Math 639: Lecture 24 The Gaussian Free Field and Liouville Quantum Gravity Bob Hough May 11, 2017 Bob Hough Math 639: Lecture 24 May 11, 2017 1 / 65 The Gaussian free field This lecture loosely follows Berestycki's `Introduction to the Gaussian Free Field and Liouville Quantum Gravity'. Bob Hough Math 639: Lecture 24 May 11, 2017 2 / 65 It^o'sformula The multidimensional It^o'sformula is as follows. Theorem (Multidimensional It^o'sformula) Let tBptq : t ¥ 0u be a d-dimensional Brownian motion and suppose tζpsq : s ¥ 0u is a continuous, adapted stochastic process with values in m d m R and increasing components. Let f : R ` Ñ R satisfy Bi f and Bjk f , all 1 ¤ j; k ¤ d, d ` 1 ¤ i ¤ d ` m are continuous t 2 E 0 |rx f pBpsq; ζpsqq| ds ă 8 then a.s.³ for all 0 ¤ s ¤ t s f pBpsq; ζpsqq ´ f pBp0q; ζp0qq “ rx f pBpuq; ζpuqq ¨ dBpuq »0 s 1 s ` r f pBpuq; ζpuqq ¨ dζpuq ` ∆ f pBpuq; ζpuqqdu: y 2 x »0 »0 Bob Hough Math 639: Lecture 24 May 11, 2017 3 / 65 Conformal maps Definition 2 Let U and V be domains in R . A mapping f : U Ñ V is conformal if it is a bijection and preserves angles. Viewed as a map between domains in C, this is equivalent to f is an analytic bijection. Bob Hough Math 639: Lecture 24 May 11, 2017 4 / 65 Conformal invariance of Brownian motion The following conformal invariance of Brownian motion may be established with It^o'sformula. Theorem Let U be a domain in the complex plane, x P U, and let f : U Ñ V be analytic. Let tBptq : t ¥ 0u be a planar Brownian motion started in x and τU “ inftt ¥ 0 : Bptq R Uu its first exit time from the domain U. There exists a planar Brownian motion tB~ptq : t ¥ 0u such that, for any t P r0; τU q, t f pBptqq “ B~pζptqq; ζptq “ |f 1pBpsqq|2ds: »0 If f is conformal, then ζpτU q is the first exit time from V by tB~ptq : t ¥ 0u. Bob Hough Math 639: Lecture 24 May 11, 2017 5 / 65 Conformal invariance of Brownian motion Proof. We assume that the domains are bounded. Recall that if f “ f1 ` if2 then the Cauchy-Riemann equations give 1 rf1 and rf2 are orthogonal, and |rf1| “ |rf2| “ |f |. Let σptq “ infts ¥ 0 : ζpsq ¥ tu Let tB~ptq : t ¥ 0u be an independent Brownian motion, and define W ptq “ f pBpσptq ^ τU qq ` B~ptq ´ B~pt ^ ζpτU qq; t ¥ 0: It suffices to check that W ptq is a Brownian motion. Since it is almost surely continuous, it remains to check the f.d.d. Bob Hough Math 639: Lecture 24 May 11, 2017 6 / 65 Conformal invariance of Brownian motion Proof. To check the f.d.d. we check for 0 ¤ s ¤ t, and λ P C, 1 E exλ,W ptqy G psq “ exp |λ|2pt ´ sq ` xλ, W psqy : 2 ” ˇ ı ˆ ˙ ˇ It suffices to check ˇ 1 E exλ,W ptqy|W psq “ f pxq “ exp |λ|2pt ´ sq ` xλ, f pxqy : 2 ” ı ˆ ˙ We evaluate this at s “ 0. Bob Hough Math 639: Lecture 24 May 11, 2017 7 / 65 Conformal invariance of Brownian motion Proof. Calculate E exλ,W ptqy W p0q “ f pxq ” ˇ ı 1 “ E exp xˇλ, f pBpσptq ^ τ qqy ` |λ|2pt ´ ζpσptq ^ τ qq : x ˇ U 2 U ˆ ˙ We use It^owith 1 F px; uq “ exp xλ, f pxqy ` |λ|2pt ´ sq : 2 ˆ ˙ Note ∆exλ,f pxqy “ |λ|2|f 1pxq|2exλ,f pxqy. Bob Hough Math 639: Lecture 24 May 11, 2017 8 / 65 Conformal invariance of Brownian motion Proof. 1 2 Recall F px; uq “ exp xλ, f pxqy ` 2 |λ| pt ´ uq . Set T “ σptq ^ τUn , with U “ tx P U : dpx; BUq ¥ 1{nu. n ` ˘ Hence T F pBpT q; ζpT qq “ F pBp0q; ζp0qq ` rx F pBpsq; ζpsqq ¨ dBpsq »0 T 1 T ` B F pBpsq; ζpsqqdζpsq ` ∆ F pBpsq; ζpsqqds: u 2 x »0 »0 Use dζpuq “ |f 1pBpuqq|2du to cancel the two terms on the bottom line. Also, the stochastic integral has mean 0. Bob Hough Math 639: Lecture 24 May 11, 2017 9 / 65 Conformal invariance of Brownian motion Proof. We thus calculate xλ,W ptqy E e W p0q “ f pxq “ Ex rF pBpσptq ^ τU q; ζpσptq ^ τU qqs ˇ ” ı 1 2 “ lim Ex rFˇ pBpT q; ζpT qqs “ F px; 0q “ exp |λ| t ` xλ, f pxqy : nÑ8 ˇ 2 ˆ ˙ Bob Hough Math 639: Lecture 24 May 11, 2017 10 / 65 Discrete case Let G “ pV ; Eq be an undirected graph, and let B be a distinguished set of vertices, called the boundary. Let V^ “ V zB be the internal vertices. For x; y P V , write x „ y if x and y are neighbors. Let tXnu be a random walk on G, in which at each step the walker chooses uniformly a random neighbor. Let P be the transition matrix, dpxq “ degpxq, which is an invariant measure for the walk, and let τ be the first hitting time to the boundary. Bob Hough Math 639: Lecture 24 May 11, 2017 11 / 65 Discrete case Definition (Green function) The Green function Gpx; yq is defined for x; y P V by putting 1 8 Gpx; yq “ Ex 1 : dpyq pXn“y;τ¡nq ˜n“0 ¸ ¸ Definition (Discrete Laplacian) The discrete Laplacian acts on functions on V by 1 ∆f pxq “ pf pyq ´ f pxqq: d x y„x p q ¸ Bob Hough Math 639: Lecture 24 May 11, 2017 12 / 65 Discrete case Proposition The following hold: 1 Let P^ denote the restriction of P to V^ . Then pI ´ P^q´1px; yq “ Gpx; yqdpyq for all x; y P V.^ 2 G is a symmetric nonnegative semidefinite function. That is, one has Gpx; yq “ Gpy; xq and if pλx qxPV is a vector then x;yPV λx λy Gpx; yq ¥ 0. Equivalently, all eigenvalues are non-negative. ř 3 Gpx; ¨q is discrete harmonic in V^ ztxu, and ∆Gpx; ¨q “ ´δx p¨q. Bob Hough Math 639: Lecture 24 May 11, 2017 13 / 65 Discrete case Definition The discrete Gaussian free field is the centered Gaussian vector phpxqqxPV with covariance given by the Green function G. Note that if x PB, then Gpx; yq “ 0 for all y P V and hence hpxq “ 0 a.s. Bob Hough Math 639: Lecture 24 May 11, 2017 14 / 65 Discrete case Theorem (Law of the GFF) The law of phpxqqxPV is absolutely continuous with respect to dx “ uPV dxu, and the joint pdf is given by ± 1 1 Probphpxq P Aq “ exp ´ py ´ y q2 dy : Z 4 u v u A ˜ u„vPV ¸ uPV » ¸ ¹ Z is a normalizing constant, called the partition function. The quadratic form appearing in the exponential is called the Dirichlet energy. Bob Hough Math 639: Lecture 24 May 11, 2017 15 / 65 Discrete case Proof. For a centered Gaussian vector pY1; :::; Ynq with covariance matrix V , the joint density is given by 1 1 exp ´ y T V ´1y Z 2 ˆ ˙ Bob Hough Math 639: Lecture 24 May 11, 2017 16 / 65 Discrete case Proof. Restrict to only variables from V^ and calculate hpx^qT G ´1hpx^q “ G ´1px; yqhpxqhpyq ^ x;¸yPV “ ´dpxqP^px; yqhpxqhpyq ` dpxqhpxq2 ^ ^ x;yP¸V :x„y x¸PV 1 “ ´ hpxqhpyq ` phpxq2 ` hpyq2q 2 x;yPV :x„y x;yPV :x„y ¸ ¸ 1 “ phpxq ´ hpyqq2: 2 x;yPV :x„y ¸ Bob Hough Math 639: Lecture 24 May 11, 2017 17 / 65 Discrete case Theorem (Markov property) Fix U Ă V . The discrete GFF hpxq can be decomposed as follows: h “ h0 ` φ where h0 is a Dirichlet boundary Gaussian free field on U and φ is harmonic on U. Moreover, h0 and φ are independent. We prove a continuum version of this theorem in the next section. Bob Hough Math 639: Lecture 24 May 11, 2017 18 / 65 Continuum case d D For D Ă R let pt px; yq be the transition kernel of Brownian motion killed on leaving D. Thus D D pt px; yq “ pt px; yqπt px; yq where |x´y|2 exp ´ 2t p px; yq “ t ´ d ¯ p2πtq 2 D and πt px; yq is the probability that a Brownian bridge of duration t remains in D. Bob Hough Math 639: Lecture 24 May 11, 2017 19 / 65 Continuum case Definition The Green function Gpx; yq “ GD px; yq is given by 8 D Gpx; yq “ π pt px; yqdt: »0 D Gpx; xq “ 8 for all x P D, since πt px; xq Ñ 1 as t Ñ 0. If D is bounded then Gpx; yq ă 8 for all x ‰ y. Bob Hough Math 639: Lecture 24 May 11, 2017 20 / 65 Continuum case Example Suppose D “ H is the upper half plane. Then H pt px; yq “ pt px; yq ´ pt px; yq and x ´ y G px; yq “ log : H x ´ y ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ Bob Hough Math 639: Lecture 24 May 11, 2017 21 / 65 Continuum case We now restrict attention to dimension 2. Proposition If T : D Ñ D1 is a conformal map (holomorphic and one-to-one), then GT pDqpT pxq; T pyqq “ GD px; yq. Bob Hough Math 639: Lecture 24 May 11, 2017 22 / 65 Continuum case Proof. Let φ be a test function and let x1; y 1 “ T pxq; T pyq.