Extreme values of two-dimensional discrete Gaussian Free Field

Transcription

1 Extreme values of two-dimensional discrete Gaussian Free Field Marek Biskup (UCLA) Joint work with Oren Louidor (Technion, Haifa) Midwest Probability Colloquium, Evanston, October 9-10, 2015

2 My collaborator

3 Plan Today: Existence and conformal properties of the random intensity measure Introduction to 2D DGFF Large-scale properties of extreme values Role and reflections of conformal invariance Tomorrow: Shape of large peaks and the freezing phenomenon Local properties of extrema: shape of large peaks Freezing & Poisson-Dirichlet limit for DGFF-REM

4 1. Intro to DGFF

5 Discrete Gaussian Free Field (DGFF) in lattice approximations of continuum domains Geometric setting: D R d (or C in d = 2) bounded, open, nice boundary D N := {x Z d : x/n D} caution needed!!! G D N := Green function of SRW on D N killed upon exit ( τ 1 GN D (x,y) = E x 1 {Xk =y} k=0 where τ := first exit time from D N )

6 Discrete Gaussian Free Field (DGFF) in lattice approximations of continuum domains Definition (DGFF) DGFF on D N := Gaussian process {h x : x D N } with E(h x ) = 0 and E(h x h y ) = G D N (x,y) N.B. May as well set zero values outside D N

7 Other ways to define same object Gaussian Hilbert space Hilbert space valued Gaussian: Vector space: H N := {f l 2 (Z d ): f = 0 on D c N, x f (x) = 0} Dirichlet inner product: f,g := x f (x) g(x) D N h x := n=1 Z n ϕ n (x) (great for simulations) where {ϕ n } ONB in H N, {Z n } i.i.d. N (0,1) Perfect setting for continuum limit... CGFF (only as random distribution)

8 Other ways to define same object Stationary law for a stochastic dynamics Dynamical equilibrium: DGFF = stationary law for Glauber dynamics with transition rule h x Z + 1 2d h y y : y x =1 where Z = N (0, 1 / 2d ). Similarly for Langevin dynamics ( 1 dh x = 2d h y h x )dt + db x y : y x =1 i.i.d. BM s

9 Other ways to define same object Gibbs measure/gradient model Explicit formula for distribution: P(h A) = 1 { exp 1 Z N A 4d } (h x h y ) 2 dh x δ 0 (dh x ) x,y x D N x D N Buzzwords: Gradient Gibbs measure Harmonic crystal Free Euclidean field theory quadratic Hamiltonian zero b.c. This definition makes it easy to verify... (OVER)

10 Gibbs-Markov property Law of DGFF = finite-volume Gibbs measure characterized by The Gibbs-Markov property: (h D = DGFF for domain D) Assume D D. Then where h D law = h D D, D + ϕ (1) h D and ϕ D, D independent Gaussian fields (2) x ϕ D, D(x) discrete harmonic on D N Proof (sketch): Set ϕ D, D(x) := E ( h D (x) h D (z): z D N D N ) Then check: h D ϕ D, D ϕ D, D and h D ϕ D, D law = h D

11 Why 2D? For x with dist(x,dn c ) > δn, N, if d = 1 scaling limit: Brownian bridge Var(h x ) = G N (x,x) log N, if d = 2 no scaling/thermodynamic limit 1, if d 3 field on Z d exists In d = 2: 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 where g := 2 (In physics, g := 1 π 2π ) The model is asymptotically scale invariant: In fact: conformally invariant G 2N (2x,2y) = G N (x,y) + o(1)

12 DGFF: a sample figure Box 35 35, range of values [ 5,5]

13 DGFF on square Uniform color system

14 DGFF on square Emphasizing the extreme values

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

16 Goal of my talks Limit theory for extreme values of DGFF Main goal: Describe limit law of extreme values of h D Question of interest: What s the role of conformal invariance? N.B.: Continuum GFF not a function! (maximum meaningless) Surprise connections: Invariant measures for independent particle systems Gaussian multiplicative chaos & Liouville Quantum Gravity

17 2. Large-scale properties of extreme values

18 Absolute maximum Setting and notation: D N := (0,N) 2 Z 2 (for now) M N := max x D N h x and m N := EM N Leading scale: (Bolthausen, Deuschel & Giacomin 00) m N 2 g logn Tightness for subsequences: (Bolthausen, Deuschel & Zeitouni 11) 2E M N m N m 2N m N Full tightness: (Bramson & Zeitouni 12) EM N = 2 g logn 3 4 g log logn + O(1) Convergence in law: (Bramson, Ding & Zeitouni 13) M N m N converges in distribution as N

19 Some known facts Extremal process tightness Ding & Zeitouni, Ding established extremal process tightness: Extreme level set: Γ N (t) := {x D N : h x m N t} c,c (0, ): lim liminf P( e ct Γ N (t) e Ct) = 1 t N and c > 0 s.t. ( ) lim limsup P x,y Γ N (c log logr): r x y N/r = 0 r N

20 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 hx m N Problem: Values in each peak strongly correlated Idea: Pick one representative value for each peak e.g. loc. max. Local maxima only: Λ r (x) := {z Z 2 : z x r} η N,r := x D N 1 {hx =max z Λr (x) h z } δ x/n δ hx m N Full extremal process to be discussed tomorrow

21 Main result Scaling limit for extreme local maxima η N,r := x D N 1 {hx =maxz Λr (x) hz } δ x/n δ hx mn Theorem (Convergence to Cox process) 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 αh dh N where α := 2/ g = 2π.

22 Corollaries η N,rN ( ) law PPP Z D (dx) e αh dh N 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 total mass of Z D measure Joint law position/value: A D open, Ẑ(A) = Z D (A)/Z D (D) ( ) P M N m N + t, N 1 argmax h D A E( ) Ẑ(A)e α 1 Ze αt N In fact: Key steps of the proof

23 Proof of Theorem, STEP 1 Key idea: Invariance of limit process under Dysonization {η N,rN : N 1} tight can extract converging subsequences Write η for a limit point. Bracket notation: η,f := η(dx,dh)f (x,h) 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,h) := logee f (x,h+b t α 2 t) with B t := standard Brownian motion.

24 Proposition explained We may write η = δ (xi,h i ) i 1 Let {B (i) t } := i.i.d. standard Brownian motions. Set η t := δ (xi,h 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 ) = E ( η,f e ) int. by parts Proposition True for all (nice) f and so η t law = η, t > 0

25 Proof of Proposition Gaussian interpolation: h,h law = h, independent Denote h law = ( 1 t ) 1/2h g logn }{{} main term ( + t ) 1/2 h g logn }{{} perturbation Θ N,r (λ) := {x D N : h x m N λ, h x = max z Λ r (x) h z} For x Θ N,r (λ) ( large r-local maximum ): ( 1 t ) 1/2h g logn x = h x 1 t 2 g logn h x + o(1) = h x t m N 2 g logn + o(1) = h x α 2 t + o(1)

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

27 Proof of Theorem, STEP 2 Invariant laws for independent particle systems Problem: Characterize point processes on D R that are invariant under independent Dysonization (x,h) (x,h + B t α ) 2 t of (the second coordinate of) its points Easy to check: PPP(ν(dx) e αh dh) okay for any ν (even random) Any other solutions? We should have talked to Tom Liggett!

28 Liggett s folk theorem Setting: Markov chain on (nice space) X w/ transition kernel P System of particles in X evolving independently by P I := loc. finite invariant measures on particle systems Question: Characterize I for a given Markov chain kerner P 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

29 Liggett s 1977 derivation adapted to our setting Fix t > 0 and define Markov kernel P on D R by (Pg)(x,h) := E 0 g ( x,h + B t α 2 t) Set g(x,h) := e f (x,h) for f 0 continuous with compact support. Proposition implies where E ( e η,f ) = E ( e η,f (n) ) f (n) (x,h) = log(p n e f )(x,h) 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

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

31 Proof of Theorem, STEP 2 concluded Proved so far: η law = PPP(M(dx,dh)) with MP law = M Key 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 with no proper closed invariant subsets N.B.: (2) covers our case and MP = M a.s. M random mixture of P-invariant laws For our chain Choquet-Deny s Theorem (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.

32 Proof of Theorem, STEP 3 Uniqueness of the limit We have shown: η Nk,r Nk law η implies η law = 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 thanks to Bramson & Ding & Zeitouni) Existence of joint limit of maxima in finite number of disjoint subsets of D uniqueness of law of Z D (dx)

33 Some literature Details for above derivation: For D := (0,1) 2 : Biskup-Louidor (arxiv: ) For general D: Biskup-Louidor (arxiv: ) Maxima for log-correlated fields: Madaule (arxiv: ), Acosta (arxiv: ) Ding, Roy and Zeitouni (arxiv: )

34 3. Reflections of conformal invariance

35 Random intensity measure: what is it, really? Some talking points At this point: Z D a.s. finite random Borel measure on D Questions: Can one characterize the law of Z D? Or even construct it directly in continuum? How do properties of GFF reflect in it? How does Z D depend on D?

36 Direct consequences of construction 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 Now some more advanced properties...

37 Gibbs-Markov for Z D measure Recall D D yields h D law = h D + ϕd, D Fact: scaling limit 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. Theorem (Gibbs-Markov property) continuum Green functions Suppose D D be such that Leb(D D) = 0. Then Z D (dx) law = e αφd, D(x) Z D(dx) Z D (D D) = 0 a.s. to Φ D, D needn t be defined outside D

38 Conformal invariance 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

39 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) Exists only as a linear functional on H 1 0 (D): {f n} ONB φ(f ) = Z n f, f n L 2 (D) {Z n} i.i.d. N (0,1) n 1 Smooth approximation: φ n (x) := n k=1 Z k f k (x) (depends on choice of basis) Smooth when basis functions smooth

40 Unifying scheme? Kahane s Multiplicative Chaosd Kahane s Multiplicative Chaos for GFF: The random measure M γ n(dx) := e γφ n(x) γ2 2 Var(φ n(x)) dx is a non-negative martingale w.r.t. F n := σ(z 1,...,Z n ). Martingale Convergence Theorem shows M γ (A) := lim n M γ n(a) exists for all Borel A R 2 and defines a loc. finite random measure Key question: For what γ do we get M γ 0? Also: does M γ depend on choice of the basis?

41 Unifying scheme? Derivative Martingale Fact: For above normalization M γ 0 γ < 2 (Kahane) What if γ = 2? Consider derivative martingale: M n(dx) = [ 2Var(φ n (x)) φ n (x) ] e 2φ n(x) 2Var(φ n (x)) dx Hard argument: n limit still exists M (A) := lim n M n(a) (Duplantier, Sheffield, Rhodes, Vargas) and defines a non-trivial, positive random Borel measure Key issue for us: Uniqueness not completely settled

42 Unifying scheme? Liouville Quantum Gravity Critical Liouville Quantum Gravity (LQG): (Our case: γ = 2) M D (dx) := rad D (x) 2 M (dx) rad D (x) = conformal radius of D from x 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) checked for LQG checked for LQG so far open (for LQG) (Biskup-Louidor, arxiv: )

43 Liouville Quantum Gravity box

44 4. Local properties: shape of large peaks

45 Full extreme process Structured description Full process: Measure η N on D R Want instead to keep track of: Position of local maximum η D N := x D N δ x/n δ hx m N Reduced value of the field at local maximum Shape of configuration around local maximum This leads to: Structured extreme process: (recall Λ r (x) := {z Z 2 : z x r}) η D N,r := x D N 1 {hx =max z Λr (x) h z } δ x/n δ hx m N δ {hx h x+z : z Z 2 } Random point measure on D R [0, ) Z2

46 Full extreme process Main limit theorem η D N,r := x D N 1 {hx =maxz Λr (x) hz } δ x/n δ hx mn δ {hx h x+z : z Z 2 } Theorem (Convergence to Cox process) There is a deterministic probability measure ν on [0, ) Z2 that, for Z D as above and any r N and N/r N, ( ) ηn,r D law N PPP Z D (dx) e αh dh ν(dφ) N such where α := 2/ g = 2π. Upshot: Shapes of highest peaks sampled independently from ν!

47 Shape of the peaks What is ν? Pinned DGFF: φ = {φ x : x Z 2 } mean-zero Gaussian with Cov(φ x,φ y ) = a(x) + a(y) a(x y) where a(x) = potential of SRW. Let ν 0 := law of φ Theorem (Cluster law) The measure ν above is given by ( ν( ) := lim ν 0 φ + 2 a r g φ x + 2 ) a(x) 0: x < r g Moreover, a.e. sample from ν has finite level sets. Limit exists by FKG, tightness hard

48 Unstructed extreme process Cluster process convergence η D N := x D N δ x/n δ hx mn Corollary (Cluster process convergence) Consider stochastically independent objects: {(x i,h i ): i N} = sample from PPP(Z D (dx) e αh dh) {φ z (i) : z Z 2 } = independent samples from ν Then η D N law N δ (xi,h i φ (i) i N z Z 2 z ) (locally finite) Analogous limit process: extrema of Branching Brownian Motion Arguin-Bovier-Kistler, Aïdekon-Berestycki-Brunet-Shi

49 Main underlying idea Heuristic argument: Condition DGFF in D N on position/value of maximum Field decomposes: centered pinned field + logarithmic cone The condition on maximum forces: we actually have the negative of the field centered pinned field + logarithmic cone 0 Technical obstacle: The limit law ν is singular w.r.t. ν 0 Theorem There is a constant c (0, ) such that ( ν 0 φ x + 2 ) c a(x) 0: x r, r. g (logr) 1/2 Actual proof: representation using random walk conditioned on 0 Bonus: local limit theorem for absolute maximum

50 5. Freezing & Poisson-Dirichlet limit for DGFF-REM

51 Random Energy Model: DGFF disorder DGFF-REM Gibbs measure: (Carpentier-Le Doussal) µ N D ( ) e βhd N x {x} := Z N (β) where Z N (β) := e βh D N x x D N Daviaud s work implies: all mass sits on a single logarithmic level set µ D N where β c = α!!! ( { x D N : h x β β c β c 2 } ) g logn 1 N Question: More precise info how µ N D distributes mass?

52 Supercritical DGFF-REM Recall: Poisson-Dirichlet law PD(s) works for all s (0,1) Take a sample from PPP(1 {x>0} x 1 s dx) Normalize by total sum and order decreasingly Law on sequences {p i : i N} with p i 0, i p i = 1 Theorem (Poisson-Dirichlet asymptotic) For all β > β c, where z D N µ D N law of {p i } = PD(β c /β) ( {z} ) δz/n (dx) law N {X i } are (conditionally on Z D ) i.i.d.(ẑ D ) with {p i } and {X i } sampled independently p i δ Xi, i N Key fact: Contribution of each cluster collapses to a point Version for overlaps: Arguin and Zindy

53 Freezing Laplace transform of normalizing constant: ( { G N,β (t) := E exp e βt x D N e βh D N } ) x Studied in different context: Derrida-Spohn, Fyodorov-Bouchaud Theorem (Freezing) For each β > β c there is c(β) R such that for all t R, G N,β ( t + mn + c(β) ) N E ( e Z D (D)e αt ) Key point: Limit independent of β should fail for β < β c Signature of Gumbel limit law: Subag-Zeitouni Details for last two sections: paper soon to be posted on arxiv

54 Thank you for attention