Extrema of discrete 2D Gaussian Free Field and Liouville quantum gravity

Transcription

1 Extrema of discrete 2D Gaussian Free Field and Liouville quantum gravity Marek Biskup (UCLA) Joint work with Oren Louidor (Technion, Haifa)

2 Discrete Gaussian Free Field (DGFF) D R d (or C in d = 2) bounded, open, nice boundary D N := {x Z d : x/n D} 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 Definition DGFF := Gaussian process {h x : x Z d } with E (h x ) = 0 and E (h x h y ) = G D N (x,y)

3 Other ways to define same object Hilbert space valued Gaussian: H := {f l 2 (Z d ) : f = 0 on D c N } with f,g := x f (x) g(x) {ϕ n } ONB in H, {Z n } i.i.d. N (0,1) Dynamical equilibrium: h x := D N Z n ϕ n (x) n=1 DGFF = stationary state for Glauber dynamics with transition rule h x Z + 1 2d h y y : y x =1 where Z = independent N (0,1)

4 Other ways to define same object Gibbs-Markov property: D D. Then h D law = h D D, D + ϕ where (1) h D and ϕ D, D independent Gaussian fields (2) x ϕ D, D(x) discrete harmonic on D N

5 Why 2D? For x with dist(x,d c N ) > δn, In d = 2: where N, if d = 1, Var(h x ) = G N (x,x) logn, if d = 2, 1, if d 3. G N (x,y) = g logn a(x,y) + o(1), N 1 a(x,y) = g log x y + O(1), x y 1 g := 2 π (In physics, g = 1 2π ) The model is nearly scale invariant: G 2N (2x,2y) = G N (x,y) + o(1)

6 DGFF on square Uniform color system

7 DGFF on square Skewed color system

8 Some known facts Absolute maximum Setting and notation: D N := (0,N) 2 Z 2 M N := max x D N h x and m N := EM N Leading scale (Bolthausen, Deuschel & Giacomin): m N 2 g logn 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)

9 Some known facts Extreme points Extreme level set: Γ N (t) := {x D N : h x m N t} Extreme point tightness (Ding & Zeitouni): 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

10 Extreme 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 one peak strongly correlated 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

11 Main results Extreme process convergence 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.

12 Corollaries Asymptotic law of maximum: Setting Z := Z D (D), P ( M N m N + t ) N E ( e α 1 Ze αt ) 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) A E( ) Ẑ (A)e α 1 Ze αt N In fact: Key steps of the proof

13 Proof of Theorem Distributional invariance Note: {η N,rN : N 1} tight, can extract converging subsequences Denote 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.

14 Proposition explained We may write η = δ xi,h i i 1 Letting {B (i) t } be independent standard Brownian motions, set η t := i 1 δ xi,h i +B (i) t α 2 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

15 Proof of Proposition I Gaussian interpolation: h,h law = h, independent ( h law = 1 t ) 1/2h ( t ) 1/2 + h g logn g logn Now let x be such that h x m N λ. Then ( 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)

16 Proof of Proposition II Concerning h, denote ( h x := t g logn ) 1/2 h x By properties of G D N Cov( h x, h y ) = we have { t + o(1), o(1), if x y r if x y N/r So we conclude: { h } x : x D L, h x m N λ, h x = max z Λ r (x) h z has asymptotically the law of {B (i) t }.

17 Proof of Theorem Question before the house Question: Characterize the point processes that are invariant under independent Dysonization of the points? x x + B t α 2 t Easy to check: PPP(ν(dx) e αh dh) okay for any ν (even random) Any other solutions?

18 Liggett s 1977 derivation For t > 0 define Markov kernel (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: For C D R compact sup P n( (x,h),c ) 0 x,h n and thus P n e f 1 uniformly. Expanding the log, f (n) 1 P n e f as n

19 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,dh) law k M(dx,dh) E ( e η,f ) = E ( e M,1 e f ) i.e., η = PPP(M(dx,dh)). Clearly, MP law = M

20 Proof of Theorem Question before the house II Question: What M can we get? 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 (2) applies in our case. Note: MP = M means M invariant for the chain. Choquet-Deny (or t 0) show M(dx,dh) = Z D (dx) e αh dh + Z D (dx) dh Tightness of maximum implies Z D = 0 a.s.

21 Proof of Theorem Finishing touches We thus know η Nk,r Nk law η implies η = PPP ( Z D (dx) e αh dh ) for some random Z D. But for Z := Z D (D), this reads P ( M Nk m Nk + t ) k E ( e α 1 Ze αt ) Hence the law of Z D (D) unique if limit law of maximum unique. An enhanced version implies uniqueness of law of Z D (dx).

22 Properties of Z-measure 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

23 Fancy properties Gibbs-Markov Recall D D yields h D law = h D + ϕd, D Fact: ϕd, D law Φ D, D on D where (1) {Φ D, D(x) : x D} Gaussian with E Φ D, D(x) = 0 and 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) Suppose D D be such that Leb(D D) = 0. Then Z D (dx) law = e αφd, D (x) Z D(dx)

24 Fancy properties Conformal invariance (CFI) Theorem (Conformal invariance) Suppose f : D D analytic bijection. Then Z 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 CFI. Consistent w/ zero-haudorff dimension conjecture (2) For D simply connected, suffices to know Z D for D := unit disc. So this is a statement of universality

25 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: h(x) = n 1 Z n ϕ n (x) Exists only as a linear functional on H 1 0 (D): Derivative martingale: h(f ) = π Z n f, ϕ n L2 (D) n 1 M (dx) = [ 2Var(h(x)) h(x) ] e 2h(x) 2Var(h(x)) dx Can be defined by smooth approximations to h or expansion in ONB (Duplantier, Sheffield, Rhodes, Vargas) KPZ relation links M -measure of sets to Lebesgue measure

26 Unifying scheme? Liouville Quantum Gravity For D simply connected, M D (dx) := rad D (x) 2 M (dx) This is the Liouville Quantum Gravity measure constructed in (Duplantier, Sheffield, Rhodes, Vargas) Theorem (B-Louidor, in progress) There is constant c (0, ) s.t. for all D Z D (dx) law = c M D (dx)

27 Full extreme process Recall we were interested in η N := x DN δ x/n δ hx m N A better representation by cluster process on D R R Z2 : η 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 } Theorem (B-Louidor, in progress) There is a measure µ on Z 2 such that (for r N, N/r N ) ( ) law η N,rN PPP Z D (dx) e αh dh µ(dφ) N where α := 2/ g = 2π. Capable of capturing universality w.r.t. short-range perturbations

28 THE END