Quantum ergodicity on regular graphs

Transcription

1 Quantum ergodcty on regular graphs Naln Anantharaman Unversté de Strasbourg 23 août 2016

2 M Remannan manfold Let (ψ k ) k N be an orthonormal bass of L 2 (M), wth QE theorem (smplfed) : ψ k = λ k ψ k, λ k λ k+1. Theorem (Shnrelman 74, Zeldtch 85, Coln de Verdère 85) Assume that the acton of the geodesc flow s ergodc for the Louvlle measure. Let a C 0 (M). Then 1 N(λ) λ k λ M a(x) ψ k (x) 2 dvol(x) M a(x)dvol(x) 2 0. λ

3 Equvalently, there exsts a subset S N of densty 1, such that a(x) ψ k (x) 2 dvol(x) a(x)dvol(x). M k S k + M Equvalently, n the weak topology. ψ k (x) 2 dvol(x) k S dvol(x) k +

4 The Quantum Unque Ergodcty conjecture QUE conjecture : Conjecture (Rudnck, Sarnak 94) On a negatvely curved manfold, we have convergence of the whole sequence : M a(x) ψ k(x) 2 dvol(x) M a(x)dvol(x) (for all a). and more generally, ψ k, a(x, D x )ψ k L 2 (M) (x,ξ) SM a(x, ξ)dxdξ.

5 Other questons or conjectures Random wave Ansatz. In the case of a chaotc geodesc flow, M. Berry Ansatz s that egenfunctons of should locally resemble a monochromatc gaussan random feld ψ λ (x) = N a α e λu α x α=1 wth the u α equdstrbuted over the unt sphere, a α d gaussan random varables, N +. Spectral statstcs : egenvalues of, λ k (E E 1/2, E + E 1/2 ) (E + ) should after sutable rescalng resemble the egenvalues of large N N gaussan symmetrc matrces.

6 QE on dscrete graphs Snce the 90s there has been the dea of usng graphs as a testng ground / toy model for quantum chaos. Smlansky, Kottos, Elon,... Bogomolny, Keatng, Berkolako, Wnn, Potet, Marklof, Gnutzmann... studed mostly metrc graphs wth Krchhoff matchng condtons for the dervatves at the vertces.

7 For a fxed metrc graph, t s known that QE does NOT hold n the lmt λ j + (at least for ratonally lengths of the edges : Coln de Verdère 2014, Tanner 2001).

8 For a fxed metrc graph, t s known that QE does NOT hold n the lmt λ j + (at least for ratonally lengths of the edges : Coln de Verdère 2014, Tanner 2001). For the lmt of large graphs, spectral statstcs have been studed (star graphs), scarrng, as well as as the valdty of random wave ansatz.

9 Here we focus on the case of large regular (dscrete) graphs (cf Smlansky). Let G = (V, E) be a (q + 1)-regular graph. Dscrete laplacan : f : V C, f (x) = (f (y) f (x)) = y x y x f (y) (q + 1)f (x). = A (q + 1)I

10 Sp(A) [ (q + 1), q + 1] Let V = N. We look at the lmt N +.

11 Sp(A) [ (q + 1), q + 1] Let V = N. We look at the lmt N +. We assume that G N has few short loops (= converges to a tree n the sense of Benjamn-Schramm).

12 Sp(A) [ (q + 1), q + 1] Let V = N. We look at the lmt N +. We assume that G N has few short loops (= converges to a tree n the sense of Benjamn-Schramm). Ths mples convergence of the spectral measure (Kesten-McKay) 1 N {, λ I } m(λ)dλ N + for any nterval I. m s a completely explct densty, supported n ( 2 q, 2 q) I

13 Theorem (Brooks-Lndenstrauss 2011) Assume that G N has few loops of length c log N. For ɛ > 0, there exsts δ > 0 s.t. for every egenfuncton φ, B V N, x B φ(x) 2 ɛ = B N δ. Proof also yelds that φ log N 1/4.

14 Theorem (A-Le Masson, 2013) Assume that G N has few short loops and that t forms an expander famly = unform spectral gap for A. Let (φ (N) ) N =1 be an ONB of egenfunctons of the laplacan on G N. Let a = a N : V N C be such that a(x) 1 for all x V N. Then lm N + 1 N N a(x) φ (N) (x) 2 a x V N =1 a = 1 a(x) N x V N Also works on shrnkng spectral ntervals 2 = 0.

15 Examples Determnstc examples : the Ramanujan graphs of Lubotzky-Phllps-Sarnak 1988 (arthmetc quotents of the q-adc symmetrc space PGL(2, Q q )/PGL(2, Z q )).

16 Examples Determnstc examples : the Ramanujan graphs of Lubotzky-Phllps-Sarnak 1988 (arthmetc quotents of the q-adc symmetrc space PGL(2, Q q )/PGL(2, Z q )). Cayley graphs of SL 2 (Z/pZ), p ranges over the prmes, (Bourgan-Gamburd, based on Helfgott 2005) Extended to all smple Chevalley groups (Breullard-Green-Tao), perfect groups and square-free p (Saleh-Golsefdy Varju)

17 Random regular graphs Our result apples n partcular to random regular graphs. In that case there also exsts a probablstc proof (Gesnger 2013) n the case where a(x) s chosen ndependently of G N.

18 Just as a comparson... the Erdös-Reny model

19 More general verson Theorem (A-Le Masson, 2013) Assume that G N has few short loops and that t forms an expander famly. Let (φ (N) ) N =1 be an ONB of egenfunctons of the laplacan on G N. Let K = K N : V N V N C be a matrx such that d(x, y) > D = K(x, y) = 0. Assume K(x, y) 1. Then lm N + 1 N N φ (N) =1, Kφ (N) K λ 2 = 0. K λ = 1 K(x, y)φ sph,λ (d(x, y)). N x,y

20 K λ = 1 K(x, y)φ sph,λ (d(x, y)). N x,y Φ sph,λ s the sphercal functon of parameter λ on the (q + 1)-regular tree.

21 K λ = 1 K(x, y)φ sph,λ (d(x, y)). N x,y Φ sph,λ s the sphercal functon of parameter λ on the (q + 1)-regular tree. Φ λ (d) = q d/2 ( 2 q + 1 cos(ds ln q) + q 1 q + 1 f λ = q 1/2+s + q 1/2 s = 2 q cos(s ln q). ) sn((d + 1)s ln q) sn(s ln q)

22 Our result says that φ (N) for most., Kφ (N) = x,y φ (N) (x)k(x, y)φ (N) (y) 1 K(x, y)φ sph,λ (d(x, y)) N x,y

23 Sketch of proof : phase space analyss on regular graphs Fourer-Helgason transform on the (q + 1)-regular tree ; phase space analyss on the tree and on a fnte regular graph ; use of the geodesc dynamcs to study egenfunctons of the laplacan.

24 Fourer-Helgason transform on the (q + 1)-regular tree f : X C. Its Fourer transform s ˆf (ω, s) = x X f (x)e s,ω (x) where s T q = R/(2π/ log q), ω X, and e s,ω (x) = q (1/2+s)hω(x) satsfes Ae s,ω = 2 q cos(s log q)e s,ω.

25 Inverson formula, Plancherel theorem f (x) = s T q,ω X f (x) 2 = x X s T q,ω X ˆf (ω, s)e s,ω (x)dm(s)dν o (ω) ˆf (ω, s) 2 dm(s)dν o (ω)

26 Paley-Wener type theorem (Cowlng-Sett) Decay of f (x) when d(x, o) smoothness of ˆf (ω, s).

27 Phase space We defne t as X X T q

28 Pseudodfferental calculus on a regular tree For a functon a(x, ω, s) on X X T q, we defne an operator Op(a) on l 2 (X) by n other words Op(a)f (x) = Op(a)e s,ω (x) = a(x, ω, s)e s,ω s T q,ω X The adjacency operator A corresponds to a(x, ω, s) = 2 q cos(s log q). a(x, ω, s)ˆf (ω, s)e s,ω (x)dm(s)dν o (ω).

29 Accordng to the Paley-Wener theorem, Decay of K a (x, y) when d(x, y) smoothness of a(x, ω, s) wrt (ω, s). Le Masson (2012) studed the operators Op(a) (behavour under composton, boundedness on l 2 (X) etc).

30 Defnton of Op(a) on a fnte regular graph G N = (V N, E N ) = Γ N \X where Γ s a group of automorphsms actng wthout fxed pont on the regular tree X. Op(a) preserves the Γ-nvarant functons on X (and thus act on l 2 (V )) ff a(x, ω, s) s Γ-nvarant (a(γ x, γ ω, s) = a(x, ω, s) for γ Γ)

31 Quantum varance G N = (V N, E N ) = Γ N \X, N = V N. Aφ (N) = λ (N) φ (N) Var(K) = 1 N N =1 φ (N), Kφ (N) 2. Trvally, Var([A, K]) = 0 for any operator K.

32 Classcal dynamcs? Trvally, Var([A, K]) = 0 for any operator K. [A, Op(a)] = Op(b)

33 Classcal dynamcs? Trvally, Var([A, K]) = 0 for any operator K. [A, Op(a)] = Op(b) where b(x, ω, s) = sn(s ln q)(a σ La) + cos(s ln q)(a σ + La 2a)

34 Although we do not know f φ (N), Op(a σ a)φ (N) s small for every, we can prove that Var(Op(a σ a)) 0. N +

35 Although we do not know f φ (N), Op(a σ a)φ (N) s small for every, we can prove that Var(Op(a σ a)) 0. N + We deduce (densty argument) that Var(Op(a)) 0 (as N ) f 1 a(x, ω, s)dν x (ω) = 0 N X T q for all s. x D N

36 Although we do not know f φ (N), Op(a σ a)φ (N) s small for every, we can prove that Var(Op(a σ a)) 0. N + We deduce (densty argument) that Var(Op(a)) 0 (as N ) f 1 a(x, ω, s)dν x (ω) = 0 = K a N λ(s) X T q for all s. x D N

37 Although we do not know f φ (N), Op(a σ a)φ (N) s small for every, we can prove that Var(Op(a σ a)) 0. N + Var(Op(a)) 1 N x D N X T q a(x, ω, s) 2 dν x (ω)dm(s) We deduce (densty argument) that Var(Op(a)) 0 (as N ) f 1 a(x, ω, s)dν x (ω) = 0 = K a N λ(s) X T q for all s. x D N

38 Do graphs teach us somethng about manfolds? By approxmaton of a manfold by graphs (trangulaton)? A pror NO.

39 Contnuous analogue of our theorem... Let (S N ) be a sequence of hyperbolc surfaces, whose genus goes to. Assume the frst egenvalue λ 1 (N) of s bounded away from 0 as N. Assume there are few short geodescs. Fx an nterval I (1/4, + ). Let (φ (N) ) be an ONB of egenfunctons of the laplacan on S N. Let a = a N : S N C be such that a(x) 1 for all x S N.

40 Contnuous analogue of our theorem... Let (S N ) be a sequence of hyperbolc surfaces, whose genus goes to. Assume the frst egenvalue λ 1 (N) of s bounded away from 0 as N. Assume there are few short geodescs. Fx an nterval I (1/4, + ). Let (φ (N) ) be an ONB of egenfunctons of the laplacan on S N. Let a = a N : S N C be such that a(x) 1 for all x S N. Then lm N + 1 Vol(S N ) λ (N) I (Le Masson Sahlsten 2016) a(x) φ (N) (x) 2 dx a S N 2 = 0.

41 Applcaton to Arthmetc Quantum Unque Ergodcty M arthmetc surface. Assume ψ k, a(x, D x )ψ k L 2 (M) λ k (x,ξ) SM a(x, ξ)dµ(x, ξ). Lndenstrauss 2001 : f the ψ k are egenfunctons of and ALL the Hecke operators, then µ s the Louvlle (unform) measure.

42 Applcaton to Arthmetc Quantum Unque Ergodcty M arthmetc surface. Assume ψ k, a(x, D x )ψ k L 2 (M) λ k (x,ξ) SM a(x, ξ)dµ(x, ξ). Lndenstrauss 2001 : f the ψ k are egenfunctons of and ALL the Hecke operators, then µ s the Louvlle (unform) measure. One step s to prove that all the ergodc components of µ have postve Kolmogorov-Sna entropy

43 Applcaton to Arthmetc Quantum Unque Ergodcty (on arthmetc surfaces) Assume ψ k, a(x, D x )ψ k L 2 (M) (x,ξ) SM a(x, ξ)dµ(x, ξ). Brooks-Lndenstrauss 2011 : f the ψ k are egenfunctons of and ONE Hecke operator, then µ s the Louvlle (unform) measure. Idea : approxmate Hecke trees by fnte regular graphs, use a modfcaton of the prevously quoted result of BL.

44 QE on the sphere, revsted (Brooks-Le Masson-Lndenstrauss 2015) For g 1,..., g k a fnte set of rotatons n SO(3), commutes wth S 2. Theorem T k f (x) = k j=1 (f (g j x) + f (g 1 j Assume that g 1,..., g k generate a free subgroup of SO(3). For each l, let (ψ (l) j ) 2l+1 j=1 be an o-n famly of egenfunctons of S 2 of egenvalue l(l + 1), that are also egenfunctons of T k. Then for any contnuous functon a on S 2, we have 1 2l + 1 2l+1 j=1 M a(x) ψ (l) j (x) 2 dvol(x) M x)) a(x)dvol(x) 2 0. l