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Zeros of random analytic functions and extreme value theory Zakhar Kabluchko University of Ulm November 9, 2012 A random equation Statement of the problem We consider an algebraic equation with random coefficients, for example z 2000 − z 1999 + z 1998 + z 1997 − z 1996 − . . . + z 3 + z 2 − z + 1 = 0 What is the distribution of the solutions of this equation in the complex plane? 2 3 Solutions of the equation Zeros of a random polynomial of degree n = 2000 Random Polynomials (Kac Ensemble) Statement of the problem Let ξ0 , ξ1 , . . . be i.i.d. random variables. Consider the equation Pn (z) := ξ0 + ξ1 z + ξ2 z 2 + . . . + ξn z n = 0. The equation has n complex roots z1 , . . . , zn . What is the distribution of roots in the complex plane? Consider the empirical measure n 1X δ(zk ). µn = n k=1 Problem: Find limn→∞ µn . Distribution of roots Theorem (Ibragimov, Zaporozhets, 2011) The following conditions are equivalent: 1 With probability 1, the sequence µn converges weakly to the uniform distribution on the unit circle. 2 E log(1 + |ξ0 |) < ∞. Remark P∞ k The series converges a.s. in the unit circle iff k=0 ξk z E log(1 + |ξ0 |) < ∞. History Hammersley (1954), Shparo und Shur (1962), Arnold (1966), Shepp und Vanderbei (1995), ... 5 Logarithmic tails Problem What happens if the coefficients ξk are heavy-tailed? Logarithmic tails We consider coefficients satisfying P[|ξ0 | > t] ∼ L(log t) , t → +∞, (log t)α where α > 0 and L is a slowly varying function. Remark ( infinite, α < 1, E log(1 + |ξ0 |) is finite, α > 1. 7 Example: Coefficients with logarithmic tails The coefficients ξk are such that P[|ξk | > t] = 1/(log t)2 and the degree is n = 2000. Distribution of roots Assume for simplicity that P[|ξ0 | > t] ∼ 1 (log t)α as t → +∞. (1) Theorem (Kabluchko, Zaporozhets, 2011) For coefficients with logarithmic tails, the following weak convergence of random probability measures holds: n 1 1X w hn1− α i δ(zk ) −→ Πα . n→∞ n k=1 The limiting random probability measure Πα is a.s. a convex combination of at most countably many uniform measures concentrated on circles centered at the origin. 9 Light and logarithmically tailed coefficients Distribution of roots Example: α = 1 If P[|ξ0 | > t] ∼ 1 log t as t → +∞, then we have n 1X w δ(zk ) −→ Π1 . n→∞ n k=1 No normalization of roots is needed. Example: α > 1 (E log |ξ0 | < ∞) The roots approach the unit circle. The distance between the 1 roots and the unit circle is of order n α −1 . Example: α < 1 (E log |ξ0 | = ∞) The roots diverge to ∞ and 0. The absolute values of the roots 1 −1 are of order O(1)n α . Extremal order statistics of the coefficients Theorem Let ξ0 , ξ1 , . . . be i.i.d. random variables with P[log |ξk | > t] ∼ t −α as t → +∞. Then, we have the following weak convergence of point processes on [0, 1] × (0, ∞): n X k log |ξk | w δ , 1/α −→ PPP αv −(α+1) dudv . n→∞ n n k=0 11 12 Extremal order statistics of the coefficients Newton polygon Idea of the proof For large n consider the equation ±e nx1 ± . . . ± e nxd = 0, where xi > 0. The most easy way for this to be true is the following: 1 Two terms, say e nxk and e nxl , cancel each other. 2 All other terms are much smaller than these two. Newton polygon Idea of the proof Apply this to the equation ξ0 + ξ1 z + . . . + ξn z n = 0. 1 Two terms cancel each other: ξk z k + ξl z l = 0. 2 All other terms are much smaller than these two. Geometrically: the points (k, log |ξk |) and (l, log |ξl |) are neighboring vertices of the least concave majorant of the set {(0, log |ξ0 |), . . . , (n, log |ξn |)}. 14 The least concave majorant Remark The number of segments is finite a.s. if and only if α < 1. 15 Limiting empirical measure of the roots Limiting measure Πα Consider the least concave majorant of the Poisson process with intensity αv −(α+1) dudv . 1 Radii of circles = exponentials of the slopes of the majorant. 2 Number of roots on a circle = length of the linearity interval. 16 Very heavy tails: α = 0 Let the coefficients be such that P[|ξ0 | > t] ∼ L(log t) as t → +∞, where L is a slowly varying function. The roots concentrate on two circles, one with a small radius, one with a large radius. The proportion of the roots lying on the small circle is uniform on [0, 1]. 17 Weyl Polynomials Let ξ0 , ξ1 , . . . be i.i.d. random variables. Consider the Weyl Polynomials Pn (z) = n X zk ξk √ . k! k=0 Let z1 , . . . , zn be the zeros of Pn . Theorem (Kabluchko, Zaporozhets, 2012) The following conditions are equivalent: P 1 The sequence of probability measures n1 k=1 δ( √zkn ) converges a.s. to the uniform distribution on the unit disk {|z| ≤ 1}. 2 E log(1 + |ξ0 |) < ∞. 18 19 Weyl Polynomials Zeros of a Weyl polynomial: Normally distributed coefficients 20 Weyl Polynomials Zeros of a Weyl polynomial: Logarithmic tails Littlewood–Offord Polynomials (1939) Let ξ0 , ξ1 , . . . be i.i.d. random variables with E log(1 + |ξ0 |) < ∞. Consider the Littlewood–Offord polynomials Pn (z) = n X ξk k=0 zk . (k!)α Let z1 , . . . , zn be the zeros of Pn . Theorem (Kabluchko, Zaporozhets, 2012) With 1, the sequence of random measures Pn probability zk 1 δ( ) converges to the probability measure with the k=1 n nα density 1 1 |z| α −2 , |z| < 1. 2πα 21 Littlewood–Offord Polynomials Zeros of the Littlewood–Offord polynomials: Normal coefficients Littlewood–Offord Polynomials Zeros of the Littlewood–Offord polynomials: Logarithmic coefficients Szegö Polynomials Szegö Polynomials: sn (z) = zk k=0 k! . Pn Theorem (Szegö, 1924) The zeros of sn (nz) cluster along the curve |ze 1−z | = 1, |z| < 1. Zeros in the Random Energy Model Random Energy Model (Derrida, 1981) A system has N states. √ The energy of the system in state i is log Nξi . ξ1 , . . . , ξN are i.i.d. standard Gaussian random variables. Consider the partition function ZN (β) = N X eβ √ log Nξk , β ∈ C. k=1 Other motivations ZN is an empirical Laplace transform. ZN is a nice random analytic function. Zeros in the Random Energy Model Complex zeros of ZN . Source: C. Moukarzel und N. Parga: Physica A 177 (1991). 26 Zeros in the Random Energy Model Theorem (Derrida, 1991) For β = σ + iτ ∈ C it holds that 1 + 1 (σ 2 − τ 2 ), β ∈ B 1 , log |ZN (β)| √ 2 lim = 2|σ|, β ∈ B 2, N→∞ log N 1 + σ2, β ∈ B 3, 2 where B1 = C\B2 ∪ B3 , B2 = {β ∈ C : 2σ 2 > 1, |σ| + |τ | > 2 2 2 √ 2}, B3 = {β ∈ C : 2σ < 1, σ + τ > 1}. 27 Zeros in the Random Energy Model Theorem (Derrida, 1991) The random measure 2π log N X δ(β) β:ZN (β)=0 converges weakly (as N → ∞) to the deterministic measure Ξ = 2Ξ3 + Ξ12 + Ξ13 . Here, Ξ3 is the Lebesgue measure on B3 . Ξ13 is the one-dimensional Lebesgue measure on the boundary between B1 and B3 . √ Ξ12 is the measure with density 2|τ | on the boundary between B1 and B2 . Rigorous proof, further results: Kabluchko und Klimovsky, 2012.