Extremal process associated with 2D discrete Gaussian Free Field

Transcription

1 Extremal process associated with 2D discrete Gaussian Free Field Marek Biskup (UCLA) Based on joint work with O. Louidor

2 Plan Prelude about random fields blame Eviatar! DGFF: definitions, level sets, maximum Extremal point process Some proofs all roads lead back to UCLA Conformal invariance rediscovered Connection to Liouville Quantum Gravity

3 Prelude Stochastic processes vs random fields A stochastic process: {X t } where t = time (t N or t R + ) Simple symmetric random walk in d = 1 and d = 2 10,000 steps 682,613 steps, box Questions: Scaling limit? Occupation time measure?

4 Prelude Stochastic processes vs random fields A random field: {φ x } where x = space (x Z d or x R d ) Linear b.c. Parabolic b.c. Questions: Asymptotic shape, correlations, fluctuations? Refined questions: Local structure, level sets, extreme values?

5 Random fields in physics Euclidean theory only General form of law: Z 1 e H(φ) Dφ where Z = e H(φ) Dφ = normalization H(φ) = Hamiltonian (energy of φ : R d R) Dφ = x ν(dφ x ) Examples: White noise: H = 0, ν standard normal GFF: H(φ) = 1 2 φ(x) 2 dx, ν = Lebesgue φ 4 -theory: ν = Lebesgue H(φ) = 1 φ(x) 2 dx + λ φ(x) 4 dx 2 Note: (most of the time) ill defined! (no product Lebesgue measure)

6 Discrete Gaussian Free Field DGFF for short Field φ : Z d R Gibbs measure: for Λ Z d finite, φ = boundary condition µ φ Λ (dφ) := 1 Z φ Λ e H Λ(φ) dφ x δ φ x (dφ x ) x Λ x Λ Hamiltonian (B(Λ) := n.n. edges incident with Λ) H Λ (φ) := κ 2 (φ x φ y ) 2 x,y B(Λ) We will choose κ = 1 2d and φ x = 0 for all x (zero b.c.) Model of: crystal deformations, random interface,...

7 Relation to simple random walk Alternative definitions Green function: For X = SRW, where τ := first exit time from Λ ( τ 1 ) G Λ (x,y) := E x 1 {Xk =y} k=0 DGFF on Λ = Gaussian process {φ x : x Z d } with E (φ x ) = 0 and E (φ x φ y ) = G Λ (x,y) Note: Zero values outside Λ Alternative def s: Gaussian Hilbert spaces, Langevin dynamics,...

8 Why 2D? V N = N N box, G N = G VN For x with dist(x,vn c ) > δn, N if d = 1 Var(φ x ) = G N (x,x) logn if d = 2 1 if d 3 In d = 2: where G N (x,y) = g logn a(x,y) + o(1) N 1 a(x,y) = g log x y + c 0 + o(1) x y 1 g := 2 π (In physics, g := 1 2π ) The d = 2 model is asymptotically scale invariant: In fact: conformally invariant G 2N (2x,2y) = G N (x,y) + o(1)

9 DGFF on square Uniform color system

10 DGFF on square Emphasizing the extreme values

11 Level set (fractal) geometry Some known facts O(1)-level sets: SLE 4 curves (Schramm & Sheffield) O(log N)-level sets: Hausdorff dimension (Daviaud) { x DN : φ x 2 g γ logn }, 0 < γ < 1 has N 2(1 γ 2 )+o(1) vertices. Note: Maximum order log N!

12 Point of focus Extreme values of DGFF Main goal: Describe statistics of extreme values of φ as N Question of interest: N.B.: Continuum GFF not a function! What s the role of conformal invariance? Surprise connections: Invariant measures for independent particle systems Liouville Quantum Gravity, multiplicative cascades,...

13 Further known facts Absolute maximum Setting and notation: V N := (0,N) 2 Z 2 M N := max x D N φ x and m N := EM N Leading scale (Bolthausen, Deuschel & Giacomin): m N 2 g logn (hard to simulate) Tightness for a subsequence (Bolthausen, Deuschel & Zeitouni): 2E M N m N m 2N m N Full tightness (Bramson & Zeitouni): EM N = 2 g logn 3 4 g log logn + O(1) Convergence in law (Bramson, Ding & Zeitouni)

14 Some known facts Extremal process tightness Ding & Zeitouni, Ding established extremal process tightness: Extreme level set: Γ N (t) := {x V N : φ x m N t} c,c (0, ) : lim liminf P( e cλ Γ N (λ ) e Cλ ) = 1 λ N and c > 0 s.t. ( ) lim limsup P x,y Γ N (c log logr) : r x y N/r = 0 r N Generalizes to bounded and open domains

15 Domains we will consider Class of domains: D = {D C: bounded, open, D = finite # of pieces } Discretized as D N := {x Z 2 : dist(x/n,d c ) > 1 / N }. Really need: Weak convergence of harmonic measures (discrete to continuous)

16 Setup for extreme order theory Extremal point process Full process: Measure η N on D R η N := x D N δ x/n δ φx m N Problem: Values in each peak strongly correlated Local maxima only: Λ r (x) := {z Z 2 : z x r} η N,r := x D N 1 {φx =max z Λr (x) φ z } δ x/n δ φx m N Including clusters: (work in progress) η N,r := x D N 1 {φx =max z Λr (x) φ z } δ x/n δ φx m N δ {φx φ z : z Z d }

17 Main result Convergence to Cox process Recall G N (x,x) = g logn + O(1) Theorem (B.-Louidor 2013, 2014) There is a random Borel measure Z D on D with 0 < Z D (D) < a.s. such that for any r N and N/r N, ( ) law η N,rN PPP Z D (dx) e αφ dφ N where α := 2/ g = 2π.

18 Corollaries Asymptotic law of maximum: Setting Z := Z D (D), P ( M N m N + t ) N E ( e α 1 Ze αt ) N.B.: Laplace transform of Z Joint law position/value: A D open, Ẑ (A) = Z D (A)/Z D (D) ( ) P M N m N + t, N 1 argmax φ A E( ) Ẑ (A)e α 1 Ze αt N In fact: Key steps of the proof

19 Proof of Theorem Invariance under Dysonization Idea: As {η N,rN : N 1} tight, extract a converging subsequence and characterize its distribution uniquely. Abbreviate: η,f := η(dx,dφ)f (x,φ) Proposition (Distributional invariance) Suppose η := a weak-limit point of some {η Nk,r Nk }. Then for any f : D R [0, ) continuous, compact support, where E ( e η,f ) = E ( e η,f t ), t > 0, f t (x,φ) := logee f (x,φ+b t α 2 t) with B t := standard Brownian motion.

20 Proposition explained We may write η = δ (xi,φ i ) i 1 Let {B (i) t } := i.i.d. standard Brownian motions. Set η t := δ (xi,φ i +B (i) t α 2 i 1 t) Well defined as t Γ N (t) grows only exponentially. Then E ( e η t,f ) = E ( e η,f t ) and so Proposition says η t law = η, t > 0 i.e., η invariant under a Dysonization of its points

21 Proof of Proposition (part 1) Gaussian interpolation: φ,φ law = φ, independent φ law = ( 1 t ) 1/2φ g logn }{{} main term For x = large r-local maximum of φ : ( + t ) 1/2 φ g logn }{{} perturbation ( 1 t ) 1/2φ g logn x = φ x 1 t 2 g logn φ x + o(1) = φ x t m N 2 g logn + o(1) = φ x α 2 t + o(1)

22 Proof of Proposition (part 2) Concerning φ, abbreviate ( φ x := Properties of Green function: t g logn ) 1/2 φ x Cov ( φ x, φ y ) {t + o(1), if x y r nearly constant = o(1), if x y N/r nearly independent So we conclude: The law of { φ x : x = large local min. of φ } is asymptotically that of independent B.M. s

23 Proof of Theorem continued Invariant laws for independent particle systems The question of what limit process we get has been reduced to: Problem: Characterize point processes on D R are invariant under independent Dysonization (x,φ) (x,φ + B t α ) 2 t of (the field coordinate of) its points Easy to check: PPP(ν(dx) e αφ dφ) okay for any ν (even random) Any other solutions?

24 Liggett s folk theorem Setting: Markov chain on (nice space) X w/ transition kernel P System of particles evolving independently by P I := loc. finite invariant measures on particle systems Theorem (Liggett 1977) Assume uniform dispersivity property: sup x X P n( x,c ) n 0 C X compact Then each µ I takes the form PPP(M(dx)), where M is a random measure satisfying MP law = M

25 Liggett s 1977 derivation For t > 0 define Markov kernel P on D R by (Pg)(x,φ) := E 0 g ( x,φ + B t α 2 t) Set g(x,φ) := e f (x,φ) for f 0 continuous with compact support. Proposition implies where E ( e η,f ) = E ( e η,f (n) ) f (n) (x,φ) = log(p n e f )(x,φ) P has uniform dispersivity property and so P n e f 1 uniformly on D R. Expanding the log, f (n) 1 P n e f as n

26 Liggett s 1977 derivation (continued) Hence But, as P is Markov, E ( e η,f ) = lim n E ( e η,1 Pn e f ) ( ) η,1 P n e f = ηp n,1 e f ( ) shows that {ηp n : n 1} is tight. Along a subsequence and so ηp n k (dx,dφ) law k M(dx,dφ) E ( e η,f ) = E ( e M,1 e f ) i.e., η = PPP(M(dx,dφ)). Clearly, MP law = M

27 Proof of Theorem Key problem II Question: What M can we get in our case? Theorem (Liggett 1977) MP law = M implies MP = M a.s. when P is a kernel of (1) an irreducible, recurrent Markov chain (2) a random walk on a closed abelian group w/o proper closed invariant subset N.B.:(2) covers our case and MP = M a.s. M random mixture of P-invariant laws For our chain Choquet-Deny (or t 0) shows M(dx,dh) = Z D (dx) e αh dh + Z D (dx) dh Tightness of maximum forces Z D = 0 a.s.

28 Proof of Theorem completed Uniqueness of the limit We thus know η Nk,r Nk law η implies η = PPP ( Z D (dx) e αh dh ) for some random Z D albeit possibly depending on {N k }. But for Z := Z D (D), this yields P ( M Nk m Nk + t ) k E ( e α 1 Ze αt ) Hence: law of Z D (D) unique if limit law of maximum unique (and we know this from Bramson & Ding & Zeitouni) Existence of joint limit of maxima in finite number of disjoint subsets of D uniqueness of law of Z D (dx) Details: B.-Louidor (arxiv: )

29 Properties of Z D -measure B.-Louidor (arxiv: ) Theorem The measure Z D satisfies: (1) Z D (A) = 0 a.s. for any Borel A D with Leb(A) = 0 (2) supp(z D ) = D and Z D ( D) = 0 a.s. (3) Z D is non-atomic a.s. Property (3) is only barely true: Conjecture Z D is supported on a set of zero Hausdorff dimension

30 Fancy properties Gibbs-Markov for Z D measure Recall D D yields φ D law = φ D + ϕd, D Fact: ϕd, D law Φ D, D on D where (1) {Φ D, D(x) : x D} mean-zero Gaussian field with Cov ( Φ D, D(x),Φ D, D(y) ) = G D (x,y) G D(x,y) (2) x Φ D, D(x) harmonic on D a.s. continuum Green functions Theorem (Gibbs-Markov property) Suppose D D be such that Leb(D D) = 0. Then Z D (dx) law = e αφd, D (x) Z D(dx)

31 Fancy properties Conformal symmetry Theorem (Conformal symmetry) Suppose f : D f (D) analytic bijection. Then Z f (D) f (dx) law = f (x) 4 Z D (dx) In particular, for D simply connected and rad D (x) conformal radius rad D (x) 4 Z D (dx) is invariant under conformal maps of D. Note: (1) Leb f (dx) = f (x) 2 Leb(dx) and so rad D (x) 2 Leb(dx) is invariant under conformal maps. (2) By GM property it suffices to know law(z D ) for D := unit disc. So this is a statement of universality

32 Unifying scheme? Continuum Gaussian Free Field Continuum GFF := Gaussian on H 1 0 (D) w.r.t. norm f π f 2 2 Formal expression: φ(x) = n 1 Z n f n (x) {f n} ONB Exists only as a linear functional on H 1 0 (D): Derivative martingale: φ(f ) = π Z n f, f n L2 (D) n 1 M (dx) = [ 2Var(φ(x)) φ(x) ] e 2φ(x) 2Var(φ(x)) dx Defined by smooth approximations to h or expansion in ONB (Duplantier, Sheffield, Rhodes, Vargas) KPZ relation links M -measure of sets to Lebesgue measure

33 Unifying scheme? Liouville Quantum Gravity/Multiplicative Chaos Liouville Quantum Gravity (LQG): M D (dx) := rad D (x) 2 M (dx) Conjecture There is constant c (0, ) s.t. for all D Z D (dx) law = c M D (dx) Current status: Law of Z D characterized by Gibbs-Markov property shift and dilation symmetry precise upper tails of Z D (A) okay for LQG okay for LQG so far open

34 Liouville Quantum Gravity box

35 THE END