Large Deviations, Linear Statistics, and Scaling Limits for Mahler Ensemble of Complex Random Polynomials

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1 Large Deviations, Linear Statistics, and Scaling Limits for Mahler Ensemble of Complex Random Polynomials Maxim L. Yattselev joint work with Christopher D. Sinclair International Conference on Approximation Theory and Applications City University of Hong Kong May 23rd, 2013

2 Mahler Measure The Mahler measure of a polynomial p(z) = a (z α n ) is given by { M(p) := exp log p(τ) dτ }, T 2π where T := { z = 1 }. It follows from Jensen s formula that M(p) = a max { 1, α n } = a exp { log + α n }.

3 How Many Polynomials Have at Most a Certain Mahler Measure? Associate to each element v C N+1 a polynomial p v. The following quantity is of number theoretic interest: Clearly, this quantity is equal to # { v Z[i] N+1 : M(p v ) T }. #Z[i] N+1 { v C N+1 : M(p v ) T }. Chern & Vaaler 1 have shown that this quantity is bounded by T 2N+2 vol { v C N+1 : M(p v ) 1 }. 1 The distribution of values of Mahler s measure, J. Reine Angew. Math., 540:1 47, 2001

4 Volume of the Star Body of Mahler Measure at Most 1 They further computed that where vol { v C N+1 : M(p v ) 1 } = H N (s) := CN M 2s( P u ) da N = πn N! π N + 1 H N(N + 1), N n=1 s s n and P u is the monic polynomial of degree N + 1 with the non-leading coefficients described by the vector u C N.

5 Mahler Ensemble of Complex Random Polynomials Question Where do the zeros of a typical polynomial from this volume lie? (Is z N 1 or (z 1) N more typical?) Definition By a random polynomial from a complex Mahler ensemble we will mean a polynomial chosen according to the density M 2s( P u ) /HN (s). Remark True interest of a number theorists lies in polynomials with integer coefficients which leads to real Mahler ensemble. Please, stay for the talk by Chris Sinclair where this more complicated case is addressed.

6 Mahler Ensemble of Complex Random Polynomials As was observed by Chern & Vaaler, a change of variables from the coefficients of polynomials to their roots, gives where H N (s) := 1 N! C N D N,s (α 1,..., α N )da N (α 1,..., α N ), D N,s := n = n { } exp 2s log + α n n<m α n α m 2 { exp 2s log τ α n dτ } α n α m 2. T 2π n<m

7 Logarithmic Capacity For any probability Borel measure on C, say ν, set 1 I [ν] := log z u dν 2 (z, u) to be its logarithmic energy. For any compact set K the logarithmic capacity of K is defined by { } cp(k) := exp inf I [ν]. supp(ν) K It is known that either cp(k) = 0 (K is polar) or else there exists the unique measure ω K, the logarithmic equilibrium distribution on K, that realizes the infimum. The measure dτ 2π T is the equilibrium distribution on both T and D.

8 Green s Function and Generalized Mahler Measure g K, Green s function with a pole at for the unbounded component of K c, the complement of a compact set K, is the unique harmonic function which is zero q.e. on K c and behaves like log z at. In particular, g D (z) = g T (z) = log + z. Put g K 0 in C \ K c. If it is continuous in C, K is called regular w.r.t. Dirichlet problem. Let K be such that cp(k) = 1. The Mahler measure of a polynomial p with respect to K is defined by { M K (p) := exp } { } log p dω K = a exp gk (α n ).

9 Density of Random Polynomials Let K be a compact set. The joint density of random configurations (zeros of random polynomials or equivalently eigenvalues of normal random matrices) is defined by { Ω N,s (z) := 1 exp 2s Z N,s } N g K (z n ) z n z m 2, n=1 where s N + 1 > 1 + c 0 for some c 0 > 0 and { Z N,s = exp C N 2s m<n } N g K (z n ) z n z m 2 da N. n=1 m<n

10 Empirical Measures Let η = {η 1,..., η N } be a random configuration chosen according to the law Ω N,s. To any such configuration we associate the empirical measure defined as ω η := 1 N N δ ηk, k=1 where δ z is the classical Dirac delta with the unit mass at z. Question Where is it most likely to find ω η when N is large? That is, where it is most likely for random polynomials to have their zeros?

11 Distance Between Measures Let ν and µ be two probability Borel measures on C. The distance between them is defined by dist(ν, µ) = sup f f dν f dµ, where the supremum is taken over all functions f that are bounded by 1 in modulus and satisfy the Lipschitz condition with constant 1 on supp(ν) sup(µ). For measures supported on a compact set it holds that dist(ν, ν n ) 0 as n if and only if ν n ν, where stands for the convergence in the weak topology of measures.

12 Large Deviation Principle The following theorem takes place. 2 Theorem (M.Y.) Let K be a compact set with connected complement which is regular with respect to the Dirichlet problem and such that K = K. Then lim lim ɛ 0 N 1 N 2 log Prob {dist (ν, ω η) < ɛ} = ( I l [ν] I [ω K ] ), for any probability Borel measure ν, supp(ν) C, where I l [ν] := I [ν] + 2 g K dν, l := lim l N s 1 N, and it holds that I l [ω K ] = I [ω K ] < I l [ν], ν ω K. 2 Large Deviations and Linear Statistics for Potential Theoretic Ensembles Associated with Regular Closed Sets, Probab. Theory Relat. Fields., 2014

13 Point Processes and Correlation Functions Let η be a random configuration chosen according to Ω N,s and ω η be the corresponding empirical measure. ω η can be considered as a simple point process on C. The correlation functions of ω η w.r.t. da are functions (if they exists) R n : C n [0, ) such that for any family of mutually disjoint subsets O 1,..., O n it holds that [ n ] E ω η (O k ) = R n (z 1,..., z n )da n (z 1,..., z n ) O 1 O n k=1 and R n (z 1,..., z n ) vanishes whenever z i = z k for i k. Thus, O R 1dA is the expected number of zeros that lie in the set O.

14 Linear Statistics Exercise R n (z 1,..., z n ) = N! Ω N,s da (N n) (z n+1,..., z N ). (N n)! C N n Theorem (M.Y.) Under the conditions of the previous theorem, it holds that (N n)! lim f R n da n = f dω n K N N! C n for each f C b (C n ), n N, where C b (C n ) is the Banach space of bounded continuous functions on C n. Remark In particular, E(ω η (O)) Nω K (O).

15 Orthogonal Polynomials Define a sequence of orthonormal polynomials {p n } such that C p n (z)p m (z)e 2sg K (z) da = δ nm. The following fact is by now standard, see Deift 3 or Mehta 4, where R n (z 1,..., z n ) = det [ K N (z i, z k ) ] n i,k=1, N 1 K N (z, w) := e s(g K (z)+g K (w)) p n (z)p n (w). n=0 3 Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. Volume 3 of Courant Lectures in Mathematics. Amer. Math. Soc., Providence, RI, Random Matrices. Volume 142 of Pure and Applied Mathematics, Elsevier/Academic Press, Amsterdam, 2004

16 Asymptotics of Orthogonal Polynomials Then the following theorem takes place 5. Theorem (M.Y and C. Sinclair) Let K be a Jordan domain whose boundary K is a Jordan curve of class C 1,α, α > 1/2. Then p n = ( 1 + o(1) ) n + 1 π ( 1 n + 1 ) Φ n Φ s uniformly on K c, where Φ is the conformal map from K c { z > 1}. Remark Observe that Φ(z) = exp { g K (z) } for z K c. 5 Universality for ensembles of matrices with potential theoretic weights on domains with smooth boundary, J. Approx. Theory, 164(5): , 2012

17 Interior Asymptotics Denote by K(z, w) the reproducing kernel for the Bergman space on K. That is, f (z) = f (w)k(z, w)da(w) K for every holomorphic f such that K f 2 da <. Theorem (M.Y. and C. Sinclair) Under the conditions of the previous theorem, K N (z, w) converges to K(z, w) locally uniformly in K K. Remark For all N large, random polynomials are expected to have a fixed number of zeros in each set of positive Lebesgue measure.

18 Exterior Asymptotics K N (z, w) = Φ(z)Φ(w) s N 1 p n (z)p n (w), z, w K c. n=0 Theorem (M.Y. and C. Sinclair) Under the conditions of the previous theorem, it holds that [ ] Φ(z)Φ(w) s K N (z, w) ( ) N Φ(z)Φ(w) s N 1 Φ (z)φ (w) c π Φ(z)Φ(w) 1 Φ(z)Φ(w) 1 locally uniformly in K c K c, where c := lim N (s N). In particular, K N (z, w) 0 when s N. Remark When c < and N is large, random polynomials are expected to have a fixed number of zeros in each set of positive Lebesgue measure.

19 Scaling Limits on the Boundary From the linear statistics we know that E [ ω η (D ε (τ)) ] Nω K (D ε (τ)) εnω K (τ). Thus, to see a non-trivial behavior around τ we need to scale ε N 1. We also know that E [ ω η (D 1 N (τ))] = = D 1N (τ) D 1 N 2 K N R 1 (z)da(z) = ( τ + z N, τ + z N D 1N (τ) K N (z, z)da(z) ) da(z). Thus, we expect integrand to converge and therefore set 1 ( K τ (z, w) := lim N N 2 K N τ + z N, τ + w ). N

20 Scaling Limits on the Boundary Theorem (M.Y. and C. Sinclair) Under the conditions of the previous theorem, it holds that K τ (z, w) = ω(τ, z)ω(τ, w) π 1 0 x(1 lx)e (a(τ,z)+a(τ,w))x dx, where a(τ, z) := zφ (τ)φ(τ) (the argument of a(τ, z) is equal to the angle between z and the outward normal to K), l = lim N s 1 N, and { ( ω(τ, z) := lim exp sg K τ + z )}. N N Remark 1 0 x(1 lx)e ηx dx = (1 l) eη (η 1) + 1 η 2 + l eη (η 2) + η + 2 η 3.

21 Plots of the Second Correlation Functions K sin (a, b) = sin(a b) a b = eη e η, η = i(a b) 2η R 2 sin 1.0 R 2 sin t t Plots of R sin 2 (a, b) = 1 K sin (a, b) 2 as a function of 2(a b). The second plot is an enlargement of the shaded region.

22 Plots of the Second Correlation Functions πk τ (z, w) ω(τ, z)ω(τ, w) = (1 (η 1) + 1 l)eη η 2 + l eη (η 2) + η + 2 η 3 where η = a(τ, z) + a(τ, w) which can be parametrized as i(a b) in the tangential direction (in which case ω(τ, z) = 1). R R t t

23 Plots of the Second Correlation Functions R t Plot of the interpolation between R2 0 and R1 2 along a tangent line.