ISSN: 889-3066 c 207 Universidad de Jaén Web site: jja.ujaen.es Jaen J. Approx. 9) 207), 24 Jaen Journal on Approximation Expected zeros of random orthogonal polynomials on the real line Igor E. Pritsker Abstract We study the expected number of zeros for random linear combinations of orthogonal polynomials with respect to measures supported on the real line. The counting measures of zeros for these random polynomials converge weakly to the corresponding equilibrium measures from potential theory. We quantify this convergence and obtain asymptotic results on the expected number of zeros located in various sets of the plane. Random coefficients may be dependent and need not have identical distributions in our results. Keywords: polynomials, random coefficients, expected number of zeros, uniform distribution, random orthogonal polynomials. MSC: Primary 30C5; Secondary 30B20, 60B0. Communicated by G. López Lagomasino Received August 26, 206 Accepted April 25, 207. Asymptotic equidistribution of zeros Zeros of polynomials of the form P n z) = n A kz k, where {A k } n are random coefficients, have been studied by Bloch and Pólya, Littlewood and Offord, Erdős and Offord, Kac, Rice, Hammersley, Shparo and Shur, Arnold, and many other authors. The early Research was partially supported by the National Security Agency grant H98230-5--0229) and by the American Institute of Mathematics.
2 Igor E. Pritsker history of the subject with numerous references is summarized in the books by Bharucha-Reid and Sambandham [5], and by Farahmand [3]. It is well known that, under mild conditions on the probability distribution of the coefficients, the majority of zeros of these polynomials accumulate near the unit circumference, being equidistributed in the angular sense. Let {Z k } n k= be the zeros of a polynomial P n of degree n, and define the zero counting measure τ n = n δ Zk. k= Equidistribution for the zeros of random polynomials is expressed via the weak convergence of τ n to the normalized arclength measure µ T on the unit circumference, where dµ T e it ) := dt/2π). Namely, w we have that τ n µt with probability abbreviated as a.s. or almost surely). More recent work on the global distribution of zeros of random polynomials include papers of Hughes and Nikeghbali [5], Ibragimov and Zeitouni [7], Ibragimov and Zaporozhets [6], Kabluchko and Zaporozhets [8, 9], etc. In particular, Ibragimov and Zaporozhets [6] proved that if the coefficients are independent and identically distributed, then the condition E[log + w A 0 ] < is necessary and sufficient for τ n µt almost surely. Here, E[X] denotes the expectation of a random variable X. A major direction in the study of zeros of random polynomials is related to the ensembles spanned by orthogonal polynomials. The classical case also falls in this category as monomials are orthogonal with respect to dt/2π) on the unit circumference, which may be used as explanation for clustering of zeros on T. These questions were considered by Shiffman and Zelditch [30] [32], Bloom [8] and [9], Bloom and Shiffman [], Bloom and Levenberg [0], Bayraktar [4], the author [26] and [25], and others. Many of the mentioned papers used potential theoretic approach to study the limiting zero distribution, including that of multivariate polynomials. Let A k, k = 0,, 2,..., be complex valued random variables. We state results on the asymptotic zero distribution under general assumptions that do not require independence or identical distribution of random coefficients. Let the distribution function of A k be defined by F k x) = P{ A k x}), x R. Suppose that there is N N, a decreasing function f : [a, ) [0, ], a >, and an increasing function g : [0, b] [0, ], 0 < b <, such that and a fx) x dx < and F kx) fx), x [a, ),.) b 0 gx) x dx < and F kx) gx), x [0, b],.2)
Expected zeros of random orthogonal polynomials on the real line 3 hold for all k N. If F x) is the distribution function of X, where X is a complex random variable, then E[log + X ] < a F x) x dx <, a 0, and b E[log F x) X ] < dx <, b > 0, 0 x see, e.g., Theorem 2.3 of Gut [4, p. 76]. Hence, when all random variables A k, k = 0,,..., are identically distributed, assumptions.).2) are equivalent to E[ log A 0 ] <. Define a sequence of orthonormal polynomials {p n } n=0 with respect to a positive Borel measure ν supported on E R and possessing finite moments, where p n z) = c n z n +... and c n > 0. We consider orthonormal polynomials with respect to a general measure with compact support, and the Freud polynomials orthonormal over E = R with respect to an exponential weight. The goal of this paper is the study of zeros for the ensembles of random orthogonal polynomials P n z) = A k p k z)..3) Our first ensemble is spanned by orthogonal polynomials with respect to a measure ν Reg with compact support E R that is regular in the sense of logarithmic potential theory. We use the notation Reg for the class of measures regular in the sense of Stahl, Totik and Ullman, which was thoroughly studied in [33]. In particular, the class Reg is characterized by the following asymptotic property for the leading coefficients of orthonormal polynomials p n : lim c/n n = /cape), where cape) is the logarithmic capacity of E, see Theorem 3..i) and Definition 3..2 in [33, pp. 60-6]. Since E is a regular set in our case, the above limit is equivalent by Theorem 3.2.3iv) of [33, p. 67] to the following: lim p n /n E =,.4) where p n E denotes the supremum norm of p n on E. We first show that the counting measures of zeros converge weakly to the equilibrium measure of E denoted by µ E, which is a positive unit Borel measure supported on E see [28]).
4 Igor E. Pritsker Theorem.. Suppose that the measure ν Reg defining the orthonormal polynomials {p k } has compact support E R that is regular in the sense of logarithmic potential theory. If the random coefficients {A k } satisfy.)-.2), then the zero counting measures of the random orthogonal polynomials.3) converge almost surely to µ E as n. The condition of regularity for ν is standard in the theory of general orthogonal polynomials. If E R is a finite union of compact intervals, and if dνx) = wx)dx with wx) > 0 a.e. on E, then ν Reg, see Corollary 4..2 in [33, p. 02] for more details. If E = [a, b] R then it is well known that dµ [a,b] x) = dx π b x)x a), x a, b), which is the Chebyshev arcsin) distribution. More generally, if E = N l= [a l, b l ] for N 2, where a < b < a 2 < b 2 <... < a N < b N are real numbers, then there exist y l b l, a l+ ), l =,..., N, such that the equilibrium measure of E is given by dµ E x) = π N N l= x y l dx N, x a l, b l ) l= x a l x b l l= see [33, p. 7]). Theorem. allows to find asymptotics for the expected number of zeros in various sets. Corollary.2. Suppose that all assumptions of Theorem. hold, and denote the number of zeros for the random orthogonal polynomials.3) in a set S C by N n S). If S C is a compact set satisfying µ E S) = 0, then lim n E [N ns)] = µ E S). A typical example of a set S is given by any rectangle of the form {x + iy C : a x b, c y d}, where a < b and c < 0 < d are real numbers. Note that Corollary.2 is not directly applicable to sets S E. In fact, Lubinsky, the author and Xie [2] proved for random orthogonal polynomials with real Gaussian coefficients that lim n E [N ns)] = µ E S), S E. 3
Expected zeros of random orthogonal polynomials on the real line 5 We now turn to the second family of orthogonal polynomials related to the Freud exponential) weights W x) = e c x λ, x R,.5) where c > 0 and λ > are constants. It is customary to define the orthogonality relation in this case by p n x)p m x)w 2 x) dx = δ mn. R We need to introduce a scaling parameter to study the asymptotic distribution of zeros for random Freud orthogonal polynomials. Define the constants γ λ = Γ 2 )Γ λ 2 ) 2Γ λ+ 2 ) and a n = γ /λ λ c /λ n /λ, and consider the contracted version of P n from.3): P ns) := P n a n s), n N..6) Consider the normalized zero counting measure τn = n n k= δ z k for the scaled polynomial Pns) of.6), where {z k } n k= are its zeros, and δ z denotes the unit point mass at z. The limiting measure for τn is described by the Ullman distribution ) λ y λ dµ λ s) = π s y2 s dy ds, s [, ]. 2 Note that µ λ is the weighted equilibrium measure for the weight w λ x) = e γ λ x λ on R see [24, 23, 29] for details). Theorem.3. If the random coefficients {A k } satisfy.)-.2), and {p k} are Freud orthogonal polynomials, then the normalized zero counting measures τn for the scaled polynomials Pns) of.6) converge weakly to µ λ with probability one. Zeros distribution for random orthogonal polynomials with varying weights was studied by Bloom and Levenberg see Section 6 of [0]). It might be possible to extend the above result to the class of superlogarithmic weights considered in [0]. As before, we find asymptotics for the expected number of zeros in sets that do not have significant overlap with the real line.
6 Igor E. Pritsker Corollary.4. Let the number of zeros for the polynomials.6) in a set S C be denoted by Nn S). If the assumptions of Theorem.3 are satisfied, and S C is a compact set such that µ λ S) = 0, then lim n E [N ns)] = µ λ S). The case S [, ] is not covered by Corollary.4. The expected number of real zeros of the random orthogonal polynomials.3) spanned by the Freud orthogonal polynomials with real Gaussian coefficients was studied by the author and Xie [27]. It was proved in [27] that lim n E [N ns)] = µ λ S), S, ). 3 2. Expected Number of Zeros: Union of Compact Intervals The main results of this paper provide quantitative estimates for the weak convergence of the zero counting measures of random orthogonal polynomials.3) to the corresponding equilibrium measure. In particular, we study the expected deviation of the normalized counting measure of zeros τ n from the equilibrium measure µ E on certain sets, which is often called the discrepancy between those measures. We use a discrepancy result of Blatt and Grothmann for deterministic polynomials see Theorem 5. in [, p. 86]), to obtain the following. Theorem 2.. Suppose that E R is a finite union of compact intervals, and that the polynomials {p k } n are orthonormal over E with respect to a weight w 0. Let {A k} n be complex random variables satisfying E[ A k t ] <, k = 0,..., n, for a fixed t 0, ], and E[log A n ] >. If I R is any interval and SI) := {z C : Re z I}, then we have for the zero counting measures τ n of the random polynomials.3) that [ n ) )] /2 E [ τ n µ E )SI)) ] 8 n t log E[ A k t ] + log max p k E + B E[log A n ], 0 k n 2.) where B is a constant that depends only on E and w. One can replace the strip SI) with other sets containing the interval I, e.g., with rectangles or sets bounded by the level curves of the Green function for C \ E see Theorem 5.3 in [, p. 90] for details). This gives essentially the same estimate as in 2.), but with constant 8 replaced by a different one.
Expected zeros of random orthogonal polynomials on the real line 7 In order to obtain more effective bounds, we consider the orthonormal polynomials p n that satisfy p n E = On p ) as n, 2.2) for a fixed positive constant p. This condition holds for many important classes of weights, as discussed below. Corollary 2.2. Under the assumptions of Theorem 2., suppose that for t 0, ] we have and If the orthogonal polynomials p n satisfy.4), then sup E[ A n t ] < 2.3) n 0 lim inf E[log A n ] >. 2.4) lim E [ τ n µ E )SI)) ] = 0. 2.5) Furthermore, if 2.2) is satisfied, then E [ τ n µ E )SI)) ] = O ) log n n as n. 2.6) Since τ n S) = N n S)/n for any set S C, where N n S) is the number of zeros for P n in S, 2.6) can be restated as E[N n SI))] = nµ E SI)) + O n log n). It is well known from the original work of Erdős and Turán [2] that the order log n/n of the right hand side in 2.6) is optimal in the deterministic case. Growth condition 2.2) holds true for polynomials orthogonal with respect to the generalized Jacobi weights of the form wx) = vx) J j= x x j αj, where vx) c > 0 a.e. on E. This follows from general results of Badkov [3] see, e.g., Theorem 3.2 on page 86). Other classes of weights that generate orthogonal polynomials satisfying 2.2) can be obtained from a Nikolskii type inequality for algebraic polynomials Q n : Q n E C n p E Q n x) 2 wx) dx) /2,
8 Igor E. Pritsker where the integral on the right is equal to for p n. In particular, the results of Mastroianni and Totik [22] can be applied here. We conclude this section by showing that the expected number of zeros for random orthogonal polynomials located at a positive distance from E is essentially negligible. Theorem 2.3. Suppose that the assumptions of Theorem 2. are satisfied. If S C \ E is any closed set, then E [τ n S)] n ) ) bn t log E[ A k t ] + log max p k E + B E[log A n ], 2.7) 0 k n where B depends only on E and w, and b > 0 depends only on E and S. In fact, b := min S g E where g E is the Green function for the complement of E with pole at infinity. Uniform assumptions give the following quantitative results for large n N. Corollary 2.4. If under the assumptions of Theorem 2.3 equations 2.2), 2.3) and 2.4) hold true, then ) log n E [τ n S)] = O as n 2.8) n and E [N n S)] = O log n) as n. 2.9) It is now easy to modify the vertical strip SI) of Theorem 2. into rather arbitrary set containing the interval I, by removing parts of this strip that are separated from I by a positive distance. Indeed, the expected number of zeros in those parts is of the order log n, which is absorbed by the right hand side terms in 2.) and 2.6). 3. Expected number of zeros: Freud weights on R We study the same questions on the expected number of zeros for random orthogonal polynomials as in the previous section, but the spanning polynomials p n are now orthogonal with respect to a Freud weight.5) defined on the whole real line. Theorem.3 shows that the normalized zero counting measures τ n for the scaled polynomials P ns) of.6) converge weakly to the Ullman distribution µ λ with probability one. Our next result gives an estimate for the rate of this convergence in terms of expected deviation of τ n from µ λ on certain test sets. Note that our assumptions on random coefficients are the same as in the previous section, and are different from those of Theorem.3.
Expected zeros of random orthogonal polynomials on the real line 9 Theorem 3.. Let {A k } n be complex random variables satisfying E[ A k t ] <, k = 0,..., n, for a fixed t 0, ], and E[log A n ] >. If I R is any interval and SI) := {z C : Re z I}, then the zero counting measures τn of the random polynomials.6) satisfy [ n ) /2 E [ τn µ λ )SI)) ] C n t log E[ A k t ] + C 2 log n E[log A n ])], 3.) where C, C 2 > 0 depend only on the constants c > 0 and λ > in the weight W of.5). Using uniform assumptions on random coefficients, we give a more concise bound. Corollary 3.2. If under the assumptions of Theorem 3. we have sup E[ A n t ] < 3.2) n 0 and then E [ τ n µ λ )SI)) ] = O lim inf E[log A n ] >, 3.3) ) log n n as n. 3.4) Letting N ns) be the number of zeros for P n in a set S C, we give an alternative form for 3.4): E[N n SI))] = nµ λsi)) + O n log n). It is also possible to establish analogs of Theorem 2.3 and Corollary 2.4 for closed sets S C \ [, ], but we do not develop this direction. 4. Proofs 4.. Proofs for Section We need to first develop results on the n-th root limits of random coefficients. These results are mostly known, see [25] [26] for example, but we include the proofs for convenience. Let {A n } n=0 be a sequence of complex valued random variables, and let F n be the distribution function of A n, n N. Our assumptions for A n below are as stated in.) and.2).
0 Igor E. Pritsker Lemma 4.. If there is N N and a decreasing function f : [a, ) [0, ], a >, such that a fx) x dx < and F nx) fx), x [a, ), holds for all n N, then lim sup A n /n a.s. 4.) Further, if there is N N and an increasing function g : [0, b] [0, ], 0 < b <, such that b 0 gx) x dx < and F nx) gx), x [0, b], holds for all n N, then lim inf A n /n a.s. 4.2) Hence, if both assumptions.) and.2) are satisfied for {A n } n=, then lim A n /n = a.s. 4.3) The almost sure limits of 4.)-4.3) follow from the first Borel-Cantelli lemma see, e.g., [4, p. 96]). Lemma Borel-Cantelli Lemma). Let {E n } n= be a sequence of arbitrary events. If n= PE n) < then PE n occurs infinitely often) = 0. Proof of Lemma 4.. We first prove 4.). For any fixed ε > 0, define events E n = { A n > e εn }, n N. Using the first assumption and letting m := maxn, ε log a ) + 2, we obtain PE n ) = P{ A n e εn })) = F n e εn )) fe εn ) n=m n=m m fe εt ) dt ε a fx) x n=m dx <. Hence PE n occurs infinitely often) = 0 by the first Borel-Cantelli lemma, so that the complementary event E c n must happen for all large n with probability. This means that A n /n e ε for all sufficiently large n N almost surely. We obtain that lim sup A n /n e ε a.s., n=m
Expected zeros of random orthogonal polynomials on the real line and 4.) follows because ε > 0 may be arbitrarily small. The proof of 4.2) proceeds in a similar way. For any given ε > 0, we set E n = { A n e εn }, n N. Using the second assumption and letting m := maxn, ε log b ) + 2, we have PE n ) = n=m F n e εn ) n=m ge εn ) n=m m ge εt ) dt ε b 0 gx) x dx <. Hence PE n i.o.) = 0, and A n /n > e ε holds for all sufficiently large n N almost surely. We obtain that lim inf A n /n e ε a.s., and 4.2) follows by letting ε 0. We also need the following simple consequence of 4.3). Lemma 4.2. If.) and.2) hold for the coefficients {A n } n=0 of random polynomials, then lim max A k /n = a.s. 4.4) 0 k n Proof. We deduce 4.4) from 4.3). Let ω be any elementary event such that lim A nω) /n =, which holds with probability one. We immediately obtain that lim inf max A kω) /n lim inf A nω) /n =. 0 k n On the other hand, elementary properties of limits imply that lim sup max A kω) /n. 0 k n Indeed, for any ε > 0 there exists n ε N such that A n ω) /n + ε for all n n ε by 4.3). Hence, ) max A kω) /n max max A k ω) /n, + ε + ε as n, 0 k n 0 k n ε and the result follows by letting ε 0.
2 Igor E. Pritsker We state a special case of Theorem 2. from Blatt, Saff and Simkani [7], which is used to prove Theorem.. Theorem BSS. Let E C be a compact set of positive capacity cape), with empty interior and connected complement. If a sequence of monic polynomials M n z) of degree n satisfy lim sup M n /n E cape), 4.5) then the zero counting measures τ n of M n z) converge weakly to µ E as n. Proof of Theorem.. Let the leading coefficient of the orthogonal polynomial p n z) be c n > 0. Then P n z) = A k p k z) = A n c n z n +..., n N. Theorem. is proved by applying Theorem BSS to the monic polynomials We first estimate the norm P n E M n z) := P nz) A n c n, n N. A k p k E n + ) max 0 k n A k max 0 k n p k E. Note that.4) implies by an elementary argument already used in the proof of Lemma 4.2) that Combining this fact with 4.4), we obtain that ) /n lim sup max p k E. 0 k n lim sup P n /n E a.s. 4.6) Since ν Reg, the leading coefficients c n of the orthonormal polynomials p n satisfy lim c/n n = /cape). 4.7)
Expected zeros of random orthogonal polynomials on the real line 3 Applying 4.2) and 4.7), we obtain that lim inf A nc n /n /cape) a.s. Using this together with 4.6), we conclude that 4.5) holds for our monic polynomials M n = P n /A n c n ) with probability one. Proof of Corollary.2. Theorem. implies that the counting measures τ n converge weakly to µ E with probability one. Since µ E S) = 0, we obtain that τ n S converges weakly to µ E S with probability one by [20, Theorem 0.5 ] and [6, Theorem 2.]. In particular, we have that the random variables τ n S) µ E S) a.s. Hence this convergence holds in L p sense by the Dominated Convergence Theorem, as τ n E) are uniformly bounded by see Chapter 5 of [4]). It follows that for any compact set E such that µ E S) = 0, and lim E[ τ ns) µ E S) ] = 0 E[τ n S) µ E S)] E[ τ n S) µ E S) ] 0 as n. But E[τ n S)] = E[N n S)]/n and E[µ E S)] = µ E S), which immediately gives the result. Proof of Theorem.3. Following [29], we call a sequence of monic polynomials {Q n } n=, with degq n) = n, asymptotically extremal with respect to the weight w λ if it satisfies lim wn λq n /n R = e F λ, where R is the supremum norm on R and F λ = log 2 + /λ is the modified Robin constant corresponding to w λ see [29, p. 240]). Theorem 4.2 of [29, p. 70] states that any sequence of such asymptotically extremal monic polynomials have their zeros distributed according to the measure µ λ. More precisely, the normalized zero counting measures of Q n converge weakly to µ λ. Denote the leading coefficient of the Freud orthonormal polynomial p n by c n. Recall the scaling paremeter a n = γ λ λ c λ n λ used to define the polynomials Pn in.6). We show that the monic polynomials Q nx) := P nx)/a n c n a n n), n N, are asymptotically extremal in the above sense with probability one, so that the result of Theorem.3 follows.
4 Igor E. Pritsker Hence, Using orthogonality, we obtain for the polynomials defined in.3) that and Lemma 4.2 implies that P n x) 2 W 2 x) dx = A k 2. /2 max A k P n x) 2 W 2 x) dx) n + ) max A k, 0 k n 0 k n /2n) ) /n lim P n x) 2 W 2 x) dx) = lim max A k = 0 k n with probability one. Applying the Nikolskii-type inequalities of Theorem 6. and Theorem 6.4 from [24], we obtain that the same holds for the supremum norm: lim wn λpn /n R = lim P nw /n R = with probability one. Recall that the leading coefficients of the Freud orthonormal polynomials p n satisfy lim n n /λ = 2c /λ γ /λ λ e /λ c/n by Theorem.2 of [29, p. 362]. We also use below that lim A n /n = with probability one by 4.3). It follows that lim wn λq n /n R = lim wn λpn /n R lim A nc n /n a n = lim = 2c /λ γ /λ λ e /λ c /λ γ /λ λ ) = e log 2+/λ) = e F λ. ) c /n n n /λ c /λ γ /λ λ Proof of Corollary.4. This result is proved in exactly the same way as Corollary.2, only replacing τ n with τn, and µ E with µ λ. The weak convergence of τn to µ λ with probability one is provided by Theorem.3.
Expected zeros of random orthogonal polynomials on the real line 5 4.2. Proofs for Section 2 We need the following consequence of Jensen s inequality. Lemma 4.3. If A k, k = 0,..., n, are complex random variables satisfying E[ A k t ] <, k = 0,..., n, for a fixed t 0, ], then [ ] E log A k n ) t log E[ A k t ]. 4.8) Proof. We first state an elementary inequality. If x i 0, i = 0,..., n, and n i=0 x i =, then x i ) t i=0 x i = for t 0, ]. Applying this inequality with x i = A i / n A k, we obtain that and E [ log i=0 n t A k ) A k t ] [ A k t n )] E log A k t. Jensen s inequality and linearity of expectation now give that [ ] [ E log A k n ] t log E A k t = n ) t log E[ A k t ]. Proof of Theorem 2.. Observe that the leading coefficient of P n is A n c n. Since E[log A n ] >, the probability that A n = 0 is zero, and P n is a polynomial of exact degree n with probability one. Theorem 5. in Chapter 2 of [, p. 86] gives the following estimate: τ n µ E )SI)) 8 n log P n E A n c n cape)) n. 4.9)
6 Igor E. Pritsker Using this estimate and Jensen s inequality, we obtain that [ ] E [ τ n µ E )SI)) ] 8 n E P n E log A n c n cape)) n 8 n E[log P n E ] logc n cape)) n ) E[log A n ]). It is clear that P n E A k p k E max 0 k n p k E A k. Hence, 4.8) yields E [log P n E ] E [ log ] A k + log max 0 k n p k E t log n ) E[ A k t ] + log max 0 k n p k E. The leading coefficient c n of the orthonormal polynomial p n provides the solution of the following extremal problem [33, p. 4]: { c n 2 = inf E } Q n x) 2 wx) dx : Q n is a monic polynomial of degree n. We use a monic polynomial Q n z) that satisfies Q n E CcapE)) n, where C > 0 depends only on E. Existence of such polynomial for a set E composed of finitely many smooth arcs and curves was first proved by Widom [34]. Hence we estimate that c n E /2 Q n x) 2 wx) dx) E ) /2 /2 wx) dx Q n E C wx) dx) cape)) n. E It follows that /2 c n cape)) n C wx) dx), where C depends only on E. Thus 2.) follows by combining the above estimates. E
Expected zeros of random orthogonal polynomials on the real line 7 Proof of Corollary 2.2. We estimate the right hand side of 2.). immediate observations that 2.3) implies while 2.4) implies If.4) is satisfied, then which implies that n ) tn log ) log n E[ A k t ] = O n n E[log A n ] O ) n For this purpose, we make two as n, as n. ) /n lim sup max p k E = lim sup p k E ) /n, 0 k n lim sup log max n p k E 0. 0 k n Hence, 2.5) follows from the above inequalities and 2.). On the other hand, if 2.2) is satisfied, then ) log n log max n p k E = O as n, 0 k n n and 2.6) follows in the same manner. Proof of Theorem 2.3. Let Q n z) = n k= z z k) be an arbitrary monic polynomial of degree n with the zero counting measure τ n = n n k= δ z k. Since µ E is a unit measure supported on E, we have that log Q n z) dµ E z) log Q n E. We need the well known representation of the Green function g E z) for the complement of E with logarithmic pole at infinity see [28, p. 07]): g E z) = log z t dµ E t) log cape).
8 Igor E. Pritsker Recall that g E z) 0, z C, and that g E z) is a positive harmonic function in C \ E. Using Fubini s theorem and the above identity, we obtain that n log Q n z) dµ E z) = = log z t dτ n t) dµ E z) = log z t dµ E z) dτ n t) g E t) dτ n t) + log cape) = n τ n S) min z S g Ez) + log cape). Denoting b := min S g E > 0, we arrive at the inequality g E z k ) + log cape) k= bτ n S) + log cape) n log Q n E. If we set Q n z) = P n z)/a n c n ) and take the expectation, then the estimate becomes E[τ n S)] bn E[log P n E ] logc n cape)) n ) E[log A n ]). The rest of this proof is identical to that of Theorem 2.. Proof of Corollary 2.4. We follow the same lines as in the proof of Corollary 2.2, but using 2.7) instead of 2.). Namely, we again obtain from 2.2). 2.3) and 2.4) that and log max n p k E = O 0 k n n ) tn log E[ A k t ] = O n E[log A n ] O ) log n n ) log n ) n n as n, as n, as n. Hence, 2.8) and 2.9) are immediate from 2.7).
Expected zeros of random orthogonal polynomials on the real line 9 4.3. Proofs for Section 3 Proof of Theorem 3.. Recall that the leading coefficient of Pn given by A nc n a n n does not vanish almost surely. Indeed, assumption E[log A n ] > gives the probability of A n = 0 is zero, and c n > 0. Hence, Pn has exactly n zeros with probability one denoted by {z k } n k=. We project the zeros of Pn onto the real line to construct the monic polynomial n M n z) := z ξ k ), ξ k = Re z k, k n. k= It is obvious that the zero counting measures ν n = n n k= ξ k for M n satisfy ν n SI)) = τn SI)). Note also that M n x) P n x) A n c n a n, x R. 4.0) We use the discrepancy estimate of Theorem. in Chapter 4 of [, p. 30] for the signed measure σ = µ λ ν n on a sufficiently large interval L R such that [, ] L and {ξ k } n k= L. In the notation of that result, we have that σ + = µ λ and σ = ν n are positive unit measures supported on L. We need to verify that the inequality µ λ I) C µ [,] I) holds for all intervals I L with a fixed constant C > 0, where dµ [,] = dx/π x 2 ) is the equilibrium measure of [, ] see.) of [, p. 30]). This inequality is satisfied because both µ λ and µ [,] are supported on [, ], the density of µ λ is uniformly bounded above for λ >, and the density of µ [,] is uniformly bounded below by a positive constant. Using Remark.2 of [, p. 3], we obtain that τ n µ λ )SI)) = µ λ ν n )SI)) D[σ] C ε, 4.) where C > 0, and ε is expressed through the logarithmic potential of σ denoted by U σ : ε = sup U σ z) = sup U µ λ z) U νn z)) = sup U µ λ z) + ) z C z C z C n log M nz). Since the function U µ λ z) + n log M nz) is subharmonic in C \ [, ], we obtain that ε = U µ λ x) + ) n log M nx). sup x [,] Recall that µ λ is the weighted equilibrium measure corresponding to the weight w λ x) = e γ λ x λ R see [24] and [29]). This implies by Theorems I..3 and IV.5. of [29] that U µ λ x) = F λ γ λ x λ, x [, ], on
20 Igor E. Pritsker where F λ = log 2 + /λ is the modified Robin constant corresponding to w λ see [29, pp. 27 and 240]). Hence, we obtain from the above and 4.0) that ε = sup F λ γ λ x λ + ) x [,] n log M nx) sup F λ γ λ x λ + x [,] n log P nx) ) A n c n a n = sup x [,] F λ + n log wn λ x) P nx) ) A n c n a n = F λ + n log wn λ P n [,] n log A n c n a n n ). Combining this estimate with 4.), we arrive at τ n µ λ)si)) C log 2 + /λ + n log wn λ P n [,] n log A n c n a n n ). Jensen s inequality now gives that ) E[ τn µ λ )SI)) ] C E[log wλ n n P n [,] ] E[log A n ] + log 2n e n/λ c n a n. 4.2) It follows from Theorem VII.2. of [29, p. 365] that c n π nγλ ) n+/2)/λ lim =. 2 n e n/λ c Since a n = γ λ λ c λ n λ, we obtain that 2 n e n/λ c n a n n ) n/λ = 2n ec = O n λ/2), c n nγ λ where the constant in the O-term depends only on c and λ. It remains to estimate the term E[log w n λ P n [,]] in 4.2). Using Theorem III.2. of [29, p. 53], we obtain that Hence, 4.8) gives w n λp n [,] = w n λp n R = P n W R E [ log w n λp n [,] ] E [ log A k p k W R max 0 k n p kw R ] A k + log max 0 k n p kw R A k.
Expected zeros of random orthogonal polynomials on the real line 2 n ) t log E[ A k t ] + log max 0 k n p kw R. Applying the Nikolskii-type inequality of Theorem VI.5.6 from [29, p. 342], we estimate p k W R O ) k λ 2λ p k x) 2 W 2 x) dx) /2 O ) n λ 2λ, 0 k n, where we used that the polynomials p k are orthonormal with respect to W on R. Thus 3.) follows from 4.2) and the above estimates. Proof of Corollary 3.2. Equation 3.4) follows from 3.) by essentially the same argument as in the proof of Corollary 2.2, where we deduce 2.6) from 2.). Acknowledgements Results of this paper were presented at the VII Jaen Conference on Approximation held in Úbeda, July 3-8, 206. The author would like to thank the organizers for the invitation and hospitality. References [] Andrievskii V. V. and H.-P. Blatt 2002) Discrepancy of Signed Measures and Polynomial Approximation, Springer Monographs in Mathematics, Springer-Verlag, New York. [2] Arnold L. 966) Über die Nullstellenverteilung zufälliger Polynome, Math. Z. 92, 2 8. [3] Badkov V. M. 992) Asymptotic and extremal properties of orthogonal polynomials corresponding to a weight with singularities, Tr. Mat. Inst. Steklova 98, 4 88 in Russian; English translation in Proc. Steklov Inst. Math. 98, 37 82 994)). [4] Bayraktar T. 206) Equidistribution of zeros of random holomorphic sections, Indiana Univ. Math. J. 655), 759 793.
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