Mandelbrot Cascades and their uses

Transcription

1 Mandelbrot Cascades and their uses Antti Kupiainen joint work with J. Barral, M. Nikula, E. Saksman, C. Webb IAS November

2 Random multifractal measures Mandelbrot Cascades are a class of random measures on R n with non-trivial multifractal properties. Cascade measure is a Borel measure µ(dx) = µ(dx; ω) on x R n, depending on ω Ω, a probability space. µ has nontrivial scaling properties: for a ball B r E µ(b r ) p r α(p) as r 0 with α(p) a quadratic polynomial.

3 Gibbs measures A one parameter family of cascade measures is given by µ β (dx) = e βφ(x) dx β [0, ) "inverse temperature". φ(x) = φ(x, ω) is a logarithmically correlated random field E φ(x)φ(x ) log 1 x x as x x 0 Proper definition requires a limiting process: µ β is not continuous w.r.t. Lebesgue measure. In 2d φ(x) is (a version) of the Gaussian Free Field, in 1d the 1/f noise.

4 Phase transition These measures exhibit a phase transition: β c s.t. For β β c, µ β is continuous, singular w.r.t. Lebesgue For β > β c, µ β is atomic They provide simple models of freezing transition believed to occur in (spin) glasses. They have also been used to shed light on The KPZ relation between dimensions of fractals in Euclidean and random geometry or more conjecturally critical exponents on regular and random surfaces (Duplantier and Sheffield) Random fractal plane curves via conformal welding (Astala, Jones, A.K. and Saksman; Sheffield)

5 β < β c Rhodes and Vargas (2013)

6 β > β c Rhodes and Vargas (2013)

7 Log correlated fields Def. Logarithmically correlated random field φ in R d : with g continuous. Eφ(x)φ(x ) = log x x 1 + g(x, y) 2d free field with covariance ( + 1) 1 1/f noise x [0, 1], α n, β n i.i.d. N(0, 1): φ(x) = n=1 1 n (α n cos 2πnx + β n sin 2πnx) Eφ(x)φ(x ) = log z z 1, z = e 2πix

8 Decomposition to scales Log correlated fields may be decomposed in scales φ(x) = φ n (x) n=0 φ n independent, fluctuations at scale 2 n E φ n (x)φ n (x ) = g n (2 n x, 2 n x ) g n (x, y) smooth, fast decay in x y Define also a regularized field with short distance cutoff 2 N N φ N (x) := φ n (x) n=0

9 Hierarchical field Let D be the set of dyadic intervals I [0, 1] Let {V I } I D be i.i.d. N(0, 1) and set φ(x) = V I = I x φ n (x) n=0 where φ n (x) = V I for the unique I s.t. I = 2 n and x I. Then Eφ(x)φ(x ) = 1 = 1 + log 2 d(x, x ) 1 I x,x d(x, x ) length of shortest dyadic interval I x, x

10 Binary trees Dyadic intervals in [0, 1] binary trees Σ = N=0 Σ N Σ N = {0, 1} N lists edges (ancestors) of level N σ = σ 0 σ 1... σ n 1 interval I σ = 2 n

11 Directed polymer and branching random walk On each edge σ of the tree random weights V σ The cutoff 2 N field φ N (x) is constant φ N (σ) on the interval corresponding to σ = σ 0 σ 1... σ N : φ N (σ) = V σ0 + V σ0 σ V σ0 σ 1...σ N Think of φ N (σ) as the energy of the directed polymer i.e. a path on the tree of length N from the root to σ We can also think of φ N (σ) as a branching random walk: at time N there are 2 N particles σ at positions φ N (σ)

12 Multiplicative chaos Let µ β,n (dx) := e βφ N(x) dx Kahane s multiplicative chaos is the random measure ν β = lim N z Nµ β,n whenever the limit exists for a (deterministic) constant z N. Density of µ β,n is a product of independent random variables e βφ N(x) = N n=0 e βφn(x)

13 Mandelbrot cascade For hierarchical field this measure is the Mandelbrot cascade (1973) It is the Gibbs measure of the directed polymer (Derrida-Spohn 1986): P(path) e βφ N(σ)

14 Martingale Let F N be the σ-algebra generated by {φ n } n N. Since φ N (x) = φ N (x) + φ N 1 (x) E(e βφ N(x) F N 1 ) = (Ee βφ N(x) )e βφ N 1(x) Normalizing the measure as (i.e. Wick ordering) we obtain In particular total mass is a martingale: ν β,n := e βφ N(x) Ee βφ N(x) dx E(ν β,n F N 1 ) = ν β,n 1. M N := ν β,n ([0, 1]) E(M N F N 1 ) = M N 1, EM N = 1.

15 Uniform Integrability M N is a positive martingale = it converges a.s. to M 0. To show M > 0 need uniform integrability, e.g. that EM p N stays bounded for some p > 1. Kahane (1985) showed there exists a critical value β c so that M N is bounded in L p for some p > 1 if and only if β < β c. In hierarchical model this is very easy to see using the tree structure.

16 Hierarchical Recursion relation M N = 1 2 e βv 1 2 β2 (M (1) N 1 + M(2) N 1 ) with V = N(0, 1), M (i) N 1 independent.

17 Uniform Integrability M N = 1 2 e βv 1 2 β2 (M (1) N 1 + M(2) N 1 ) Let p > 1. Using (a + b) p a p + b p get EM p N ( 1 2 )p e 1 2 (p2 p)β 2 2E M p N 1 Thus, if M N converges in L p then necessarily ( 1 2 )p 1 e 1 2 (p2 p)β 2 1 i.e. β 2 (2 log 2)/p and so, if β 2 log 2, M N can not converge in any L p, p>1. Converse is not much harder. Also, the argument extends to Kahane s log correlated chaos.

18 Phase transition Kahane: β c M > 0 almost surely for β < β c M = 0 almost surely for β β c Moreover lim N ν β,n = ν β almost surely and ν β 0, (singular) continuous for β < β c ν β = 0 for β β c We have also M L p (Ω) for p < (β c /β) 2 and with φ(p) = p ( β β c ) 2 (p p 2 ) Eν(I) p C I φ(p) Is it possible to obtain a nontrivial measure ν β for β β c? Is it continuous? Atomic?

19 Liouville model Find z N s.t. the random variable 1 z N e βφ N(x) dx 0 converges or, equivalently that its Laplace transform, i.e. the partition function of the "Liouville model" F(λ, N) := E e λz 1 N 0 e βφ N (x) dx converges for all λ 0 and is nontrivial. We saw that for β < β c Wick ordering z N = 1/Ee βφ N(x) = e 1 2 β2 log 2 N = e log 2 2 β2 N works. (also, Hoegh-Krohn (1971): β < β c / 2)

20 Hierarchical Recursion relation Consider the total mass of 2 N e βφ N(x) dx i.e. the partition function of the directed polymer Z N = σ Σ N e βφ N(σ). It satisfies the recursion Z N d = e βv (Z (1) N 1 + Z (2) N 1 ), Look at Laplace transform of Z N in the variable λ = e βy, y (, ): G N (y) := Ee e βy Z N For β < β c we saw 2 N e β2n Z N converges i.e. G N (y + c β N) has a limit as N if c β = 1 2 β + log 2/β.

21 Recursion relation imply G N (y) := Ee e βy Z N, Z N d = e βv (Z (1) N 1 + Z (2) N 1 ) G N+1 (y) = E(exp( e β(y+v ) (Z (1) N + Z (2) N ))) = ρ(v)e(exp( e β(y+v) Z N )) 2 dv = ρ(v)g N (y + v) 2 dv. ρ(v) density of V. Continuum limit N by iteration, initial data { G 0 (y) = exp( e βy 0 if y ) 1 if y

22 Traveling wave G N+1 (y) = ρ(v)g N (y + v) 2 dv. G N 0 is a linearly stable solution G N 1 is a linearly unstable solution G N (y) = w c (y cn) traveling wave solutions Given an initial datum G 0, G N tends to a traveling wave with speed c(β) selected by asymptotics of of G 0 at i.e. by the parameter β. Indeed, our recursion is finite time version of the Fischer-Kolmogorov PDE.

23 Asymptotics Recall G N (x) = E exp( e βy Z N ) Following Bramson s analysis for the PDE (1983) one gets Theorem (C.Webb 2011) lim G N(y + m β,n ) g(y) N with (β c = 2 log 2) βn m β,n = β c N 1 2β c log N β c N 3 2β c log N if β < β c if β = β c if β > β c Freezing: g(y) is independent of β for β β c Transition from typical to extreme configurations dominating the measure.

24 Convergence of the total mass This gives the desired normalization since as N : E exp( e β(y+m β,n) Z N ) = G N (y + m β,n ) g(y) Hence e βm β,n Z N z β as N in distribution. In particular at the critical point this becomes N e βφ N(x) Ee βφ N(x) dx z β c i.e. the martingale is renormalized by N 1 2. Similar results by Aïdekon & Shi, Madaule (2012).

25 Critical point and low temperature Consequences of N 1 2 (J. Barral, A.K, M. Nikula, E. Saksman, C. Webb): ν βc is a.s. continuous, Hausdorff dimension zero Logarithmic modulus of continuity: for γ < 1 2, almost surely ν βc (I) C(ω) log I γ Consequence of freezing (Barral, Rhodes, Vargas): ν β purely atomic for β > β c.

26 Law of the low temperature measures Recall the (renormalized) total mass has the law E exp( e βy z β ) = g(y) β β c. Put t = e βy. Then t β c β = e βcy and thus E exp( tz β ) = E exp( t β c β z βc ) β β c. Let for α (0, 1) L α (s), s 0, be the stable Lévy process independent on z βc. Then Ee tlα(s) = e stα. z β d = L β c β (z βc )

27 Law of the low temperature measures Extends to measures (Barral, Rhodes and Vargas): L βc β ν β ([0, t]) d = L βc β (ν βc ([0, t])) for all t [0, 1] pure jump process = ν β, β > β c, a.s. purely atomic. The critical measure determines the low temperature one.

28 Exponential of the Free field In multiplicative chaos the renormalization group becomes non-local. However much can be done using convexity and the fact that the covariances of cascade and chaos are comparable. β < β c : Martingale normalization gives nontrivial limit (Kahane, Bacry & Muzy) β = β c : N 1 2 martingale normalization gives continuous measure (Duplantier, Rhodes, Sheffield, Vargas) of zero dimension (Barral, A.K., Nikula, Saksman,Webb). β > β c : Normalization, freezing, atomicity (Madaule, Rhodes and Vargas) β = Let M N be the maximum of φ N (x). Then m N Em N converges in distribution (Bolthausen, Bramson, Zeitouni, Ding)

29 Random Geometry: KPZ KPZ formula relates Hausdorff dimension of fractals in the Euclidean metric and their dimension in a random metric. Define on [0, 1] random metric ρ β (x, y) = ν β ([x, y]) Let K [0, 1] ζ 0 Hausdorff dimension of K w.r.t. Euclidean metric ζ β Hausdorff dimension of K w.r.t. random metric ρ β Then For β β c, ζ 0 = ζ + ( β β c ) 2 ζ(1 ζ) For β > β c, ζ β = βc β ζ β c Duplantier and Sheffield, Benjamini and Schramm: β < β c

30 Random Curves and Surfaces Conformally invariant random plane curves Glueing discs with the random metric ν β produces random curves, loop version of SLE κ(β), κ(β) < 4 if β < β c (Astala, Jones, A.K. Saksman; Sheffield). How about β β c? Random surfaces Riemannian metric e βφ N(z) (dz) 2 on a domain or S 2. Do we get as N a random metric space M β? What is the Hausdorff dimension of M β? (? = 4 for β = 8/3)? Is it a scaling limit of random triangulations weighted with Potts or O(N) models?

31 Critical random band matrices Critical random band matrices Inverse participation rates E H ij 2 = (1 + i j /b) 2 P q = i expected to scale with volume as ψ i q P q N τ(q) with (localized) 0 < τ(q) < q 1 (extended). Let H ij (n) := H ij 1 i j n For b small (small off-diagonal terms) Levitov derived a renormalization group equation for P q (n): P q (n + 1) = d ξp (1) q (n) + (1 ξ)p (2) q (n) ξ certain random variable on [0, 1].

32 Freezing transition Mirlin and Evers used this to compute EP q (N) N τ(q) with τ(q) = 2Γ(q 1) πγ(q 1 2 ) Moreover, studying the tail of the pdf of P q they concluded a transition at q = : τ(q) = τ(q), q q, τ(q) = αq, q q Looks like a freezing transition as in cascade with logarithmic corrections at q q (Fyodorov). Real challenge is to justify the RG!

33 Characteristic polynomial Characteristic polynomial of N N unitary matrix where x [0, 1], Then log p N (x) = 1 2 p N (x) := det(1 e 2πix U N ) (e 2πinx trun n + e 2πinx tru n n=1 N ) Diaconis and Shahshahani: if U N is CUE then for any M { ntru n N } n M { 1 2 (a n + ib n )} n M as N with a n, b n i.i.d. N(0, 1). Thus, formally 2 log p N (x) the 1/f noise. n=1 1 n (a n cos 2πnx + b n sin 2πnx)

34 Characteristic polynomial Does the limit lim z N p N (x) β dx = ν β (dx) N exist? Can it be realized as a martingale in N? (Bourgade, Hughes, Nikeghbali, Yor, showed for fixed x) Does it exhibit a freezing transition? Applications to ζ-function: (Fyodorov and Keating)

35 Conformal Welding Conformal welding gives a correspondence between: Closed curves in Ĉ Homeomorphisms φ : S 1 S 1 Jordan curve Γ Ĉ splits plane Ĉ to inside R and outside R c. Riemann mappings f + : D R and f : D c R c f and f + extend continuously to S 1 = D = D c = φ = (f + ) 1 f : S 1 S 1 Homeomorphism Welding problem: invert this: Given φ : S 1 S 1, find Γ and f ±.

36 Continuity Continuity follows from ν βc (I σ ) = d e βcφ N(σ) z βc, for σ Σ n and n 1 2 and a tail estimate for z βc. e βcφ N(σ) z βc σ Σ n

37 Proof of KPZ For the upper bound need to control 1-point functions E (ρ β (x, y)) s x y φ(s) where the multi-fractal exponent is explicit: φ(s) = s ( β ) 2 (s s 2 ) β c For the lower bound need to estimate the 2-point function E (dν β (x)dν β (y)) using hierarchical structure and scale invariance. In low temperatures the Levy process induces a natural scaling.