Zeros of random analytic functions and extreme value theory Zakhar - PowerPoint PPT Presentation

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

Solutions of the equation Zeros of a random polynomial of degree n = 2000 3

Random Polynomials (Kac Ensemble) Statement of the problem Let ξ 0 , ξ 1 , . . . be i.i.d. random variables. Consider the equation P n ( z ) := ξ 0 + ξ 1 z + ξ 2 z 2 + . . . + ξ n z n = 0 . The equation has n complex roots z 1 , . . . , z n . What is the distribution of roots in the complex plane? Consider the empirical measure n µ n = 1 � δ ( z k ) . n k =1 Problem: Find lim n →∞ µ n . 4

Distribution of roots Theorem (Ibragimov, Zaporozhets, 2011) The following conditions are equivalent: 1 With probability 1, the sequence µ n converges weakly to the uniform distribution on the unit circle. 2 E log(1 + | ξ 0 | ) < ∞ . Remark k =0 ξ k z k converges a.s. in the unit circle iff The series � ∞ E log(1 + | ξ 0 | ) < ∞ . History Hammersley (1954), Shparo und Shur (1962), Arnold (1966), Shepp und Vanderbei (1995), ... 5

Logarithmic tails Problem What happens if the coefficients ξ k are heavy-tailed? Logarithmic tails We consider coefficients satisfying P [ | ξ 0 | > t ] ∼ L (log t ) (log t ) α , t → + ∞ , where α > 0 and L is a slowly varying function. Remark � infinite , α < 1 , E log(1 + | ξ 0 | ) is finite , α > 1 . 6

Example: Coefficients with logarithmic tails The coefficients ξ k are such that P [ | ξ k | > t ] = 1 / (log t ) 2 and the degree is n = 2000. 7

Distribution of roots Assume for simplicity that 1 P [ | ξ 0 | > t ] ∼ as t → + ∞ . (1) (log t ) α Theorem (Kabluchko, Zaporozhets, 2011) For coefficients with logarithmic tails, the following weak con- vergence of random probability measures holds: n 1 δ ( z � n 1 − 1 α � � w ) n →∞ Π α . − → k n k =1 The limiting random probability measure Π α is a.s. a convex combination of at most countably many uniform measures con- centrated on circles centered at the origin. 8

Light and logarithmically tailed coefficients 9

Distribution of roots Example: α = 1 1 If P [ | ξ 0 | > t ] ∼ log t as t → + ∞ , then we have n 1 w � δ ( z k ) n →∞ Π 1 . − → n k =1 No normalization of roots is needed. Example: α > 1 ( E log | ξ 0 | < ∞ ) The roots approach the unit circle. The distance between the 1 α − 1 . roots and the unit circle is of order n Example: α < 1 ( E log | ξ 0 | = ∞ ) The roots diverge to ∞ and 0. The absolute values of the roots 1 α − 1 . are of order O (1) n 10

Extremal order statistics of the coefficients Theorem Let ξ 0 , ξ 1 , . . . be i.i.d. random variables with P [log | ξ k | > t ] ∼ t − α as t → + ∞ . Then, we have the following weak convergence of point pro- cesses on [0 , 1] × (0 , ∞ ): n � k � n , log | ξ k | w � � α v − ( α +1) dudv � δ n →∞ PPP − → . n 1 /α k =0 11

Extremal order statistics of the coefficients 12

Newton polygon Idea of the proof For large n consider the equation ± e nx 1 ± . . . ± e nx d = 0 , where x i > 0. The most easy way for this to be true is the following: 1 Two terms, say e nx k and e nx l , cancel each other. 2 All other terms are much smaller than these two. 13

Newton polygon Idea of the proof Apply this to the equation ξ 0 + ξ 1 z + . . . + ξ n z n = 0 . 1 Two terms cancel each other: ξ k z k + ξ l z l = 0. 2 All other terms are much smaller than these two. Geometrically: the points ( k , log | ξ k | ) and ( l , log | ξ l | ) are neigh- boring vertices of the least concave majorant of the set { (0 , log | ξ 0 | ) , . . . , ( n , log | ξ n | ) } . 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 Π α Consider the least concave majorant of the Poisson process with intensity α v − ( α +1) dudv . 1 Radii of circles = exponentials of the slopes of the majo- rant. 2 Number of roots on a circle = length of the linearity inter- val. 16

Very heavy tails: α = 0 Let the coefficients be such that P [ | ξ 0 | > 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 ξ 0 , ξ 1 , . . . be i.i.d. random variables. Consider the Weyl Polynomials n z k � P n ( z ) = ξ k √ . k ! k =0 Let z 1 , . . . , z n be the zeros of P n . Theorem (Kabluchko, Zaporozhets, 2012) The following conditions are equivalent: 1 The sequence of probability measures 1 k =1 δ ( z k � √ n ) con- n verges a.s. to the uniform distribution on the unit disk {| z | ≤ 1 } . 2 E log(1 + | ξ 0 | ) < ∞ . 18

Weyl Polynomials Zeros of a Weyl polynomial: Normally distributed coefficients 19

Weyl Polynomials Zeros of a Weyl polynomial: Logarithmic tails 20

Littlewood–Offord Polynomials (1939) Let ξ 0 , ξ 1 , . . . be i.i.d. random variables with E log(1 + | ξ 0 | ) < ∞ . Consider the Littlewood–Offord polynomials n z k � P n ( z ) = ξ k ( k !) α . k =0 Let z 1 , . . . , z n be the zeros of P n . Theorem (Kabluchko, Zaporozhets, 2012) With probability 1, the sequence of random measures � n 1 k =1 δ ( z k n α ) converges to the probability measure with the n density 1 1 α − 2 , | z | < 1 . 2 πα | z | 21

Littlewood–Offord Polynomials Zeros of the Littlewood–Offord polynomials: Normal coefficients 22

Littlewood–Offord Polynomials Zeros of the Littlewood–Offord polynomials: Logarithmic coefficients 23

Szeg¨ o Polynomials o Polynomials: s n ( z ) = � n z k Szeg¨ k ! . k =0 Theorem (Szeg¨ o, 1924) The zeros of s n ( nz ) cluster along the curve | ze 1 − z | = 1, | z | < 1. 24

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 N ξ i . ξ 1 , . . . , ξ N are i.i.d. standard Gaussian random variables. Consider the partition function N e β √ log N ξ k , β ∈ C . � Z N ( β ) = k =1 Other motivations Z N is an empirical Laplace transform. Z N is a nice random analytic function. 25

Zeros in the Random Energy Model Complex zeros of Z N . Source: C. Moukarzel und N. Parga: Physica A 177 (1991). 26

Zeros in the Random Energy Model Theorem (Derrida, 1991) For β = σ + i τ ∈ C it holds that 2 ( σ 2 − τ 2 ) ,  1 + 1 β ∈ B 1 , √  log |Z N ( β ) |  lim = 2 | σ | , β ∈ B 2 , log N N →∞  1 2 + σ 2 , β ∈ B 3 ,  where B 1 = C \ B 2 ∪ B 3 , √ B 2 = { β ∈ C : 2 σ 2 > 1 , | σ | + | τ | > 2 } , B 3 = { β ∈ C : 2 σ 2 < 1 , σ 2 + τ 2 > 1 } . 27

Zeros in the Random Energy Model Theorem (Derrida, 1991) The random measure 2 π � δ ( β ) log N β : Z N ( β )=0 converges weakly (as N → ∞ ) to the deterministic measure Ξ = 2Ξ 3 + Ξ 12 + Ξ 13 . Here, Ξ 3 is the Lebesgue measure on B 3 . Ξ 13 is the one-dimensional Lebesgue measure on the boundary between B 1 and B 3 . √ Ξ 12 is the measure with density 2 | τ | on the boundary between B 1 and B 2 . Rigorous proof, further results: Kabluchko und Klimovsky, 2012. 28