Quantum ergodicity Nalini Anantharaman Universit e de Strasbourg - - PowerPoint PPT Presentation

QE on manifolds QE on discrete graphs Perspectives on random matrices

Quantum ergodicity

Nalini Anantharaman

Universit´ e de Strasbourg

24 mars 2016

QE on manifolds QE on discrete graphs Perspectives on random matrices

Quantum ergodicity on manifolds (comparing < 0 curvature, > 0 curvature and 0 curvature). QE on large graphs.

QE on manifolds QE on discrete graphs Perspectives on random matrices

QE on manifolds QE on discrete graphs Perspectives on random matrices

Figure: A few eigenfunctions of the Bunimovich billiard (Heller, 89).

QE on manifolds QE on discrete graphs Perspectives on random matrices

A survey of QE on manifolds

M a compact riemannian manifold, of dimension d. ∆ψk = −λkψk ψkL2(M) = 1 in the limit λk − → +∞. We study the weak limits of the probability measures on M, |ψk(x)|2dVol(x) λk − → +∞.

QE on manifolds QE on discrete graphs Perspectives on random matrices

This question is linked with the ergodic theory for the geodesic flow / billiard flow. Hence the name quantum ergodicity.

QE on manifolds QE on discrete graphs Perspectives on random matrices

Let (ψk)k∈N be an orthonormal basis of L2(M), with −∆ψk = λkψk, λk ≤ λk+1. QE theorem (simplified) : Theorem (Shnirelman 74, Zelditch 85, Colin de Verdi` ere 85) Assume that the action of the geodesic flow is ergodic for the Liouville measure. Let a ∈ C 0(M). Then 1 N(λ)

• λk≤λ
• M

a(x)|ψk(x)|2dVol(x) −

• M

a(x)dVol(x)

• 2

− →

λ− →∞ 0.

QE on manifolds QE on discrete graphs Perspectives on random matrices

Equivalently, there exists a subset S ⊂ N of density 1, such that

• M

a(x)|ψk(x)|2dVol(x)

k∈S

− − − − − →

k− →+∞

• M

a(x)dVol(x).

QE on manifolds QE on discrete graphs Perspectives on random matrices

Equivalently, there exists a subset S ⊂ N of density 1, such that

• M

a(x)|ψk(x)|2dVol(x)

k∈S

− − − − − →

k− →+∞

• M

a(x)dVol(x). Equivalently, |ψk(x)|2dVol(x)

k∈S

− − − − − →

k− →+∞ dVol(x)

in the weak topology.

QE on manifolds QE on discrete graphs Perspectives on random matrices

The full statement uses analysis on phase space, i.e. T ∗M = {(x, ξ), x ∈ M, ξ ∈ T ∗

x M}.

For a = a(x, ξ) a “reasonable” function on T ∗M, we can define an

• perator on L2(M),

a(x, Dx) (Dx = 1 i ∂x) Say a ∈ S0(T ∗M) if a is smooth and 0-homogeneous in ξ (i.e. a is a smooth fn on the sphere bundle SM).

QE on manifolds QE on discrete graphs Perspectives on random matrices

−∆ψk = λkψk, λk ≤ λk+1. For a ∈ S0(T ∗M), we consider ψk, a(x, Dx)ψkL2(M).

QE on manifolds QE on discrete graphs Perspectives on random matrices

−∆ψk = λkψk, λk ≤ λk+1. For a ∈ S0(T ∗M), we consider ψk, a(x, Dx)ψkL2(M). This amounts to

• M a(x)|ψk(x)|2dVol(x) if a = a(x).

QE on manifolds QE on discrete graphs Perspectives on random matrices

Let (ψk)k∈N be an orthonormal basis of L2(M), with −∆ψk = λkψk, λk ≤ λk+1. QE theorem : Theorem (Shnirelman, Zelditch, Colin de Verdi` ere) Assume that the action of the geodesic flow is ergodic for the Liouville measure. Let a(x, ξ) ∈ S0(T ∗M). Then 1 N(λ)

• λk≤λ
• ψk, a(x, Dx)ψkL2(M) −
• |ξ|=1

a(x, ξ)dxdξ

• 2

− → 0.

QE on manifolds QE on discrete graphs Perspectives on random matrices

Recall : the geodesic flow φt : SM − → SM (t ∈ R) is the flow generated by the vector field

ξ |ξ| · ∂x.

Ergodicity of the geodesic flow means that the only φt-invariant L2 functions on SM are the constant functions. Equivalently : the only L2-functions a on SM such that

ξ |ξ| · ∂xa = 0 are the constant functions.

QE on manifolds QE on discrete graphs Perspectives on random matrices

Idea of proof.

For any bounded operator K on L2(M), define the quantum variance Varλ(K) = 1 N(λ)

• λk≤λ
• ψk, KψkL2(M)
• 2 .

QE on manifolds QE on discrete graphs Perspectives on random matrices

Idea of proof.

For any bounded operator K on L2(M), define the quantum variance Varλ(K) = 1 N(λ)

• λk≤λ
• ψk, KψkL2(M)
• 2 .

The proof start from the trivial observation that Varλ([f (∆), K]) = 0 for any K, for any function f . [f (∆), K] = f (∆)K − Kf (∆).

QE on manifolds QE on discrete graphs Perspectives on random matrices

The proof start from the trivial observation that Varλ([ √ −∆, K]) = 0 for any K. In addition, if K = a(x, Dx) is a pseudodifferential operator with a ∈ S0(T ∗M), then [ √ −∆, a(x, Dx)] = b(x, Dx) where b(x, ξ) = ξ |ξ| · ∂xa

• (x, ξ) + r(x, ξ)

where r is −1-homogeneous in ξ and

ξ |ξ| · ∂x is the derivative along

the geodesic flow.

QE on manifolds QE on discrete graphs Perspectives on random matrices

This implies that Varλ

• ( ξ

|ξ| · ∂xa)(x, Dx)

• −

→

λ− →+∞ 0.

QE on manifolds QE on discrete graphs Perspectives on random matrices

This implies that Varλ

• ( ξ

|ξ| · ∂xa)(x, Dx)

• −

→

λ− →+∞ 0.

In addition, for any a, Varλ(a(x, Dx)) ≤ C

• |ξ|=1

|a(x, ξ)|2dxdξ.

QE on manifolds QE on discrete graphs Perspectives on random matrices

This implies that Varλ

• ( ξ

|ξ| · ∂xa)(x, Dx)

• −

→

λ− →+∞ 0.

In addition, for any a, Varλ(a(x, Dx)) ≤ C

• |ξ|=1

|a(x, ξ)|2dxdξ. If the geodesic flow is ergodic, this implies Varλ (a(x, Dx)) − →

λ− →+∞ 0

if a has zero mean.

QE on manifolds QE on discrete graphs Perspectives on random matrices

[Arnd B¨ acker]

QE on manifolds QE on discrete graphs Perspectives on random matrices

QE on manifolds QE on discrete graphs Perspectives on random matrices

QUE conjecture : Conjecture (Rudnick, Sarnak 94) On a negatively curved manifold, we have convergence of the whole sequence : ψk, a(x, Dx)ψkL2(M) − →

• (x,ξ)∈SM a(x, ξ)dxdξ.

QE on manifolds QE on discrete graphs Perspectives on random matrices

QUE conjecture : Conjecture (Rudnick, Sarnak 94) On a negatively curved manifold, we have convergence of the whole sequence : ψk, a(x, Dx)ψkL2(M) − →

• (x,ξ)∈SM a(x, ξ)dxdξ.

Proven by E. Lindenstrauss in the special case of arithmetic congruence surfaces, for joint eigenfunctions of the Laplacian and the Hecke operators.

QE on manifolds QE on discrete graphs Perspectives on random matrices

A-Nonnenmacher (06) proved a weaker statement valid in greater generality. Let M have negative curvature. Assume ψk, a(x, Dx)ψkL2(M) − →

• (x,ξ)∈SM

a(x, ξ)dµ(x, ξ) Then µ must have positive Kolmogorov-Sinai entropy. For constant negative curvature, our result implies that the support

• f µ has dimension ≥ d = dim M.

QE on manifolds QE on discrete graphs Perspectives on random matrices

Other geometries

The sphere

QE on manifolds QE on discrete graphs Perspectives on random matrices

The flat torus, Rd/2πZd

(Jakobson-Bourgain 97, Jaffard 90, A-Fermanian-Maci` a, A-Maci` a 2012) It’s not possible for a sequence of eigenfunctions to concentrate on a closed geodesic. φλ(x) =

• k∈Zd,|k|2=λ

ckeik.x. Assume |φλ(x)|2dx − →

λ− →+∞ ν(dx)

Then ν is absolutely continuous, i.e. it has a density ν(dx) = ρ(x)dx. Moreover, for any non-empty open Ω ⊂ Td, there exists c(Ω) > 0 (independent on the sequence φλ) such that ν(Ω) ≥ c(Ω).

QE on manifolds QE on discrete graphs Perspectives on random matrices

QE on discrete graphs

Since the 90s there has been the idea of using graphs as a testing ground/toy model for quantum chaos. Smilansky, Kottos, Elon,... Keating, Berkolaiko, Winn, Piotet, Gnutzmann Marklof...

QE on manifolds QE on discrete graphs Perspectives on random matrices

Here we focus on the case of large regular (discrete) graphs. Let G = (V , E) be a (q + 1)-regular graph. Discrete laplacian : f : V − → C, ∆f (x) =

• y∼x

(f (y) − f (x)) =

• y∼x

f (y) − (q + 1)f (x). ∆ = A − (q + 1)I

QE on manifolds QE on discrete graphs Perspectives on random matrices

Sp(A) ⊂ [−(q + 1), q + 1] Let |V | = N. We look at the limit N − → +∞.

QE on manifolds QE on discrete graphs Perspectives on random matrices

Sp(A) ⊂ [−(q + 1), q + 1] Let |V | = N. We look at the limit N − → +∞. We assume that GN has “few” short loops (= converges to a tree in the sense of Benjamini-Schramm).

QE on manifolds QE on discrete graphs Perspectives on random matrices

Sp(A) ⊂ [−(q + 1), q + 1] Let |V | = N. We look at the limit N − → +∞. We assume that GN has “few” short loops (= converges to a tree in the sense of Benjamini-Schramm). This implies convergence of the spectral measure (Kesten-McKay) 1 N ♯{i, λi ∈ I} − →

N− →+∞

• I

ρ(λ)dλ for any interval I. ρ is a completely explicit density, supported in (−2√q, 2√q)

QE on manifolds QE on discrete graphs Perspectives on random matrices

Theorem (A-Le Masson, 2013) Assume that GN has “few” short loops and that it forms an expander family = uniform spectral gap for A. Let (φ(N)

i

)N

i=1 be an ONB of eigenfunctions of the laplacian on GN.

Let a = aN : VN − → C be such that |a(x)| ≤ 1 for all x ∈ VN. Then lim

N− →+∞

1 N

N

• i=1
• x∈VN

a(x)|φ(N)

i

(x)|2 − a

• 2

= 0. a = 1 N

• x∈VN

a(x)

QE on manifolds QE on discrete graphs Perspectives on random matrices

Theorem (Brooks-Lindenstrauss 2011) Assume that GN has “few” loops of length ≤ c log N. For ǫ > 0, there exists δ > 0 s.t. for every eigenfunction φ, B ⊂ VN,

• x∈B

|φ(x)|2 ≥ ǫ = ⇒ |B| ≥ Nδ. Proof also yields that φ∞ ≤ | log N|−1/4.

QE on manifolds QE on discrete graphs Perspectives on random matrices

Examples

Deterministic examples : the Ramanujan graphs of Lubotzky-Phillips-Sarnak 1988 (arithmetic quotients of the q-adic symmetric space PGL(2, Qq)/PGL(2, Zq)). Cayley graphs of SL2(Z/pZ), p ranges over the primes, (Bourgain-Gamburd, based on Helfgott 2005)

QE on manifolds QE on discrete graphs Perspectives on random matrices

Random regular graphs

QE on manifolds QE on discrete graphs Perspectives on random matrices

Theorem (A-Le Masson, 2013) Assume that GN has “few” short loops and that it forms an expander family = uniform spectral gap for A. Let (φ(N)

i

)N

i=1 be an ONB of eigenfunctions of the laplacian on GN.

Let a = aN : VN − → C be such that |a(x)| ≤ 1 for all x ∈ VN. Then lim

N− →+∞

1 N

N

• i=1
• x∈VN

a(x)|φ(N)

i

(x)|2 − a

• 2

= 0. a = 1 N

• x∈VN

a(x)

• Also works on shrinking spectral intervals
• Applies to random regular graphs. In that case there also exists a

probabilistic proof (Geisinger 2013) in the case where a(x) is chosen independently of GN.

QE on manifolds QE on discrete graphs Perspectives on random matrices

More general version

Theorem (A-Le Masson, 2013) Assume that GN has “few” short loops and that it forms an expander family. Let (φ(N)

i

)N

i=1 be an ONB of eigenfunctions of the laplacian on GN.

Let K = KN : VN × VN − → C be a matrix such that d(x, y) > D = ⇒ K(x, y) = 0. Assume |K(x, y)| ≤ 1. Then lim

N− →+∞

1 N

N

• i=1
• φ(N)

i

, Kφ(N)

i

− Kλi

• 2

= 0.

QE on manifolds QE on discrete graphs Perspectives on random matrices

A formula for Kλ

Kλ = 1 N

• x,y

K(x, y)Φsph,λ(d(x, y)). Φsph,λ is the spherical function of parameter λ on the (q + 1)-regular tree.

QE on manifolds QE on discrete graphs Perspectives on random matrices

A formula for Kλ

Kλ = 1 N

• x,y

K(x, y)Φsph,λ(d(x, y)). Φsph,λ is the spherical function of parameter λ on the (q + 1)-regular tree. Φλ(d) = q−d/2

• 2

q + 1 cos(ds ln q) + q − 1 q + 1 sin((d + 1)s ln q) sin(s ln q)

• if λ = q1/2+is + q1/2−is = 2√q cos(s ln q).

QE on manifolds QE on discrete graphs Perspectives on random matrices

Our result says that φ(N)

i

, Kφ(N)

i

=

• x,y

φ(N)

i

(x)K(x, y)φ(N)

i

(y) ∼ 1 N

• x,y

K(x, y)Φsph,λi(d(x, y)) for most i.

QE on manifolds QE on discrete graphs Perspectives on random matrices

Anisotropic case

We can adapt the proof to Apf (x) =

• y∼x

p(x, y)f (y) for “homogeneous” probability weights (anisotropic random walks, cf Fig` a-Talamanca–Steger).

QE on manifolds QE on discrete graphs Perspectives on random matrices

With the “same” proof, we obtain the same result Theorem (2015) Assume that GN has “few” short loops and that it forms an expander family. Let (φ(N)

i

)N

i=1 be an ONB of eigenfunctions of Ap on GN.

Let K = KN : VN × VN − → C be a matrix such that d(x, y) > D = ⇒ K(x, y) = 0. Assume |K(x, y)| ≤ 1. Then lim

N− →+∞

1 N

N

• i=1
• φ(N)

i

, Kφ(N)

i

− Kλi

• 2

= 0.

QE on manifolds QE on discrete graphs Perspectives on random matrices

Kλ = 1 N

• x,y

K(x, y)ℑm Gλ+i0(x, y) ℑm Gλ+i0(x, x). (the Green function of the infinite (q + 1)-regular tree)

QE on manifolds QE on discrete graphs Perspectives on random matrices

What kind of graphs / operators would we like to deal with next ?

∆ + V (x) on a regular graph (for instance V (x) random iid, i.e. Anderson model). In progress with Mostafa Sabri. regular graph with non-homogeneous (for instance i.i.d random) weights on the edges large “quantum” graphs (cf. B. Winn) with arbitrary boundary conditions some non-regular graphs ? e.g. percolation graphs based on regular graphs

QE on manifolds QE on discrete graphs Perspectives on random matrices

Wigner matrices

Hemitian matrices of size N × N, random iid entries. Law is centered, has a density and gaussian tails. Erd¨

• s, Schlein, Yau, Yin, Bourgade (2009...), Tao-Vu : for any

eigenvector φ,

1 φ∞ ≤ N−1/2+ǫφ2 (“full delocalization”, 2009) 2 ∃η, ν > 0 s.t.

B ⊂ {1, . . . , N},

• x∈B

|φ(x)|2 ≥ 1 − η = ⇒ |B| ≥ νN.

3 QUE (2013) : for any fixed k, aN : {1, . . . , N} −

→ [−1, 1],

• x

aN(x)|φ(N)

k

(x)|2 − aN

• ≤ δ|aN|

N with overwhelming probability.

QE on manifolds QE on discrete graphs Perspectives on random matrices

Erd¨

• s-R´

enyi graphs

Random graph with N vertices. Edges are chosen independently with probability p = p(N). The adjacency matrix is then a random N × N symmetric matrix (containing only 0 and 1). Erd¨

• s, Knowles, Yau, Yin 2013 : if pN ≫ (ln N)C, then with high

probability there is full delocalization of all eigenvectors : φ∞ ≤ C (ln N)c √ N φ2.

QE on manifolds QE on discrete graphs Perspectives on random matrices

Random regular graphs with dN − → +∞.

Dumitriu-Pal, Tran-Vu-Wang, Geisinger. Bauerschmidt-Knowles-Yau 2015 : if dN ≥ (log N)4, then with proba ≥ 1 − e−ξ log ξ all eigenvectors have φ∞ ≤ C ξ √ N φ2. + QUE result