Some new estimates on the Liouville heat kernel

Transcription

1 Some new estimates on the Liouville heat kernel Vincent Vargas 1 2 ENS Paris 1 first part in collaboration with: Maillard, Rhodes, Zeitouni 2 second part in collaboration with: David, Kupiainen, Rhodes

2 Outline 1 Gaussian multiplicative chaos theory 2 Set-up and Notations Liouville Brownian motion Liouville heat kernel 3 Off-diagonal bounds on the heat kernel Lower bound on the heat kernel Upper bound on the Liouville heat kernel 4 Liouville quantum gravity on the Riemann sphere Construction Basic properties

3 Notations We equip the two dimensional torus T with: d T standard volume distance and dx volume form the Laplace-Beltrami operator on T p t (x, y) the standard heat kernel of the Brownian motion B on T Recall that: p t (x, y) = 1 T + e λnt e n (x)e n (y) n 1 where (λ n ) n 1 (increasing) eigenvalues and (e n ) n 1 (normalized) eigenvectors: e n = 2πλ n e n, e n (x)dx = 0. T

4 Log-correlated field X Notations: G standard Green function of the Laplacian : G(x, y) = n 1 1 λ n e n (x)e n (y) X GFF on T under P X (expectation E X ): E X 1 [X (x)x (y)] = G(x, y) = ln + + g(x, y) d T (x, y)

5 Gaussian multiplicative chaos (Liouville measure) Gaussian multiplicative chaos associated to X : γ2 γx (x) M γ (dx) = e 2 E[X (x)2] dx. Theorem (Kahane, 1985) M γ can be defined by regularizing the field X and a limit procedure. M γ 0 if and only if γ < 2. If γ < 2, the measure M γ lives almost surely on a set of Hausdorff dimension 2 γ2 2 (the set of thick points).

6 Liouville Brownian motion Framework: Standard Brownian motion B = (B t ) t 0 on T PB x (and E B x ) probability (expectation) of B starting from x. P x y t B (and E x y t B ) law (expectation) of the Brownian bridge (B s ) 0 s t from x to y with lifetime t. Liouville Brownian motion starting from x T formally defined by: db t = e γ 2 X (Bt) db t

7 Liouville heat kernel: definition Liouville Brownian motion starting from x T: B t = B F (t) 1 where F (t) = t 0 γ2 γx (Br ) e 2 EX [X 2 (B r )] dr. Liouville heat kernel p γ t defined for all f by: E x B [f (B t)] = E x B [f (B F (t) 1)] = T f (y)p γ t (x, y) M γ (dy), t > 0

8 Liouville heat kernel: representation and regularity Consider the Hilbert-Schmidt operator: T γ : f G γ (x, y)f (y)m γ (dy) with T G γ (x, y) = G(x, y) T G(z, y)m γ(dz) M γ (T) Let (λ γ,n ) n 1 be the (increasing) eigenvalues of Tγ 1 associated to the eigenvectors (e γ n) n 1. We have: n 1 < +. 1 λ 2 γ,n

9 Liouville heat kernel: representation and regularity We have the following representation: Theorem (Maillard, Rhodes, V., Zeitouni) p γ t (x, y) = 1 M γ (T) + e λγ,nt e γ n(x)e γ n(y). n 1 Furthermore, it is of class C,0,0 (R + T 2 ). If γ < 2 2, it is even of class C,1,1 (R + T 2 ).

10 Speculations and heuristics Watabiki (1993) conjectures that one can construct a metric space (T, d γ ) which is locally monofractal with intrinsic Hausdorff dimension d H (γ) = 1 + γ2 4 + (1 γ 2 ) γ 4 2. The literature on diffusion on fractals suggests that the heat kernel p γ t (x, y) then takes the following form for small t: ( ) p γ t (x, y) C t exp C d γ(x, y) d H(γ)/(d H (γ) 1) t 1 d H (γ) 1

11 Summary of our bounds within these heuristics

12 The lower bound: fixed points Theorem (Maillard, Rhodes, V., Zeitouni) Fix x y. For all η > 0, there exists some random variable T 0 = T 0 (x, y, η) such that for all t T 0, ( p γ t (x, y) exp t 1 ) 1+γ 2 /4 η, P X -a.s.

13 The lower bound: typical points Theorem (Maillard, Rhodes, V., Zeitouni) Conditioned on the Gaussian field X, let x,y be sampled according to the measure M γ (T) 1 M γ. For all η > 0, there exists some random variable T 0, such that for all t T 0, ( ) p γ t (x, y) exp t 1 ν(γ) η, P X -a.s., where 1 + γ2 4 γ 2 [0, 8/3] ( ) 1 ν(γ) = 1 + γ 2 γ2 4 1 γ2 4 γ 2 (8/3, 3] 4 γ 2 γ 2 (3, 4).

14 Strategy of the proof for fixed points Work with the resolvent which has the explicit representation: 0 e λt p γ t (x, y)dt = 0 E x t y B [ e λf (t)] p t (x, y)dt, λ > 0. Goal: give a lower bound of E x y t [ B e λf (t) ] (Brownian bridge in random environment) Strategy: find an event A t that costs P x y t B (A t ) = e c/t such that [ e λf (t) ] A t is big and use Jensen: E x t y B E x t y B [e λf (t) A t ] e λe x t y B [F (t) A t]

15 Strategy of the proof for fixed points Figure: Strategy followed by the bridge in the boxes (S t k ) k 1 t of side length t Event A t : the (B s ) s t stays in the corridor [x, y] [ t, t] and is accelerated on the δ-thick boxes of M γ. Multifractal Analysis: {k; M γ (S t k ) t2+δγ γ2 /2 } t δ2 /2 1.

16 Strategy of the proof for fixed points Spending t δ time in all δ-thick boxes costs (e c t 2 t δ ) tδ2 /2 1 = e c t 1+δ 2 /2 t δ. The cost is e c/t for t δ = t 2+δ2 /2. Therefore, we will consider the event A t that the bridge spends t 2+δ2 /2 time in each δ-thick boxes of M γ. The contribution on F (t) of the δ-thick boxes is then: t δ2 /2 1 t 2+δ2 /2 t γ2 /2 δγ = t 1+(δ γ/2)2 +γ 2 /4 t 1+γ2 /4 Conclusion: F (t) t 1+γ2 /4 on the event A t.

17 Strategy of the proof for fixed points Going back to the Resolvent: 0 e λt p γ t (x, y)dt = E x t y B E x t y B ce cλ 1 2+γ 2 /4 [ e λf (t)] p t (x, y)dt [e λf (t) A t ] e c/t dt e λe x y t B [F (t) A t] e c/t dt e λt1+γ2 /4e c/t dt

18 The upper bound: a useful general lemma In the context of Liouville Brownian motion, we have the following lemma: Lemma (Barlow,Grygorian) Let β > 1, α > 0 and τ B(y,r) denotes the LBM exit time from the Euclidean ball B(y, r). Assume that: ) 1) For all x, y and t > 0, we have p γ t (x, y) C( 1 t + 1. α 2) lim sup y T P y (τ B(y,r) r β ) = 0 r 0 Then, for all t > 0 and M γ almost all x, y T, p γ t (x, y) C ( 1 ) ( ( ) β d(x, y) t α + 1 exp C β 1 ) t 1/β.

19 The upper bound ( ) 2 ( Set α = 2 1 γ 2 and u > 0, β(u) = γ u + γ 2 u γ2 2 Theorem (Maillard, Rhodes, V., Zeitouni) For each δ > 0, we set α δ = α δ, β δ = β(α δ ) + δ. Then, there exist two random constants c 1 = c 1 (X ), c 2 = c 2 (X ) > 0 such that x, y T, t > 0, p γ t (x, y) c ( ( ) 1 t 1+δ exp dt (x, y) β δ β δ 1 ) c 2 t 1/β. δ ) 2. Remark Using similar techniques, these bounds were improved recently by S. Andres, N. Kajino to β = (γ+2)2 2.

20 The upper bound: strategy of the proof By definition, for a fixed y T τ B(y,r) = F (T B(y,r) ) = TB(y,r) 0 γ2 γx (Br ) e 2 EX [X 2 (B r )] dr. where T B(y,r) is the exit time of a standard Brownian motion B starting from y. One has for all q < 4 γ 2 P X P y (τ B(y,r) r β ) r βq E X E y 1 [ F (T B(y,r) ) q ] Using this relation on a fine grid and a union bound entails the bound β.

21 Other works on the Liouville heat kernel N. Berestycki, C. Garban, R. Rhodes, V.(2014): KPZ formula derived from the Liouville heat kernel S. Andres, N. Kajino (2014): Continuity and estimates of the Liouville heat kernel with applications to spectral dimensions M. Biskup, J. Ding : work in progress.

22 Liouville quantum gravity on the Riemann sphere Why study Liouville quantum gravity on the Riemann sphere? Important conformal field theory (indexed by a continuum set of parameters) which is exactly solvable Scaling limit of random planar maps Link with 4d-gauge theories

23 Liouville quantum gravity on the Riemann sphere References: N. Seiberg (1990): Notes on Quantum Liouville Theory and Quantum Gravity Y. Nakayama (2004): Liouville field theory: a decade after the revolution D. Harlow, J. Maltz, E. Witten (2011): Analytic continuation of Liouville theory

24 Liouville quantum gravity on the Riemann sphere Consider the following partition function on the sphere (Polyakov 1981) Z = e S L(X,ĝ) DX where S L is the Liouville action: S L (X, ĝ) := 1 ( ĝ X 2 (x)+qrĝ (x)x (x)+4πµe γx (x)) ĝ(x)dx 4π R 2 and ĝ some reference metric on the sphere. Goal: construct a CFT independent of the reference metric (within the same conformal equivalence class) with action given by S L. Here, we will choose ĝ(x) = 4 (1+ x 2 ) 2.

25 Liouville quantum gravity on the Riemann sphere: the Gaussian Free Field We denote: ĝ Laplacian G Green function with vanishing mean, GFF X : E[X (x)x (y)] = G(x, y). Liouville field: X (x) + Q 2 ln ĝ(x) Regularized GFF: X ɛ (x) = 1 2π 2πɛ 0 X (x + ɛe iθ )dθ Vertex operator: V α,ɛ (x) := ɛ α2 /2 e α(xɛ(x)+ Q 2 ln ĝ(x))

26 Liouville quantum gravity on the Riemann sphere: the n-point correlation function Goal: construct a CFT on the sphere. Problem: if ψ Mobius transform, X ψ (Law) X.

27 Liouville quantum gravity on the Riemann sphere: the n-point correlation function Goal: construct a CFT on the sphere. Problem: if ψ Mobius transform, X ψ (Law) X. In order to ensure conformal invariance, we need to integrate with respect to the Lebesgue measure. Hence, we get the following definition: C((α i ), (z i ), µ, F (.)) [ ( ) = lim E ɛ α 2 i 2 e α i (c+x ɛ+ Q 2 ln ĝ)(z i ) ɛ 0 R i ( F (.) exp µe γc ɛ γ2 2 R 2 e γ(xɛ(x)+q/2 ln ĝ(x)) dx e Q 4π R 2 2(c+X (x)+ Q ln ĝ(x)) ĝ(x)dx 2 )] dc.

28 Liouville quantum gravity on the Riemann sphere C((α i ), (z i ), µ, F (.)) [ = lim E ɛ α 2 i 2 e α i (c+x ɛ+ Q 2 ln ĝ)(z i ) e Q 4π R 2 2(c+X (x)+ Q ln ĝ(x)) ĝ(x)dx 2 ɛ 0 R F (.) exp i ( µe γc ɛ γ2 2 R 2 e γ(xɛ(x)+q/2 ln ĝ(x)) dx )] dc. We set C((α i ), (z i ), µ) = C((α i ), (z i ), µ, F (.) = 1) and the probability E (z i ),(α i ),µ [F (.)] = C((α i), (z i ), µ, F (.)). C((α i ), (z i ), µ)

29 Liouville quantum gravity on the Riemann sphere: existence of the n-point correlation Theorem (David, Kupiainen, Rhodes, V.) If i α i > 2Q and α i < Q (Seiberg bound), then C((α i ), (z i ), µ) ( = ĝ(z i ) α 2 ) i 4 + Q 2 α i e i j α i α j G(z i,z j ) e i ( Γ γ 1 ( i ) [ α i 2Q) E 1 Z (zi )(R 2 ) 2Q C(ĝ) µ γ i α i 2Q γ ] γ i α i where Γ is the standard gamma function, C(ĝ) a global constant and γ2 γx (x) Z (zi )(dx) = e 2 E[X (x)2 ]+γ i α i G(x,z i ) dx

30 Liouville quantum gravity on the Riemann sphere: conformal invariance of the n-point correlation Theorem (David, Kupiainen, Rhodes, V.) If ψ is a Mobius transform then C((α i ), (ψ(z i )), µ) = i ψ (z i ) 2 α i C((αi ), (z i ), µ) where αi are the conformal weights: αi = α i 2 (Q α i 2 ). Proof: use definition of C((α i ), (z i ), µ) as a limit and then: Girsanov+ computations involving change of metrics + X ψ 1 4π R X ψ(x)ĝ(x)dx (Law) = X 2

31 Liouville quantum gravity on the Riemann sphere: the Liouville measure The Liouville measure Z L (dx) = lim ɛ e γ(xɛ(x)+ Q 2 ln ĝ(x)) dx is conformally invariant (with respect to Mobius) and its total mass has a Γ distribution E (z i ),(α i ),µ [F (Z L (R 2 ))] = Γ( i α i µ 2Q γ i α i 2Q γ ) 0 F (v)v Conditioning on the volume, we get the distribution ( E[F E (z i ),(α i ),µ [Z L (dx) Z L (R 2 ) = A] = A Z (z i )(dx) Z (zi )(R 2 ) i α i 2Q γ 1 e µv dv ) Z (zi )(R 2 ) i α i 2Q γ ] E[Z (zi )(R 2 ) i α i 2Q γ ]

32 Liouville quantum gravity on the Riemann sphere: perspectives Perspectives: Compute the semi-classical limit of your system, i.e. γ 0 (Liouville equation on the sphere with conical singularities) Give explicit expressions for the 3 point correlation function (conjectured to be the 3 point correlations of numerous 2d-statistical physics systems at criticality)