Planar maps, circle patterns, conformal point processes and two dimensional gravity

Transcription

1 Planar maps, circle patterns, conformal point processes and two dimensional gravity François David joint work with Bertrand Eynard (+ recent work with Séverin Charbonnier) IPhT, CEA-Saclay and CNRS 1

2 1. Continuum and discrete 2D gravity: what remains to be understood? 2. Circle packings and circle patterns 3. Delaunay circle patterns and planar maps 4. A measure over planar triangulations 5. Spanning 3-trees representation 6. Kähler geometry over triangulation space and 3D hyperbolic geometry 7. Discretized Faddev-Popov operator and Polyakov s 2D gravity 8. Local uniform bounds and the continuum limit 2

3 1. a: Continuum formulations of 2D quantum gravity Quantization = Feynman path integral over 2D Riemannian metrics (+ matter fields) Z D[g ab ]exp µ 0 Z d 2 x p g Z D[X] e S[X,g] For physicists, this is a fairly well understood theory 3

4 Polyakov functional integral, use the conformal gauge g ab =ĝ ab e A. Polyakov 1981 The Faddeev-Popov ghost systems leads to the effective action for the remaining Z conformal Z factor. The effective theory is the Liouville theory (from conformal anomaly consistency condition), a CFT and integrable quantum theory D[g ab ]= S L ['] = 1 For pure gravity 2 Real positive action for Dĝe [ ] det(r FP ĝ )= Z pĝ 1 2 ( ˆr') 2 + Q ˆR ' + µ R e ' 2 Q = c L =1+6Q 2 = 26 c M c M =0 1 <c M apple 1 Z Dĝ['] e S L['] 4

5 1.b: Discretized formulations of 2D quantum gravity * Random planar lattices (maps) F. D., J. Fröhlich, V. Kazakov & A. Migdal (circa ) N ' 10 3 not an isometric embedding! Also fairly well understood 5

6 1.b: Discretized formulations of 2D quantum gravity * Random planar lattices (maps) F. D., J. Fröhlich, V. Kazakov & A. Migdal (circa ) N ' 10 4 Also fairly well understood not an isometric embedding! 6

7 1.b: Discretized formulations of 2D quantum gravity * Random planar lattices (maps) F. D., J. Fröhlich, V. Kazakov & A. Migdal (circa ) N ' 10 5 Also fairly well understood not an isometric embedding! 7

8 2 Discrete 2d gravity Random matrices recursion relations integrability Continuous 2d gravity Topological gravity QFT CFT integrability combinatorics Cori-Vauquelin- Schaeffer-... bijections combinatorics probabilities Planar maps (diagrams) Brownian maps & well labelled trees KPZ relations Liouville theory & conformal field theories conformal invariance probabilities Gaussian free field, SLE Still lacking: a constructive & geometrical link between the discrete and the continuous formulations 8

9 1. Continuum and discrete 2D gravity: what remains to be understood? 2. Circle packings and circle patterns 3. Delaunay circle patterns and planar maps 4. A measure over planar triangulations 5. Spanning 3-trees representation 6. Kähler geometry over triangulation space and 3D hyperbolic geometry 7. Discretized Faddev-Popov operator and Polyakov s 2D gravity 8. Local uniform bounds and the continuum limit 9

10 2.a: Circle packings The Koebe-Andreev-Thurston theorem There is a bijection between simple triangulations and circle packings, modulo SL(2,C) Möbius transformations v 1 v3 v 3 v 2 v3 Illustrations borrowed to Schramm & Mishenko 10

11 11

12 Planar triangulation N '

13 Circle packing embedding 13

14 Dual spherical triangle embedding 14

15 2.b: A generalisation of circle packings: circle patterns Circles meeting at common points. The angles of intersection of the circles are given. Find the planar pattern and the radii of the circles. theorem of existence and unicity (Igor Rivin 1994, Ann. of Math.) circle packing = circle patterns with angles 0 or /2 only 15

16 1. Continuum and discrete 2D gravity: what remains to be understood? 2. Circle packings and circle patterns 3. Delaunay circle patterns and planar maps 4. A measure over planar triangulations 5. Spanning 3-trees representation 6. Kähler geometry over triangulation space and 3D hyperbolic geometry 7. Discretized Faddev-Popov operator and Polyakov s 2D gravity 8. Local uniform bounds and the continuum limit 16

17 T =(T,{ }) { } angles 1) the angles are positive 2) around each vertex 3) around each closed cycle «Dressed» planar maps T an abstract triangulation of the sphere, with e attached to the edges e such that 0 < e < X e =2 e!v X e2c e Sphere Sphere 17

18 Delaunay triangulations and planar maps There is a bijection* between such «dressed» planar maps and Delaunay triangulations in the plane, such that the circle intersection angles are e = * modulo global SL(2, C) transformations (as usual) e Sphere Plane 18

19 Voronoï tessellation and Delaunay triangulation in the plane Delaunay condition: no vertex of a triangle must be inside the circumscribed circle to any another triangle Local flatness (p.l. manifold) X e!v e =2 >0 From Wikimedia Commons Last condition on cycles no change of orientations or foldings (to be discussed later) This excludes some triangulations but it is conjectured that one keeps generic ones (universality) 19

20 1. Continuum and discrete 2D gravity: what remains to be understood? 2. Circle packings and circle patterns 3. Delaunay circle patterns and planar maps 4. A measure over planar triangulations 5. Spanning 3-trees representation 6. Kähler geometry over triangulation space and 3D hyperbolic geometry 7. Discretized Faddev-Popov operator and Polyakov s 2D gravity 8. Local uniform bounds and the continuum limit 20

21 ( ) = ( )) T + Take as initial measure on triangulations the uniform measure on ( ) triangulations and the flat measure on the angles (+ inequalities) µ( T ) = µ(t,d )) = uniform(t ) d (e) e E(T ) v V(T ) (e) 2 ev Question: which measure does this induce on Delaunay triangulations? For T this consider N+3 points, 3 fixed by ( ) D N+3 = C N+3 /SL(2, C) ' C N dµ(z 4,,z N+3 ) =? T + SL(2, C) ( ) = E( ) E( ) F. David, June 26, 2015 GGI, Firenze, Italy 21

22 Moving the points allows to explore the whole space of Delaunay triangulations and of dressed abstract triangulations Transition between Delaunay triangulations by edge flips the flip of link e occurs when (e) =0 22

23 Elementary example: the hexahedron (5 points) moving the 5 th point

24 Question 1: Which sets of edges form independent basis for the angles? Answer: The sets whose complementary form a cycle-rootedspanning-tree of the triangulation with odd length cycle 24

25 Question 2: What is the Jacobian of the change of angle variables between two basis of edges? Answer: Jacobian = 1! Indeed... G T + { } = = ( µ(t,d ) = 1 2 uniform(t + ) ) ( ) = e E 0 (T ) So, the measure over the points is given locally (for a given = E ( ) E( ) Delaunay triangulation) by a simple Jacobian E µ(t,d ) = dµ(z) = ( ) ( ) N+3 v=4 d 2 z v + { ( ) d (e) det J T (z) {1,2,3} Ē e J T (z) v, z v ) e E(T E ) v V(T )( ) ciated to the three fixed vertices ( ) { } E 25

26 { } { } E ( ) = ( ( ) + ) Question 3: What is the local form of the measure? The matrix elements = of the Jacobian are made of simple poles J v,e e, J v,e e z v tex is = a vertex of one of the tw J v1,e = i z v4 z v1 z v3 z v1 J v3,e = i 2 1 z v3 z v1 J v2,e = i 2 1 z v3 z v2 D( ) J v4,e = i 2 1 z v4 z v2 ( ) = ( ) 1 z v3 z v2 1 z v4 z v2 1 z v4 z v1 ( D The determinant { } of the Jacobian matrix is locally a rational function ( of the ) s and z v s z v D T (z) {1,2,3} = det J T (z) {1,2,3} Ē 0 + = ( + ) 26

27 1. Continuum and discrete 2D gravity: what remains to be understood? 2. Circle packings and circle patterns 3. Delaunay circle patterns and planar maps 4. A measure over planar triangulations 5. Spanning 3-trees representation 6. Kähler geometry over triangulation space and 3D hyperbolic geometry 7. Discretized Faddev-Popov operator and Polyakov s 2D gravity 8. Local uniform bounds and the continuum limit 27

28 Definition 2.3 (triangle rooted spanning 3-tree) Let T be a planar triangulation with N + 3 vertices, and = f 0 be a face (triangle) of T,with3verticesV( ) = (v 1,v 2,v 3 ) and 3 edges E( ) = (e 1,e 2,e 3 ). Let E(T ) = E(T ) E( ) be the set of 3N edges of T not in. We call a -rooted 3-tree of T ( R3T )afamilyf of three disjoint subsets (I, I, I ) of edges of E(T ) such that: 1. (I, I, I ) are disjoint and disjoint of E( ) 2. Each I E( ), I E( ), I E( ) is a cycle rooted spanning tree of T with cycle. I I I ( ) 5: 2 inequivalent triangle-rooted spanning 3-trees of a = planar triang I I I nb: -rooted spanning 3-trees are NOT Schnyder woods! 28

29 Theorem 2.3 Let T be a planar triangulation of the plane with N +3 vertices. If the 3 fixed points (v 1,v 2,v 3 ) belong to a triangle (face) of T,themeasuredeterminanttakes the form D T (z) {1,2,3}) = 4 N F=(I,I,I ) R3T of T (F) e=(v v ) I 1 z v z v e=(v v ) I 1 z v z v (2.10) where (F) = ±1 is a sign factor, coming from the topology of T and of F, thatwillbe defined later. The measure can be written as a sum over these spanning 3- ( ) trees D Tree 1: analytic part I I - Tree I 2: anti-analytic part This is a non trivial extension of the spanning tree representation of the determinant of scalar Laplacians. This representation is specific to this Jacobian matrix D. It is useful for proof of convergence, factorization properties, etc. Can we use it for more? 29

30 1. Continuum and discrete 2D gravity: what remains to be understood? 2. Circle packings and circle patterns 3. Delaunay circle patterns and planar maps 4. A measure over planar triangulations 5. Spanning 3-trees representation 6. Kähler geometry over triangulation space and 3D hyperbolic geometry 7. Discretized Faddev-Popov operator and Polyakov s 2D gravity 8. Local uniform bounds and the continuum limit 30

31 Hyperbolic volume of triangle = volume of ideal tetrahedron above the triangle in hyperbolic Poincaré half-space ( ) ) = ( ) Vol(f) = L( 1 ) + L( 2 ) + L( 3 ) ki-milnor = Im(Li function 2 (z)) + ln(z)arg (1 z) z = z 3 z 1 z 2 z1 ( ) = ( ( )) ( action of a triangulation = (minus) sum over h-volumes A T = triangles f F(T ) Vol(f) A 31

32 The mesure as a Kähler form! Define the N x N matrix ) = ( + ) D u v @ z v A T (z) 1. D u v is a Kähler form on D N+3 i.e. D>0 D u v 2. is countinuous (no discontinuity when a flip occurs) 3. The measure determinant is the Kähler volume form D T (z) {1,2,3}) = det (D u, v ) u,v {1,2,3} The (2N x 2N) Jacobian has been reduced to a N x N Kähler determinant! D ( ) { }) But it is not a determinantal process! 32

33 It is clear that 1 2 surprising, theuinitial measure d v,f independent = 2 v,f+ = 1 v,f+ over v = 2 4 f v 4 f v Sv Sv ˆ ˆ This is not too angles can be written as a combination of Chern classes In other words, uv is a connection globally defined on the bundle Lv along the fiber is 1, its curvature duv is then the Chern class: 1 n f 1 c1 (Lv )apparently = duv = depend d v,f Notice that the coefficient Cv T=might triangulation T, on the + d v,f+ 2 4 f =2 f =1 2 ever, it was proved by Kontsevich that it doesn t 4 v N 2 v = ± N! 2, and we have: 22 2N +1 d e e This shows that our measure DT (z) {1,2,3} is also the measure of topological gr and the 4 angleproofs measure isthe a measure on the moduli space of the of results fiber of the circle bundle L T, is a circle centered at v. A point M0,N +3 punctured sphere a point on the circle, and Riemann a coordinate for that point is the angle the segment [v, f ] where f is a face adjacent to v.a basis of edges 4.1 Chosing So this measure is related (and quite possibly equivalent) to the Proof of Th.over 2.1 M0,N +3 and our model is related Weil-Peterson measure of the bundle L is a 2-form on T, notice that it is independent s tolabelling topological gravity Wearound havev.2d defined the measure on T N +3 to be the uniform measure on triangula rigin of of faces v v v N +3 f,v N +3 ds, if we label the edges around v in the trigonometric order e1,..., en, tensored with the flat Lebesgue measure on angles e s (constrained by (2.1)): F. David, June 26, n e 1 d e d e v = 2 4 e=2 e =1 uniform(t ) d e (3.6) ( 2 + e ) GGI, Firenze, Italy 33

34 The 2-d metric on the boundary of the hyperbolic tetrahedra induced by the 2d-metric in H3 makes it a constant negative curvature ball with cusps singularities (punctured sphere) H 3 If our measure is the W-P measure, this implies that our model is in the same universality class (pure gravity) than planar maps string = 1/2 34

35 Conformal invariance is exact and explicit dµ(z) =d 2 z det(d) is a conformal point process Independence of the 3 fixed points and H = det D \ a,b,c (z) 3(z a,z b,z c ) 2 SL(2, C) invariance 3(z a,z b,z c )=(z a z b )(z a z c )(z b z c ) is independent of the choice of points z! w = az + b cz + d H(z) = N+3 Y i=1 w 0 (z i ) with ad bc =1 2 H(w) = N+3 Y i=1 1 cz i + d 2 H(w) 35

36 Consequence: D is a very singular but integrable measure One expects large fluctuations of the density of points at all scales, consequence of conformal invariance Poisson process on the sphere collapse of half of the points X The sum of angles around the collapsed points e! 2 + e!v 36

37 1. Continuum and discrete 2D gravity: what remains to be understood? 2. Circle packings and circle patterns 3. Delaunay circle patterns and planar maps 4. A measure over planar triangulations 5. Spanning 3-trees representation 6. Kähler geometry over triangulation space and 3D hyperbolic geometry 7. Discretized Faddev-Popov operator and Polyakov s 2D gravity 8. Local uniform bounds and the continuum limit 37

38 Local geometrical representation of D as a sum over the triangles of local operators D u, v = f D u, v (f), D u, v (f) z v Vol(f) ( ) D(f) = 1 cot( 2 ) + cot( 3 ) cot( 3 ) i cot( 2 ) + i cot( 8 R(f) 2 3 ) + i cot( 3 ) + cot( 1 ) cot( 1 ) i cot( 2 ) i cot( 1 ) + i cot( 1 ) + cot( 2 ) ( 38

39 This suggest D V( is ) a discretized Fadeev-Popov determinant! The Hermitean form D can be written as D(f) = i,j vertices of f V( ) F( ) with local derivative operator from (f) = 1 2i (f) = 1 2i ( ) (v i )D i (f) (v j ) = Area(f) R(f) 2 V( ) = ( = ) ( ) (v 1 )( z 3 z 2 ) + (v 2 )( z 1 z 3 ) + (v 3 )( z 2 z 1 ) Area(f) (v 1 )(z 3 z 2 ) + (v 2 )(z 1 z 3 ) + (v 3 )(z 2 z 1 ) Area(f) (f) (f) (vertices)! (faces) = simple geometric characterization of r r f,v v = 1 z v z v + log(radius(f)) z v 0 z v 39

40 ( ) = ( ) ( ) = If complex functions are identified with real vector fields = z, = z ( ) = ( ) Area(f) = d 2 w f 1 R(f) 2 = e (w f ) One gets with D(f) X Area(f) = r ˆ R(f) 2 (f) r (f) = ( ) = ( ) f D = d 2 w e z z z z (w) (f) = 2log(R(f)) ( One can see the Kähler form as a discretized version of the Faddeev-Popov determinant in Polyakov s formulation of two dimensional gravity and of non-crititical string theory! Indeed... 40

41 ) = ( ) = ( ) = ( ) Back to the functional integral over 2d Riemannian metrics, in the ( ) conformal gauge g ab (z) = ab e (z) Faddev-Popov determin with Faddeev-Popov ghost systems det( FP ) = D[ = ] = D[ ] ( ˆ ˆ ˆ = ( ) = ( ) D[g ab = ]( = ) D[ = ( ] ) det( ) FP = D[ ] ( ( ) + ( ) ˆ D[c, b] exp D[ ] ( ) = ( ) ˆ ˆ det( FP ) = d 2 z e (b zz ( c) zz + b z z ( c) z z ) Integrating over the b s (the ghosts) only gives = ( ) = + ( ) D[c] exp d 2 z c z z c z = = ( that ) appears in Therefore the = ( ) Kähler form The D operator is nothing but a discretised FP determinant D = = FP = (b, c) This suggest to identify the scalar function on faces ) = (f) = ( 2log(R(f)) ( with a discretized Liouville field on the Voronoï lattice... ( 41

42 1. Continuum and discrete 2D gravity: what remains to be understood? 2. Circle packings and circle patterns 3. Delaunay circle patterns and planar maps 4. A measure over planar triangulations 5. Spanning 3-trees representation 6. Kähler geometry over triangulation space and 3D hyperbolic geometry 7. Discretized Faddev-Popov operator and Polyakov s 2D gravity 8. Local uniform bounds and the continuum limit 42

43 Up to now, existence results at finite N (number of points) We are interested in the continuum, large N limit. First step, get bounds as a function of N. Theorem 1: Delaunay triangulations extremize the Kähler volume measure. If T Delaunay and T not Delaunay with same vertices {z} D T (z) \{1,2,3} > D T 0(z) \{1,2,3} Proof: Use the Lawson Flip Algorithm to go from T to T and show that the measure always increases during a flip. 43

44 Consider the total integral over N(+3) points Z A N = C N N+3 Y v=4 d 2 z v D T D N (z) \{1,2,3} Theorem 2: The sequence is growing as A N+1 (N + 1) 2 8 A N Proof: Compare integrands when adding/removing a point, using theorem 1 and proper decomposition of the plane into some kind of conformal Delaunay triangulation Local inequalities potentially interesting to prove convergence for probability measures, not only integrals 44

45 Corollary: The partition function defined by the series Z(g) = X N A N /N! g N has a finite non-zero radius of convergence g <g c and a critical point on the real axis, which is expected to correspond to continuum 2D gravity in the scaling limit g/g c =e µ Ra 2 a! 0 45

46 We have an explicit quasi-conformal embedding of planar «dressed admissible» 2-dimensional maps onto the complex plane This point process is well defined for finite number of points It has many interesting properties (conformal invariance) It is what is expected from continuum 2d quantum gravity But we would like to be able to characterise its continuum limit, namely the limit when the density of points become infinite, and the corresponding statistical system This is more difficult... renormalization group methods needed (work in progress) Any help and ideas from mathematicians is welcome Thank you for your interest! Conclusion 46