LIOUVILLE QUANTUM MULTIFRACTALITY

Transcription

1 LIOUVILLE QUANTUM MULTIFRACTALITY Bertrand Duplantier Institut de Physique Théorique Université Paris-Saclay, France 116TH STATISTICAL MECHANICS CONFERENCE HONOREES: JOHN CARDY & SUSAN COPPERSMITH Rutgers University, Hill Center December 18 20, 2016 Based on joint work with Gaëtan Borot (MPI Bonn), Jérémie Bouttier (ENS-Lyon) Jason Miller (Cambridge), Scott Sheffield (MIT)

2 Random Planar Map & Conformal Map [Courtesy of N. Curien] Left: A random triangulation of the sphere. Right: Conformal map to the sphere. In the continuum scaling limit: Liouville Quantum Gravity A.M. Polyakov 81 (See also J. Teschner s talk)

3 Random Planar Maps & Statistical Models Percolation hulls [Courtesy of N. Curien].

4 LIOUVILLE QG RANDOM MEASURE µ= e γh dz

5 Gaussian Free Field (GFF) [Courtesy of J. Miller] Distribution h with Gaussian weight exp [ 1 2 (h,h) ], and Dirichlet inner product in domain D ( f 1, f 2 ) := (2π) 1 f 1 (z) f 2 (z)dz D = Cov ( ) (h, f 1 ),(h, f 2 )

6 LIOUVILLE QUANTUM MEASURE µ γ := lim ε 0 exp [ γh ε (z) ] ε γ2 /2 dz, where h ε (z) is the GFF average on a circle of radius ε, converges weakly for γ<2 to a random measure, denoted by e γh(z) dz. [Høegh-Krohn 71; Kahane 85; D. & Sheffield 11] For γ=2, the renormalized one, log(1/ε) [ exp [ γh ε (z) ] ε γ2 /2 ] γ=2 dz, converges, as ε 0, to a positive non-atomic random measure. [D., Rhodes, Sheffield, Vargas 14]

7 O(n) model on a Random Planar Map g h A disk triangulationc, and its local weights (α=1 here). α 1 Z l = C u V(C) w(c), w(c)=n L g T 1 h T 2, the sum runs over all configurationsc of a disk of fixed perimeter l; u is an auxiliary weight per vertex, V(C) denotes the number of vertices of the map ofc (volume); T 1, T 2 are the numbers of empty or occupied triangles; the number of loopsl ofc is weighted by n [0,2].

8 Phase diagram and critical loci h supercritical dense dilute subcritical generic Qualitative phase diagram of the O(n) loop model (n [0,2]) on a random map. For u=1, a line of critical points separates the subcritical and supercritical phase. Critical points may be in three different universality classes: generic (pure gravity), dilute and dense. g

9 Random Map Nesting Theorem [Borot, Bouttier, D. 16] Fix(g,h) and n (0,2) such that the model reaches a dilute or dense critical point for the vertex weight u=1. In the ensemble of random pointed disks of volume V and perimeter L, the probability distribution of the numbern of separating loops between the marked point and the boundary behaves as: [ clnv P N. = p V,L=l] (lnv) 2 1 V π c J(p) (sphere), π [ clnv ] P N = 2π p V,L= V 2l c. (lnv) 2 1 V 2π c J(p) (disk), where l>0is fixed, and lnv p, and: ( ) 2 p J(p)= pln + arccot(p) arccos(n/2). n 1+ p 2 with c=1 (dilute), c=1/[1 1 π arccos( n 2 ) ] (dense), which decreases from 2 to 1 as n increases from 0 to 2.

10 Large Deviations Function J(p) p J(p) for n=1 (blue), n= 2 (Ising, green) and n= 3 (3-Potts, orange).

11 The Conformal Loop Ensemble (CLE) [Sheffield 09, Sheffield & Werner 12] The critical O(n)-model on a regular planar lattice is expected to converge in the continuum limit to SLE κ /CLE κ, for n= 2cos ( 4π/κ ) κ (8/3,4], dilute phase, n (0,2], κ [4,8), dense phase, (Loop-erased random walk & spanning trees [Lawler, Schramm, Werner], Ising & percolation [Smirnov], GFF contour lines [Schramm-Sheffield].) The same is expected in the critical dilute or dense phase on a random planar map, the random area measure being in the scaling limit the Liouville quantum measure for γ=min{ κ,4/ κ}.

12 Nesting in the Conformal Loop Ensemble (CLE) ερ ε 1 N z (ε) is the number of nested loops of a CLE κ, κ (8/3,8) surrounding the ball B(z,ε) in the unit disk.

13 Extreme nesting in CLE [Miller, Watson & Wilson 14] LetN z (ε) be the number of loops of a CLE κ, κ (8/3,8) surrounding the ball B(z,ε), and Φ ν the set of points z where dim H Φ ν = 2 γ κ (ν) lim N z(ε)/ln(1/ε)=ν. ε 0 γ κ (ν)=νλ κ(1/ν),ν 0; Λ κ(x) := sup(λx Λ κ (λ)) Λ κ (λ)=ln λ R cos(4π/κ) ( (1 ) ) λ cos π κ κ Moment generating function of the loop log-conformal radius [Cardy & Ziff 02; Kenyon & Wilson 04; Schramm, Sheffield & Wilson 09]

14 Large Deviations Euclidean case: for a ball of radius ε P ( N z νln(1/ε) ε ) =P ( N z νt t ) ε γ κ(ν) = exp[ tγ κ (ν)]. Liouville Quantum Gravity: t := lnε; A := γ 1 lnδ, δ := B(z,ε) µ γ (quantum ball) Conditioned on δ, hence A, perform the convolution P Q (N z A) := 0 P ( N z t ) P(t A)dt, where P(t A) is the probability distribution of the random Euclidean log-radius t, given the quantum log-radius A.

15 Probability Distribution (γ= 8/3) [A=2; 20; 200] AP A ( t) t/a t = lnε, A= γ 1 lnδ, δ= P(t A)= A [ exp 1 ( A aγ t ) ] 2 2πt 3 2t a γ := 2/γ γ/2 µ γ B(z,ε)

16 Large Deviations t = lnε, A= γ 1 logδ (quantum ball), N νlnε=νt,n p lnδ=γpa It implies ν=γpa/t. P Q (N z γpa A) := 0 P ( N z γpa=νt t ) P(t A)dt The asymptotic result above then yields, for A +, ( dt A P Q (N z γpa A) exp (A a γt) 2 ) γ κ (ν)t 0 2πt 3 2t exp[ A Θ(p)] (saddle point at νt constant) Θ(p) is the large deviation function for the loop number around a δ-quantum ball scaling as p log(1/δ).

17 Liouville Quantum Large Deviations In the plane, the Legendre transform gave γ κ (ν)=λ νλ κ (λ), In Liouville Quantum Gravity Θ(p)= Uγ 1 (λ/2) pλ κ (λ), 1 ν = Λ κ(λ) λ. 1 p = Λ κ(λ) Uγ 1 (λ/2), where Uγ 1 (λ/2) := ( ) aγ+ 2 2λ a γ /γ is the inverse KPZ function, with γ=min { κ,4/ κ }, a γ = 2/γ γ/2.

18 Nesting in Liouville Quantum Gravity Theorem [Borot, Bouttier, D. 16] The quantum nesting probability of a CLE κ in a simply connected domain, for the numbern z of loops surrounding a ball centered at z and conditioned to have a given Liouville quantum area δ, has the large deviations form, P Q ( N z cp 2π ln(1/δ) δ ) δ c 2π J(p), δ 0, where c and J are the same as in the combinatorial result obtained for the critical O(n) model in the scaling limit of large random maps.

19 CLE on the Riemann sphere [Kemppainen & Werner 14] Theorem [Borot, Bouttier, D. 16] The nesting probability in CLE κ (Ĉ) between two balls of radius ε 1 and ε 2 and centered at two distinct punctures, has the large deviations form, PĈ[ N (ε 1,ε 2 ) νln(1/(ε 1 ε 2 )) ] (ε 1 ε 2 ) γ κ(ν), ν 0, ε 1,ε 2 0, where γ κ (ν) is the large deviations function of the disk topology. Corollary The nesting probability in CLE κ (Ĉ) between two balls of same radius ε and centered at two distinct punctures, has the large deviations form, PĈ( N (ε,ε) νln(1/ε) ) ε γ κ(ν), ν 0, ε 0, where γ κ (ν) is related to the disk large deviations function by γ κ (2ν)=2γ κ (ν).

20 Riemann sphere Theorem [Borot, Bouttier, D. 16] On the Riemann sphere Ĉ, the large deviations function Θ which governs the quantum nesting probability between two non-overlapping δ-quantum balls, PĈQ (N pln(1/δ) δ) δ Θ(p), δ 0, is related to the similar function Θ for the disk topology by Θ(2p)=2Θ(p), so that ( cp ) PĈQ N δ 0 π ln(1/δ) δ π c J(p), where c and J are the same as before. Perfect matching of LQG results for CLE κ with those for the O(n) model on a random planar map, with the correspondence δ 1/V, with δ 0, V +.

21 HAPPY BIRTHDAYS, JOHN & SUSAN!