The maximum of the characteristic polynomial for a random permutation matrix

Transcription

1 The maximum of the characteristic polynomial for a random permutation matrix Random Matrices and Free Probability Workshop IPAM, 2018/05/16 Nick Cook, UCLA Based on joint work with Ofer Zeitouni

2 Model and previous work Let P N be a uniform random N N permutation matrix and let χ N (z) = det(zi N P N ) denote its characteristic polynomial. Consider the random field X N : R/Z [, ) X N (t) = log χ N (e( t)) = log det(i N e(t)p N ) where e(t) := exp(2πit). Hambly Keevash O Connell Stark 99 obtained a CLT: For fixed t R/Z of finite type, X N (t) π 2 12 log N N(0, 1) (and similarly for the imaginary part of log χ N ). Note X N is badly behaved at rational points (atom at ). Multidimensional CLT obtained by Dang Zeindler 13. 1

3 Numerical simulations Figure 1: Simulated X N (t) with N = 10 4 for t [0.1, 0.11]. Cycle structure of P N : 6310, 1914, 909, 668, 79, 47, 33, 19, 12, 5, 3, 1. (Generated using the Chinese restaurant process.) 2

4 Numerical simulations Figure 2: Simulated X N (t) with N = 10 9 for t [0.2, 0.35]. Cycle structure of P N : 892,060,223, 78,087,020, 19,479,718, 9,152,317, 630,684, 352,623, 114,502, 104,059, 8,973, 8,193, 1,641, 33, 5, 3, 2, 2, 1, 1. 3

5 Main result: Law of large numbers for the maximum of X N Recall: X N (t) = log χ N (e(t)) = log det(i N e(t)p N ). Theorem (C., Zeitouni 18) 1 log N sup X N (t) x c t R/Z in probability. (Informally: sup χ N (z) = N xc +o(1) with high probability.) z =1 4

6 Related work: Maximum of the CUE field Replacing P N with a Haar unitary U N, we obtain the CUE field: X cue N (t) = log det(i N e(t)u N ). Conjecture (Fyodorov Hiary Keating 12): ( M N := sup XN cue (t) log N 3 ) t R/Z 4 log log N converges in distribution. sup XN cue (t) = log N + o P (log N) Arguin Belius Bourgade 15 t R/Z Related work of Arguin Belius Bourgade Soundararajan Radziwi l l 16 and Najnudel 16 on the Riemann ζ function. All proofs proceed by exposing an underlying branching structure. Note that, unlike X N, distribution of X cue N is invariant under rotations. 5

7 Related work: Maximum of the CUE field Replacing P N with a Haar unitary U N, we obtain the CUE field: X cue N (t) = log det(i N e(t)u N ). Conjecture (Fyodorov Hiary Keating 12): ( M N := sup XN cue (t) log N 3 ) t R/Z 4 log log N converges in distribution. sup XN cue (t) = log N + o P (log N) Arguin Belius Bourgade 15 t R/Z = log N 3 4 log log N + o P(log log N) Paquette Zeitouni 15 Related work of Arguin Belius Bourgade Soundararajan Radziwi l l 16 and Najnudel 16 on the Riemann ζ function. All proofs proceed by exposing an underlying branching structure. Note that, unlike X N, distribution of X cue N is invariant under rotations. 5

8 Related work: Maximum of the CUE field Replacing P N with a Haar unitary U N, we obtain the CUE field: X cue N (t) = log det(i N e(t)u N ). Conjecture (Fyodorov Hiary Keating 12): ( M N := sup XN cue (t) log N 3 ) t R/Z 4 log log N converges in distribution. sup XN cue (t) = log N + o P (log N) Arguin Belius Bourgade 15 t R/Z = log N 3 4 log log N + o P(log log N) Paquette Zeitouni 15 = log N 3 4 log log N + O P(1) Chhaibi Madaule Najnudel 16 Related work of Arguin Belius Bourgade Soundararajan Radziwi l l 16 and Najnudel 16 on the Riemann ζ function. All proofs proceed by exposing an underlying branching structure. Note that, unlike X N, distribution of X cue N is invariant under rotations. 5

9 Maximal displacement for branching random walk (BRW) X N and XN cue are logarithmically correlated fields Archetypical log-correlated field is (binary, Gaussian) BRW, which we view as a random field Xn brw (t), t [0, 1]. For each t [0, 1], Xn brw (t) N(0, n). Decorrelation of increments: Denoting by anc(s, t) [1, n] the generation where lineages of s, t split, we have Cov ( X n (s) X m (s), X n (t) X m (t) ) = 0 for any anc(s, t) < m < n. 6

10 Maximal displacement for branching random walk (BRW) Theorem (Hammersley 74, Kingman 75, Biggins 77) 1 n sup Xn brw (t) b c = 2 log 2 in probability. t [0,1) We recall some key proof ideas going back to Bramson 78. Let T n = {k2 n [0, 1)} and put S n (b) = {t T n : Xn brw (t) bn}. Upper bound. First moment method (union bound): E S n (b c + ɛ) = 2 n P(X brw n (1) (b c + ɛ)n) e c(ɛ)n. Then apply Markov s inequality. Lower bound. Same computation shows E S n (b c ɛ) e c (ɛ)n, so we d be done if we can establish concentration, i.e. Var S n (b c ɛ) (E S n (b c ɛ) ) 2 = o(1). But this is false. (See the board...) 7

11 Maximal displacement for branching random walk (BRW) Modified second moment: Rather than counting high points S n (b), we should only count points that make steady progress toward b c n. Toy computation: Let X n (1) = Xn/2 brw, X n (2) = Xn brw Xn/2 brw, and put E 2 (t) = { X n (1) (t), X n (2) (t) bn }. 2 For pairs (s, t) with anc(s, t) < n/2, we can bound ( P(E 2 (s) E 2 (t)) = P P X (1) n/2 (1) (s), X n/2 (t) bn ) 2 ) P ( X (1) n/2 (s) bn 2 ( P X n (2) (s), X (2) ( X (2) n (s) bn 2 n (t) bn 2 where we used the decorrelation of increments. Proceeding in a similar manner, we can show the greatest contribution to the second moment of the number of steadily advancing particles comes from pairs (s, t) whose lineages split early on. 8 ) 2 )

12 First steps: Discretize and pass to Poisson model X N (t) = log det(i N e(t)p N ), t R/Z. It is enough to show max X N (t) = (x c + o(1)) log N t T N w.h.p. where T N is a mesh for R/Z of size T N = O(N). X N only depends on the cycle structure of P N : X N (t) = C l (P N ) log 1 e(lt), 1 l N where C l (P N ) is the number of cycles of length l in P N. Arratia Tavare 92: Let Z l be independent Poi(1/l) variables. Then ( ) d TV (Cl (P N )) l L, (Z l ) l L 0 if L = o(n). 9

13 First steps: Discretize and pass to Poisson model Let ω(1) W N o(1) be a slowly growing function of N and split X N (t) = C l (P N ) log 1 e(lt) + C l (P N ) log 1 e(lt) l N/W N/W <l N =: X N (t) + X > N (t). Letting Y N (t) = l N Z l log 1 e(lt), we have X N (t) Y N/W (t) in distribution. Second moment computation shows X > N (t) N/W <l N C l (P N ) O(log W ) = o(log N) w.h.p. so for the upper bound it suffices to consider the Poisson field Y N. 10

14 Upper bound: first attempt Let S(Y N, x) = {t T N : Y N (t) x log N}. We want to show that for any fixed ɛ > 0, S(Y N, x c + ɛ) = 0 w.h.p. First moment: E S(Y N, x) = t T N P(Y N (t) x log N). To estimate tail events, estimate MGFs: E e βy N (t) = ( 1 ( E exp(βz l log 1 e(lt) ) = exp 1 e(lt) 1) ) β. l l N l N For t irrational, the sum is (log N) ( e(u) β du 1 ). This eventually shows E S(Y N, 1 + ɛ) = o(1). 11

15 Fix: Condition on a typical distribution of cycle lengths Problem: First moment is thrown off by clumping events, where there is an atypically large number of cycles of comparable length (the corresponding waves t log 1 e(lt) add constructively). Solution: Consider a sequence of windows I k = [e δk, e δ(k+1) ) [N] with δ small (actually o(1)) and put N (I k ) = l I k Z l, Y Ik (t) = l I k Z l log 1 e(lt). Expect E N (I k ) δ cycles in each window. We condition on a typical sequence (N (I k )) k 1 (with N (I k ) {0, 1} for most k). Let K = {k : N (I k ) = 1}. Then K log N, and for k K, the increments Y Ik (t) = log 1 e(l k t), P(l k = l) 1 l I k l are still independent and have comparable contribution to the fluctuations of Y N. 12

16 Upper bound Under the conditioning on (N (I k )) k we have E e βy N (t) k K If t is irrational then E exp ( βy Ik (t) ) = k K l I e(lt) β k l l l I k = 1 0 ( 1 l I k 1 l l I k ) 1 e(lt) β. l 1 e(u) β du + o β,t (1). As we ll be taking a union bound over t T N, we need to quantify dependence of the error on t. We get satisfactory errors outside low-frequency Bohr sets B ξ (η) = {t R/Z : ξt R/Z η}. We call t Maj(ξ 0, η) := ξ ξ 0 B ξ (η) major arc points. 13

17 Upper bound: Major and minor arcs While Y N (t) is badly behaved on major arcs, we can show sup Y N (t) c(ξ 0, η) log 2 N t Maj(ξ 0,η) w.h.p. On the complementary set of minor arcs, Y N behaves like a sequence of iid variables: Y Ik (t) = log 1 e(l k t) log 1 e(u k ), u k Unif (R/Z). Letting λ(β) = log E exp ( β log 1 e(u) ) = log e(u) β du, for major arcs we have a first moment estimate for S(Y N, x) \ Maj : t T N \Maj(ξ 0,η) P (Y N (t) x log N) T N exp ( λ (x) K ) N 1 λ (x). where λ is the Legendre transform of λ. x c is the unique solution to λ (x) = 1. 14

18 Lower bound: Early, middle, and late generations For M N, denote the truncated field X M (t) = l M C l (P N ) log 1 e(lt), and the set of survivors S M (x) = {t T N : X M (t) x log M}. We show S N (x c ɛ) is nonempty by tracking the population of survivors across three epochs: 1. Early generations: S N c ( C) N w.h.p. 2. Middle generations: S N/W (x c ɛ) N c(ɛ) w.h.p. 3. Late generations: S N (x c 2ɛ) 1 w.h.p. For early and middle generations we can replace X M with Y M. 15

19 Lower bound: Middle generations We follow the second moment argument for BRW. Requires approximate decorrelation of tail events for increments m,n (t) := k K [m,n) Y I k (t). For ξ 0 N put d ξ0 (s, t) = min ξs + ξ,ξ { ξ ξ t R/Z. 0,...,ξ 0}\{0} Lower bound on d ξ0 (s, t) gives quantitative linear indepedence of 1, s, t over Z. If m = ω(1), s, t / Maj(ξ 0, η) and d ξ0 (s, t) > η for ξ 0 = ω(n 2 ) and η e δm, then we have an estimate of the form P ( m,n (s), m,n (t) x ) (1 + o(1)) P( m,n (s) x) 2. 16

20 Lower bound: Late generations ( S N (x c 2ɛ) 1) Poisson model is no longer available. We condition on the N c(ɛ) survivors of Middle Ages S N/W := S N/W (x c ɛ) and want to show they aren t all wiped out by the high frequency tail X > N = X N X N/W. Note that log 1 e(lt) ɛ log N for t in the Bohr set B l (N ɛ ) = {t R/Z : lt R/Z N ɛ }. So if S N/W B l (N ε ) for some l (N/W, N], there is a chance the whole population gets wiped out. We employ a structural dichotomy for S N/W : Either (Structured case) S N/W has large overlap with B l (N ɛ ) for N/W 3 different frequencies l (N/W, N], or (Unstructured case) it doesn t. In the unstructured case, we can show it is unlikely any of the exceptional frequencies are selected (using the switchings method). 17

21 Lower bound: Late generations (Structured case) S N/W has large overlap with B l (N ɛ ) for N/W 3 different frequencies l (N/W, N]. In this case we can use pigeonholing and a Vinogradov-type lemma to show the S N/W must actually contain an element that is major arc, specifically an element of B ξ (η) for some ξ = W O(1), η = W O(1) N ɛ 1. But such major arc points are unlikely to be in S N/W in the first place. 18