended the results of Kac to the case where B is any Borel subset of the complex plane. They noticed that as the degree of the polynomials gets large, the zeros tend to cluster near the unit circle are approximately uniformly distributed around the circle. I. Ibragimov O. Zeitouni [0, using different techniques, have obtained similar results for i.i.d. coefficients in the domain of attraction of the stable law. They again observed the clustering of the zeros near the unit circle. However, this result about the clustering of the complex roots of rom polynomials has already been observed by Šparo Šur [9 in a general setting. They considered i.i.d. complex coefficients a n n 0 such that, P [a k = 0, E [ log + a k <, k = 0,,..., k = 0,,...,
ZEROS OF RADOM POLYOMIALS 5 where log + a k = max {0, log a k }. They proved that ν ρ = ν θ, φ = φ θ 2π where the convergence holds in probability. Arnold [ improved this result proved that the convergence holds in fact almost surely in the p th mean if the moduli of a k are equidistributed E [ log a k < for k = 0,,...,. Recently, Shmerling Hochberg [8 have shown that the condition on equidistribution can be dropped if a n n 0 is a sequence of independent variables which have continuous densities f n which are uniformly bounded in some neighborhood of the origin with finite means µ n stard deviations σ n that satisfy the condition { } n n max sup µn, sup σn =, n n P [a 0 = 0 = 0 Finally, let us mention that the distribution of roots of rom polynomials has also been investigated in physics, which among others, appear naturally in the context of quantum chaotic dynamics. Bogomolny et. al. [5 studied self-inversive polynomials, with a k = a k, a k k = 0,,..., 2 complex independent Gaussian variables with mean zero, they proved that not only do the zeros cluster near the unit circle, but a finite proportion of them lie on it. This case is very interesting since it shows that at least in some special cases, we can drop the independence equidistribution assumptions on the coefficients. Theorems 6 7 of this paper includes extends the above mentioned results on uniform clustering of zeros. 3. Basic estimates In the first part of this paper, we will apply Jensen s formula repeatedly, so we recall it here see [4, for example. Lemma 2. Let f be a holomorphic function in a neighborhood of the closed disc D r = {z C, z r}, such that f0 0. Let z i be the zeros of f in D r = {z C, z < r}, repeated according to their multiplicities, then 2π 2π 0 log f re iϕ dϕ = log f0 + z i log r z i We also use Jensen s inequality repeatedly, which states that if X is a positive rom variable, such that E[log X exists, then E[log X log E[X 3
6 C.P. HUGHES AD A. IKEGHBALI Before considering rom polynomials, we will first state some fundamental results about zeros of deterministic polynomials. For, let a k 0 k be a sequence of complex numbers satisfying a 0 a 0. From this sequence construct the polynomial P Z = a k Z k, denote its zeros by z i where i ranges from to. For 0 ρ, we are interested in estimates for ν ρ = # {z j, z j < ρ} { ν / ρ = # z j, z j > } ρ { ν ρ = # z j, ρ z j } ρ which counts the number of zeros of the polynomial P Z which lie respectively inside the open disc of radius ρ, outside the closed disc of radius / ρ, inside the closed annulus bounded by circles of radius ρ / ρ. Lemma 3. For, let a k 0 k be an sequence of complex numbers which satisfy a 0 a 0. Then, for 0 < ρ < ν ρ log a k log a 0, 4 ρ n ν / ρ log a k log a 5 ρ ν ρ 2 log a k ρ 2 log a 0 2 log a Proof. An application of Jensen s formula, 3, with r = yields 2π 2π 0 log P e iϕ dϕ log P 0 = z i < log z i where the sum on the right h side is on zeros lying inside the open unit disk. We have the following minorization for this sum: z i < log z i z i < ρ log z i ρ ν ρ 6
ZEROS OF RADOM POLYOMIALS 7 since if 0 ρ, then for all z i ρ, log/ z i ρ, by definition there are ν ρ such terms in the sum. We also have the following trivial upper bound max P e iϕ a k, ϕ [0,2π so ρ ν ρ 2π log P e iϕ dϕ log a 0 2π 0 log a k log a 0 which gives equation 4. To estimate the number of zeros lying outside the closed disc of radius ρ, note that if z 0 is a zero of the polynomial P Z = a kz k, then /z 0 is a zero of the polynomial Q Z := Z P Z = a +a Z +...+a 0 Z. Therefore, the number of zeros of P Z outside the closed disc of radius / ρ equals the number of zeros of Q Z inside the open disc of radius ρ. Therefore, from 4 we get ν / ρ log a k log a ρ which gives equation 5. Since ν ρ = ν ρ + ν / ρ we immediately get 6. To deal with the asymptotic distribution of the arguments of the zeros of rom polynomials that is, to show the angles are uniformly distributed we use a result of Erdős Turan [7: Lemma 4 Erdős-Turan. Let a k 0 k be a sequence of complex numbers such that a 0 a 0. For 0 θ < φ 2π, let ν θ, φ denote the number of zeros of P Z = a kz k which belong to the sector θ arg z < φ. Then ν θ, φ φ θ [ 2 2π C log a k 2 log a 0 2 log a for some constant C Remark. By considering the example P Z = z, we observe that C > / log 2. Combining Lemmas 3 4 yields the following proposition:
8 C.P. HUGHES AD A. IKEGHBALI Proposition 5. Let a k 0 k be a sequence of complex numbers which satisfy a 0 a 0. Denote the zeros of the polynomial P Z = a k Z k by z i for i from to, for 0 ρ let ν ρ := # {z i for 0 θ < φ < 2π let : ρ z i / ρ} ν θ, φ := # {z i : θ arg z i < φ} Then, for 0 α α ν 2 log a k α 2 log a 0 2 log a 7 there exists a constant C such that ν θ, φ φ θ 2 2π C log a k 2 log a 0 2 log a Remark. ote again that the same function log a k 2 log a 0 2 log a controls both the clustering of the zeros near the unit circle, the uniform distribution of the arguments of the zeros. Remark. ote that for any complex coefficients a k, log a k 2 log a 0 2 log a log 2 Therefore, this method cannot detect when all the zeros are on the unit circle. Remark. ote that if a k λa k for some λ 0, then the zeros of P Z are unchanged, log λa k 2 log λa 0 2 log λa = log a k 2 log a 0 2 log a so, in some sense, this is a natural function to control the location of the zeros. 8
ZEROS OF RADOM POLYOMIALS 9 We are interested in the zeros of sequences of rom polynomials. Let Ω, F, P be a probability space on which the array of rom variables, a,k, is defined. From this sequence we construct the rom polynomial 0 k P Z = a,k Z k. We require no independence restriction on our rom variables. We only assume that P [a,0 = 0 = 0 9 P [a, = 0 = 0, 0 for all. We recap the various types of convergence which we will see in this project: we say that X converges in probability to X if for all ɛ > 0, P{ X X > ɛ} 0 as ; we say that X converges in the p th mean to X if E [ X X p 0 as ; we say that X converges almost surely to X if for all ω Ω \ E where E, called the exceptional set, is a measure zero subset of the measurable sets Ω, X ω = Xω. The fact that almost sure convergence for bounded variables implies the convergence in the p th mean is a classical result in probability theory see, for example, [. The fact that convergence in the mean square implies convergence in probability follows from Chebyshev s inequality. 4. Uniform clustering results for roots of rom polynomials ow we give several results for the uniform clustering of the zeros of rom polynomials. Theorem 6 Main theorem. For, let a,k 0 k be an array of rom complex numbers such that P [a,0 = 0 = 0 P [a, = 0 = 0 for all. Denote the zeros of the polynomial P Z = a,k Z k by z i, for 0 ρ, let ν ρ := # {z i for 0 θ < φ < 2π, let Let : ρ z i / ρ} ν θ, φ := # {z i : θ arg z i < φ} F := log a,k 2 log a,0 2 log a,
0 C.P. HUGHES AD A. IKEGHBALI If E [F = o as 2 then there exists a positive function α satisfying α = o such that [ α E ν = [ E ν θ, φ = φ θ 2π In fact the convergence also holds in probability in the p th mean, for all positive p. Furthermore, if there exists a deterministic positive function α satisfying α for all, such that then F = oα almost surely 3 ν α =, a.s. ν θ, φ = φ θ 2π, Remark. It is clear that the only way for a sequence of polynomials not to have zeros which cluster uniformly to the unit circle is if there exists a constant c > 0 such that E [F > c for an infinite number of. Proof. The convergence in mean for ν α/ is a consequence of 7. We have [ α E ν 2 α E [F Therefore we see that the result follows for any positive function α satisfying α for all such that E [F /α 0, such a function exists by assumption 2. a.s. Similarly from 8 2 we have that [ E ν θ, φ φ θ 2 C 2π E [F = o ote that the mean square convergence implies convergence in the mean, as in the theorem, also convergence in probability. ote further, that since the rom variables are uniformly bounded 0 ν θ, φ, mean convergence implies convergence in the p th mean for all positive p. In the same way, the almost sure convergence of ν α/ ν θ, φ follows immediately from 7 8, using 3.
ZEROS OF RADOM POLYOMIALS We shall now give some examples for which the hypotheses of Theorem 6 are satisfied. Corollary 6.. Let a,k be an array of rom complex numbers which satisfy 9 0. Assume that E [log a,0 = o, E [log a, = o, that there exists a fixed s > 0 a sequence ε tending to zero such that sup E [ a,k s expε 0 k Then, there exists an α = o such that F, defined in, satisfies E [F = oα, so [ α E ν = [ E ν θ, φ φ θ 2π = 0 Proof. It is a consequence of Theorem 6 the following chain of concavity inequalities: [ [ E log a,k = s s E log a,k [log s E a,k s s log E [ a,k s s log + expε = s log + + ε = o since we assume ε 0 as. Therefore F, defined in, satisfies F = o, the result follows from Theorem 6. Remark. The Corollary shows that under some very general conditions just some conditions on the size of the expected values of the modulus of the coefficients, without assuming any independence or equidistribution condition, the zeros of rom polynomials tend to cluster uniformly near the unit circle. We can also remark that we do not assume that our coefficients must have density functions: they can be discrete-valued rom variables. Example. Let a,k be a rom variables distributed according to the Cauchy distribution with parameter k +. The first moment does not
2 C.P. HUGHES AD A. IKEGHBALI exist but some fractional moments do, in particular we have for 0 s < Moreover, E [ a,k s = k + π x s x 2 + 2 k + 2 dx = π s k + s Γ 2 + s 2 Γ 2 s 2 E [log a,k = logk + Hence we can apply Corollary 6. deduce that the zeros of the sequence of rom polynomials with coefficients a,k where a,k are chosen 0 k from the Cauchy distribution with parameter k + cluster uniformly around the unit circle. Example. We can also interpret this result for sequences of deterministic polynomials, since then E [ a,k = a,k. For example, for every sequence of polynomials with nonzero bounded integer coefficients, we have for all ρ 0,, ν ρ = similarly ν θ, φ = φ θ 2π. Indeed, one can take ρ = α/ for any sequence α such that log /α 0. Corollary 6.2. Let ε be a sequence of positive real numbers, which satisfies ε = 0. Let a,k be an array of complex rom variables such that for each, exp ε a,k exp ε for all k. Then there exists a deterministic positive function α = o such that α ν =, a.s. ν θ, φ φ θ 2π, a.s. The convergence also holds in the p th mean for all positive p. Proof. With the hypotheses of the corollary, we have that F := log a,k 2 log a,0 2 log a, log + expε logexp ε 2ε + log + so for any positive function α satisfying α 2ε + log = oα for example, α = ε + log 2, the result follows from the second half of Theorem 6.
ZEROS OF RADOM POLYOMIALS 3 Example. Let a,k, for fixed, 0 k, be discrete rom variables taking values in {±,..., ±}, not necessarily having the same distribution; then ν log +γ =, a.s., γ > 0 ν θ, φ φ θ 2π, a.s. As a special case, we have the well known rom polynomials of the form µ kz k, with µ k = ±, with probabilities p p. Moreover, we have from the Markov inequality, the following rate for the convergence in probability: where ε > 0. [ P [ P ν α ν θ, φ φ θ 2π > ε > ε C log ε α C log ε 2 4.. Self inversive polynomials. The Theorem 6 also gives us an interesting result for self-inversive polynomials. These polynomials are of interest in physics see [5 in rom matrix theory characteristic polynomials of rom unitary matrices. A polynomial P Z = a kz k is said to be self-inversive if a P Z = a 0 Z P /Z where z denotes the complex conjugate of z, P Z = P Z. This implies a k = a 0 a a k for all k. One can see that the zeros of self-inversive polynomials lie either on the unit circle or are symmetric with respect to it, that is, if z is a zero, so is /z. So, with the notations of Theorem 3, we just have to check that ν α tends to zero. Corollary 6.3. Let P Z = a kz k be a sequence of rom = self inversive polynomials satisfying P [a,0 = 0 = 0 for all E [log a,0 = o There exists a fixed s > 0 a positive sequence ε tending to zero as such that for all, E [ a,k s expε for all k
4 C.P. HUGHES AD A. IKEGHBALI then there exists a function α such that [ α E ν [ E ν θ, φ φ θ 2π =, = 0 In fact the convergence holds in the p th mean for any positive p. Proof. It is a consequence of Corollary 6.. Remark. Usually, one is interested in the case where a,0 = ; in this case there is only one condition to check: that for all, E [ a,k s expε for all k. Remark. We can also prove results about almost sure convergence as in the general case. 4.2. The derivative of the characteristic polynomial. The characteristic polynomial of a rom unitary matrix was introduced by Keating Snaith [3 as a model to underst statistical properties of the Riemann zeta function, ζs. We can apply the methods developed in this paper to study the location of the zeros of the derivative of the characteristic polynomial, first considered by Mezzadri [6 in order to model the horizontal distribution of the zeros of ζ s. Having a good understing of the location of the zeros of ζ s is important, because if there are no zeros to the left of the vertical line Res = /2, then the Riemann Hypothesis would be true. Denote the characteristic polynomial of an unitary matrix M by Λ M Z = det M ZI = k Sc k MZ k, where Sc j denotes the j th secular coefficient of the matrix M. Since all the zeros of Λ M Z lie on the unit circle, it follows that Λ M Z is self-inversive. The derivative is given by Λ M Z = k+ k + Sc k MZ k. We will use the following fact about secular coefficients averaged over Haar measure, due to Haake et. al. [9: { [ if j = k E Sc j MSc k M = 0 otherwise
so E [ ZEROS OF RADOM POLYOMIALS 5 k + 2 Sc k M 2 = + 2 + 6 Furthermore, Sc M = det MTr M, so E [log Sc M = E [log Tr U 2 log E [ Tr U 2 = 2 log 2 Therefore, Theorem 6. allows us to deduce that if α tends to infinity faster than log, then [ α E ν = By completely different methods, which are special to Λ M Z, Mezzadri [6 has previously shown that the zeros cluster in fact his results give an asymptotic expansion for the rate of clustering. 5. Classical Rom Polynomials Let us now consider the special, but very important, case of the classical rom polynomials as mentioned in the first section, that is P Z = a k Z k 4 These polynomials have been extensively studied see, for example, [2 or [8 for a complete account. The uniform clustering of the zeros have often been noticed in some special cases of i.i.d. coefficients, as in [0, [2, [7 for example but these papers are concerned with the density distribution of the zeros as is mentioned in section 2, it has been proved in more general cases by Arnold [ in the case of equidistributed coefficients, by Shmerling Hochberg [8 in the case of independent non equidistributed coefficients. We shall now see that we can recover improve the results in [ [8. The results of the previous section take a simpler form in the special case of rom polynomials of the form 4. The conditions 9 0 become P [a = 0 = 0, for all 0 5 We will restate Theorem 6 for this classical case. Theorem 7. Let a k k 0 be a sequence of complex rom variables which satisfy 5. Let F := log a k 2 log a 0 2 log a
6 C.P. HUGHES AD A. IKEGHBALI If E [F = o as then there exists a positive function α satisfying α = o such that [ α E ν = [ E ν θ, φ = φ θ 2π In particular, the convergence also holds in probability in the p th mean, for all positive p, since 0 ν α/ 0 ν θ, φ. Furthermore, if there exists a deterministic positive function α satisfying α for all, such that then F = oα ν almost surely α =, a.s. ν θ, φ = φ θ 2π, Remark. Again, we can observe that the result holds for the special case ν ρ, for fixed ρ 0,. One simply takes α = ρ. Theorem 7 shows that under some very general conditions assuming neither independence nor equidistribution we have a uniform clustering of the zeros of rom polynomials near the unit circle. Example. We shall now present two examples to show that our results are not completely sharp. Denote by ν r the number of zeros in the disc of radius r centered at 0. Using classical results about rom polynomials with coefficients a n which are i.i.d. stard Gaussian [8, [7, it can be shown that as [ E ν α 2α 2α a.s. if α exp2α if α α 0 /2 if α 0 [ If the zeros are to cluster, then we must have E ν α 0. Hence in this case we must have α. However, there exists a constant c > 0 such that E[F > c log, we can only deduce clustering from our results when α/ log. This is not surprising since our results are presented in great generality, if one knows specific information about the distribution of the a k it is plausible that specialized techniques would give more information about clustering.
ZEROS OF RADOM POLYOMIALS 7 Our second example concerns polynomials which have all their roots on the unit circle, for example Z. Since for any polynomial, F log 2, our results can only deduce clustering when α, despite the fact that in this case it holds true for any α 0. We will now show some cases where Theorem 7 allows us to deduce almost sure convergence of the zeros to the unit circle. Theorem 8. Let a n n 0 be a sequence of complex rom variables. Assume that there exists some s 0, such that k µ k := E [ a k s < for some 0 < δ there exists t > 0, such that for all [ ξ := E a t { a δ} = O q 6 for some q > 0. Assume further that: sup µ k /k = k or, equivalently, there exists a sequence ε tending to zero such that µ k = expε. Then for any deterministic positive sequence α satisfying α = o α ε +log, α ν =, a.s. ν θ, φ = φ θ 2π, a.s. In fact the convergence also holds in the p th mean for every positive p. Proof. ote that 6 implies P{ a k = 0} = 0. Therefore, from Theorem 7 it is sufficient to prove that for the choice of α = o given in the theorem, α log a k 2 log a 0 2 log a 0 a.s.
8 C.P. HUGHES AD A. IKEGHBALI For 0 < δ we have log 2 log a k 2 log a 0 2 log a s log + a k s + 2t log a 0 { a0 δ} + log 2t a { a δ} so since α, it is sufficient to show that α log + a k s = 0 a.s + log δ log α a t { a δ} = 0 a.s. We are first going to prove that α log + a k a.s. for our sequence α. = 0, Consider first the case when µ k is finite. By the monotone convergence theorem, the sum a k s converges almost surely as to an integrable rom variable X. Therefore, since α tends to infinity as, we see that α s log + a k s = 0 a.s. We can thus assume that µ k =. Given ε > 0, take β > 0 such that log + β ε/3. As sup k µ k k =, µk < for all k, there exists a constant C = Cβ such that for all k we have µ k C + β k. Hence, for sufficiently large, 0 log µ k log C + + log + β log β Thus, 0 + log µ k There exists such that for, log C + ε/3 log β + ε/3 log C + log + β log β + +
ZEROS OF RADOM POLYOMIALS 9 Hence, for all ε > 0, we found 0 = max, k 0, such that for all 0 we have + log µ k ε, which implies log µ k = o. We can thus write for 0: log µ k = ε with ε 0 ε. Since + 2 /k + 2 for all 0 k, we have log + a k s ow, as we have log + + 2 a k s exp ε exp ε k + 2 a k s exp ε 2 log + + ε + log + k + 2 µ k = exp ε, [ ak s exp ε E k + 2 k + 2 < We deduce from the monotone convergence theorem that a k s exp ε k + 2 converges almost surely to an integrable rom variable. α to be any positive function such that we have α ε + log α log + a k s = 0, a.s. ow, let us show that for the same sequence α, we have log α a t { a δ} = 0, a.s. Hence, taking
20 C.P. HUGHES AD A. IKEGHBALI From 6 we have 0 log a t { a δ} log + q + 2 log + + log a t { a δ} + + q+2 a t { a δ} From the Markov inequality, we have, for any ε > 0: [ P + q+2 a t { a δ} > ε ξ ε + q+2 As ξ = O q, for large enough, [ P + q+2 a t { a δ} > ε ε C + 2 for some positive constant C. Hence by the Borel-Cantelli lemma, + q+2 a t { a δ} 0 a.s. We can conclude that if α goes to infinity faster than log which our choice of α does, then log α the theorem follows. a t { a δ} = 0, a.s. Corollary 8.. Let a n n 0 be a sequence of complex rom variables such that the moduli a n are from p different probability distributions on the positive real line, say F j dx j p. Assume that there exists some s > 0 such that 0 x s F j dx < that there exists some 0 < δ such that there exists some t > 0 such that δ 0 x t F j dx < for any δ 0,. Then for any deterministic positive sequence α = o such that α/ log, α ν =, a.s. ν θ, φ = φ θ 2π, a.s. In fact the convergence also holds in the p th mean for every positive p. Proof. This is immediate from Theorem 8.
ZEROS OF RADOM POLYOMIALS 2 Corollary 8.2. Let a n n 0 be a sequence of complex rom variables such that the moduli a n have densities which are uniformly bounded in a neighborhood of the origin. Assume that there exists some s 0, such that, µ E [ a s < sup µ k k = k Then there exists a deterministic sequence α = o such that α ν =, a.s. ν θ, φ = φ θ 2π, a.s. In fact the convergence also holds in the p th mean for every positive p. Proof. It suffices to notice that in this special case, sup ξ C for some positive constant C. Example. Let P Z = a kz k, with a k being distributed on R + with Cauchy distribution with parameter k σ, σ > 0. This distribution has density 2 πk σ x 2 + k 2σ on the positive real line. The conditions of Theorem 8 are satisfied since [ [ µ k := E C k ξ σ := E Ck σ. Therefore, if a /2 k a /2 { a } α = o is such that α/ log, then α ν =, a.s. ν θ, φ = φ θ 2π, a.s. Again, the convergence also holds in the p th mean for every positive p. We can still weaken the hypotheses still have mean convergence. Proposition 9. Let a n n 0 be a sequence of complex rom variables. Assume that there exists some s 0, such that some t > 0, such that, µ E [ a s < sup µ k k = k, ξ E [ a t { a δ} <
22 C.P. HUGHES AD A. IKEGHBALI for any δ 0,, Then: [ E ν log + ξ = o α p = 0, p > 0 for some sequence α = o, 0 < α < [ E ν θ, φ φ θ p 2π = 0, p > 0 Proof. We first go through the same arguments as previously for the mean convergence then conclude to the p th mean convergence because of the boundedness of ν ν θ, φ. α Again, as in the previous section, we can specialize our results to the special case of deterministic coefficients. Proposition 0. Let ε n be a sequence of positive real numbers tending to zero, let a n n 0 be a sequence of complex numbers such that for all n exp ε n n a n exp +ε n n Then there exists a positive function α = o such that zeros of the polynomial a kz k satisfy α ν = ν θ, φ = φ θ 2π, 6. Convergence of the empirical measure In this section, we study the convergence of the rom empirical measure associated with the zeros of a rom polynomial. We use elementary results about convergence of probability measures that can be found in textbooks such as [5, [3. Let P Z = a,kz k be a sequence of rom polynomials, = such that P{a,0 = 0} = 0 P{a, = 0} = 0. Let z k k denote the zeros of P Z. Let µ k= δ zk
ZEROS OF RADOM POLYOMIALS 23 denote the empirical rom probability measure associated with the zeros on C = C\ {0}. For every continuous bounded function, f dµ f, µ = f z k Recall that a sequence of probability measures λ n is said to converge weakly to a probability measure λ if for all bounded continuous functions f, f dλ f dλ or with our notations: f, λ f, λ 7 As C, endowed with the metric d z, z 2 z z 2 + z, z 2 is a locally compact polish space, we can in fact take the space C K C of continuous functions with compact support as space of test functions in 7. Following Bilu [4 we call a function f : C C stard, if k= f re iϕ = g r exp ipϕ where g : R + C is continuous compactly supported, p Z. Lemma. The linear space, generated by the stard functions, is dense in the space of all compactly supported functions C C with the supnorm. Proof. See [4. Corollary.. Let λ n be a sequence of probability measures on C, λ one more probability measure on C. Assume that f, λ f, λ, for any stard function. Then λ n converges weakly to λ. Proposition 2. Let P Z = a kz k be a sequence of rom = polynomials. If α ν =, a.s. ν θ, φ φ θ 2π a.s. then the sequence of rom measures µ converges almost surely weakly to the Haar measure on the unit circle, that is to say for all bounded continuous functions f : C C, we have: f, µ 2π 2π 0 fe iϕ dϕ, a.s
24 C.P. HUGHES AD A. IKEGHBALI 2 If [ E ν α = [ E ν θ, φ φ θ 2π = 0 then the sequence of measures µ converges in mean weakly to the Haar measure on the unit circle, that is to say for all continuous functions f : C C, we have: E [ f, µ 2π 2π 0 fe iϕ dϕ = 0 In fact the convergence holds in the p th mean, for all positive p. Proof. By corollary., we need only prove this result for stard functions. We will first prove part, the almost sure convergence case. Let fz be a stard function, that is f re iϕ = g r e ipϕ where g : R + C p Z. We must distinguish between two cases when p = 0 when p 0. When p = 0, we have to prove that f, µ g almost surely. As g is continuous, ε > 0, there exists δ > 0, such that for all r [ δ, / δ, g r g < ε. Denoting D = [ δ, / δ g = sup r>0 g r, we then have f, µ g = g z k g + g z k g z k D z k / D ν δ ε + 2 g ν δ The result then follows from the assumption that ν δ =, a.s. ow consider the case when p 0. As 2π g exp ipϕ dϕ = 0 we must show f, µ = 0 a.s. For this, we notice that: f z k = g z k exp ipϑ k k= k= g exp ipϑ k + k= g z k g k=
ZEROS OF RADOM POLYOMIALS 25 where ϑ k is the argument of z k in [0, 2π. As ν θ, φ φ θ 2π a.s., we can apply Weyl s theorem for uniformly distributed sequence of real numbers to deduce that exp ipϑ k = 0, a.s. k= We have already shown that k= g z k g = 0, a.s., this completes the proof of part. Part 2 of the theorem concerns the case of mean convergence. As before, we let f re iϕ = g r exp ipϕ be a stard function, again we must distinguish between p = 0 p 0. The proof in the case p = 0 does not change. Indeed, we still have f, µ g ν δ ε + 2 ν δ g leading to [ [ E [ f, µ g E ν δ ε + 2 E ν δ g 0 as For the case p 0, we still have g f z k exp ipϑ k + Hence [ E k= k= g z k g k= f z k [ [ g E exp ipϑ k + E g z k g k= k= k= Again, the case p = 0 shows that [ E g z k g = 0 k= so to complete the proof, we just have to show that [ E exp ipϑ k = 0 k= which follows from the following lemma.
26 C.P. HUGHES AD A. IKEGHBALI Lemma 3. If [ E ν θ, φ φ θ 2π = 0 if fz = garg z where g is a continuous function defined on the torus R/2πZ, we have [ E f, µ 2π fe iϕ dϕ 2π = 0 8 0 Proof. We can assume that f is real valued; otherwise we would consider the real imaginary parts. In the special case when fz = [θ,φ arg z, [ 8 is exactly our assumption: E ν θ, φ φ θ 2π = 0. It is easy to see that 8 holds for finite linear combination of such functions, hence for step functions. ow, if g is continuous, for any ε > 0, there exist two step functions g g 2 such that g g g 2, 2π g 2 ϕ g ϕ dϕ ε 2π 0 For simplicity, let g := 2π 2π 0 g ϕ dϕ. Letting fz = garg z, f z = g arg z f 2 z = g 2 arg z we then have Hence: f, µ g g g f, µ g f 2, µ g 2 g g 2 f, µ g f 2, µ g 2 + f, µ g + g g + g 2 g E f, µ g E f 2, µ g 2 + E f, µ g + 2ε The lemma follows from the fact that E [ f j, µ g j = 0, for j =, 2 by the assumption of the lemma. Corollary 3.. In each case of convergence of the proposition, we have in fact: [ E f, µ 2π fe iϕ p dϕ 2π = 0 0 Acknowledgments This work was carried out at the Rom Matrix Approaches in umber Theory programme held at the Isaac ewton Institute, where the first author was supported by EPSRC grant 0976, the second author was partially supported by a SF Focussed Research Group grant 0244660. We wish to thank David Farmer, Steve Gonek Marc Yor for useful conversations.
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