La fonction à deux points et à trois points des quadrangulations et cartes. Éric Fusy (CNRS/LIX) Travaux avec Jérémie Bouttier et Emmanuel Guitter

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1 La foncton à deux ponts et à tros ponts des quadrangulatons et cartes Érc Fusy (CNRS/LIX) Travaux avec Jéréme Boutter et Emmanuel Gutter Sémnare Caln, LIPN, Ma 4

2 Maps Def. Planar map = connected graph embedded on the sphere = Easer to draw n the plane (by choosng a face to be the outer face)

3 Maps as random dscrete surfaces Natural questons: Typcal dstance between (random) vertces n random maps the order of magntude s n /4 ( n / n random trees) random quadrang. { - [Chassang-Schaeffer 4] probablstc - [Boutter D Francesco Gutter 3] exact GF expressons How does a random map (rescaled by n /4 ) look lke? convergence to the Brownan map [Le Gall 3, Mermont 3]

4 Countng (rooted) maps wth a marked corner Very smple countng formulas ([Tutte 6s]), for nstance Let q n = #{rooted quadrangulatons wth n faces} m n = #{rooted maps wth n edges} Then m n = q n = n+ 3n (n)! n!(n+)!

5 Countng (rooted) maps wth a marked corner Very smple countng formulas ([Tutte 6s]), for nstance Let q n = #{rooted quadrangulatons wth n faces} m n = #{rooted maps wth n edges} Then m n = q n = n+ 3n (n)! n!(n+)! Proof of m n = q n by easy local bjecton:

6 Countng (rooted) maps wth a marked corner Very smple countng formulas ([Tutte 6s]), for nstance Let q n = #{rooted quadrangulatons wth n faces} m n = #{rooted maps wth n edges} Then m n = q n = n+ 3n (n)! n!(n+)! Proof of m n = q n by easy local bjecton: But ths bjecton does not preserve dstance-parameters (only bounds)

7 The k-pont functon Let M = n M[n] be a famly of maps (quadrangulatons, general,...) where n s a sze-parameter (# faces for quad., # edges for gen. maps) Let M (k) = famly of maps from M wth k marked vertces v,..., v k

8 The k-pont functon Let M = n M[n] be a famly of maps (quadrangulatons, general,...) where n s a sze-parameter (# faces for quad., # edges for gen. maps) Let M (k) = famly of maps from M wth k marked vertces v,..., v k Refnement by dstances : For D = (d,j ) <j k any ( k let M (k) D ) -tuple of postve ntegers := subfamly of M(k) where dst(v, v j ) = d j for < j k The countng seres G D G D (g) of M (k) D s called the k-pont functon of M wth respect to the sze v v quadrangulaton k = d = 3 v v v 3 general map k = 3 d = d 3 = d 3 = 3

9 Exact expressons for the k-pont functon For the two-pont functons: - quadrangulatons [Boutter D Francesco Gutter 3] - maps wth prescrbed (bounded) face-degrees [Boutter Gutter 8] - general maps [Ambjørn Budd 3] - general hypermaps, general constellatons [Boutter F Gutter 3] For the three-pont functons - quadrangulatons [Boutter Gutter 8] - general maps & bpartte maps [F Gutter 4]

10 Exact expressons for the k-pont functon Outlne of the talk For the two-pont functons: - quadrangulatons uses Schaeffer s bjecton - maps wth prescrbed (bounded) face-degrees [Boutter D Francesco Gutter 3] [Boutter Gutter 8] 3 - general maps based on clever observaton [Ambjørn Budd 3] on Mermont s bjecton - general hypermaps, general constellatons [Boutter F Gutter 3] For the three-pont functons - quadrangulatons uses Mermont s bjecton [Boutter Gutter 8] 4 - general maps & bpartte maps [F Gutter 4] uses AB bjecton

11 Computng the two-pont functon of quadrangulatons usng the Schaeffer bjecton

12 Well-labelled trees Well-labelled tree = plane tree where - each vertex v has a label l(v) Z - each edge e = {u, v} satsfes l(u) l(v) -

13 Ponted quadrangulatons, geodesc labellng Ponted quadrangulaton = quadrangulaton wth a marked vertex v Geodesc labellng wth respect to v : l(v) = dst(v, v) 3 Rk: two types of faces v stretched + confluent

14 The Schaeffer bjecton 3 3 [Schaeffer 99], also [Cor-Vauqueln 8] Ponted quadrangulaton well-labelled tree wth mn-label= n faces n edges 3 Local rule n each face:

15 The -pont functon of quadrangulatons () Denote by G d G d (g) the two-pont functon of quadrangulatons bjecton G d (g) = GF of well-labelled trees wth mn-label= and wth a marked vertex of label d d = Rk: G d = F d F d = d F d where F d F d (g) = GF of well-labelled trees wth postve labels and wth a marked vertex of label d

16 The -pont functon of quadrangulatons () - - > + - wth R = F = log g(r +R +R + ) > GF rooted well-labelled trees wth postve labels and label at the root

17 The -pont functon of quadrangulatons () - - > + - wth R = F = log g(r +R +R + ) GF rooted well-labelled trees wth postve labels and label at the root Equ. for R : R = g(r +R +R + ) (so F = log(r ), G d = log( R d R d )) >

18 The -pont functon of quadrangulatons () - - > + - wth R = F = log g(r +R +R + ) GF rooted well-labelled trees wth postve labels and label at the root Equ. for R : R = g(r +R +R + ) (so F = log(r ), G d = log( R d R d )) Exact expresson for R [BDG 3] R = R [] x[ + 3] x wth the notaton [] x = x [ + ] x [ + ] x wth R R(g) and x x(g) gven by R(g) = S 6g x(g) = 6 S / (+6g)S S 4g+ +S+6g > { x R = + 3gR x = gr ( + x + x ) wth S = g

19 The -pont functon of quadrangulatons () - - > + - wth R = F = log g(r +R +R + ) GF rooted well-labelled trees wth postve labels and label at the root Equ. for R : R = g(r +R +R + ) (so F = log(r ), G d = log( R d R d )) Exact expresson for R [BDG 3] R = R [] x[ + 3] x wth the notaton [] x = x [ + ] x [ + ] x wth R R(g) and x x(g) gven by R(g) = S 6g x(g) = 6 Fnal -pont functon expresson: > { x R = + 3gR x = gr ( + x + x ) S / (+6g)S S 4g+ +S+6g ( [d] x G d = log [d+3] x [d ] x [d+] x wth S = g )

20 Asymptotc consderatons Two-pont functon of (plane) trees: G d (g) = (gr ) d wth R = + gr = 4g g G d s the d th power of a seres havng a square-root sngularty d = 5 d/n / converges n law (Raylegh law, densty α exp( α ))

21 Asymptotc consderatons Two-pont functon of (plane) trees: G d (g) = (gr ) d wth R = + gr = 4g g G d s the d th power of a seres havng a square-root sngularty d = 5 d/n / converges n law (Raylegh law, densty α exp( α )) Two-pont functon of quadrangulatons: G d (g) d a x d + a x d + where x = x(g) has a quartc sngularty d/n /4 converges to an explct law [BDG 3]

22 Asymptotc consderatons Two-pont functon of (plane) trees: G d (g) = (gr ) d wth R = + gr = 4g g G d s the d th power of a seres havng a square-root sngularty d/n / converges n law (Raylegh law, densty α exp( α )) Two-pont functon of quadrangulatons: G d (g) d a x d + a x d + where x = x(g) has a quartc sngularty d = 5 d/n /4 converges to an explct law [BDG 3] Convergence n the two cases follows from (proof by Hankel contour) [Banderer, Flajolet, Louchard, Schaeffer 3]: x(g) ( g) s [g n ]x αns for < s <, e t Im(exp( αt s e πs ))dt g πn

23 Computng the two-pont and three-pont functon of quadrangulatons usng Mermont s bjecton

24 Well-labelled maps Well-labelled map = map where - each vertex v has a label l(v) Z - each edge e = {u, v} satsfes l(u) l(v) a well-labelled map M wth 3 faces Rk: Well-labelled tree = well-labelled map wth one face

25 Very-well-labelled quadrangulatons Very-well-labelled quadrangulaton = quadrangulaton where - each vertex v has a label l(v) Z - each edge e = {u, v} satsfes l(u) l(v) = - a very-well-labelled quadrangulaton Q wth 3 local mn - Rk: two types of faces stretched + + confluent Def: local mn= vertex wth all neghbours of larger label Rk: Geodesc labellng there s just one local mn, of label

26 The Mermont bjecton [Mermont 7], [Ambjørn, Budd 3] Very-well labelled quadrangulaton Q well-labelled map M n faces n edges local mn v face f non-local mn l(v) = mn(f) same label vertex

27 The Mermont bjecton [Mermont 7], [Ambjørn, Budd 3] Very-well labelled quadrangulaton Q well-labelled map M n faces n edges recover the Schaeffer bjecton (case of one local mn, of label ) local mn v face f non-local mn l(v) = mn(f) same label vertex

28 Proof of the stated propertes mples - (follows from the local rules) c From each corner c n a face of M starts a label-decreasng path of Q that stays n the face and ends at a local mn of Q

29 Proof of the stated propertes mples - (follows from the local rules) c From each corner c n a face of M starts a label-decreasng path of Q that stays n the face and ends at a local mn of Q

30 Proof of the stated propertes mples - (follows from the local rules) - c From each corner c n a face of M starts a label-decreasng path of Q that stays n the face and ends at a local mn of Q

31 Proof of the stated propertes mples - (follows from the local rules) - c - From each corner c n a face of M starts a label-decreasng path of Q that stays n the face and ends at a local mn of Q

32 Proof of the stated propertes mples - (follows from the local rules) - c - -3 From each corner c n a face of M starts a label-decreasng path of Q that stays n the face and ends at a local mn of Q

33 Proof of the stated propertes mples - (follows from the local rules) - c From each corner c n a face of M starts a label-decreasng path of Q that stays n the face and ends at a local mn of Q

34 Proof of the stated propertes mples - (follows from the local rules) - c From each corner c n a face of M starts a label-decreasng path of Q that stays n the face and ends at a local mn of Q Let n = # faces of Q, p = # local mn of Q, f = # faces of M Q M #V #E #F n + n n n + p n f = k + p Euler s relaton, wth k = # connected comp. of M

35 Proof of the stated propertes mples - (follows from the local rules) - c From each corner c n a face of M starts a label-decreasng path of Q that stays n the face and ends at a local mn of Q Let n = # faces of Q, p = # local mn of Q, f = # faces of M Q M #V #E #F n + n n n + p n f = k + p Euler s relaton, wth Drawng above f p k = # connected comp. of M Hence k = (M connected) f = p, and there s exactly one local mn of Q n each face of M

36 The case of two local mn d = 5 Γ the boundary, here mn Γ = f Γ - v - v - Mermont f dst(v, v ) = mn Γ l(v ) l(v )

37 The case of two local mn d = 5 Γ the boundary, here mn Γ = f Γ - v - v - Mermont f dst(v, v ) = mn Γ l(v ) l(v )

38 The case of two local mn d = 5 Γ the boundary, here mn Γ = f Γ - v - v - Mermont f dst(v, v ) = mn Γ l(v ) l(v ) Proof: v Γ, a shortest path v v v has length l(v) l(v ) l(v ) (because of the exstence of a label-decreasng path on each sde) - v l(v) l(v ) l(v) l(v ) - v v

39 Another way of computng the -pont functon [Boutter, Gutter 8] Let d and let s, t such that s + t = d A b-ponted quadrangulaton Q where d = d has a unque very-well labellng l(.) wth two local mn, at v, v, and l(v ) = s, l(v ) = t. l(.) s gven by l(v) = mn(dst(v, v) s, dst(v, v) t) d = 5 - v -3 v

40 Another way of computng the -pont functon [Boutter, Gutter 8] Let d and let s, t such that s + t = d A b-ponted quadrangulaton Q where d = d has a unque very-well labellng l(.) wth two local mn, at v, v, and l(v ) = s, l(v ) = t. l(.) s gven by l(v) = mn(dst(v, v) s, dst(v, v) t) d = v v - - -

41 Another way of computng the -pont functon [Boutter, Gutter 8] Let d and let s, t such that s + t = d A b-ponted quadrangulaton Q where d = d has a unque very-well labellng l(.) wth two local mn, at v, v, and l(v ) = s, l(v ) = t. l(.) s gven by l(v) = mn(dst(v, v) s, dst(v, v) t) d = 5 f - - v v f - -

42 Another way of computng the -pont functon [Boutter, Gutter 8] Let d and let s, t such that s + t = d A b-ponted quadrangulaton Q where d = d has a unque very-well labellng l(.) wth two local mn, at v, v, and l(v ) = s, l(v ) = t. l(.) s gven by l(v) = mn(dst(v, v) s, dst(v, v) t) d = 5 f Γ - - v v f - - The assocated well-labelled map wth two faces f, f satsfes: mn(f ) = s +, mn(f ) = t + mn Γ = (by precedng slde)

43 Another way of computng the -pont functon [Boutter, Gutter 8] Let d and let s, t such that s + t = d A b-ponted quadrangulaton Q where d = d has a unque very-well labellng l(.) wth two local mn, at v, v, and l(v ) = s, l(v ) = t. l(.) s gven by l(v) = mn(dst(v, v) s, dst(v, v) t) d = 5 f Γ - - v v f - - The assocated well-labelled map wth two faces f, f satsfes: mn(f ) = s +, mn(f ) = t + mn Γ = (by precedng slde)

44 nother way of computng the -pont functon We conclude that, for d = s + t (s, t ) G d (g) s the seres of mn(f )= s f s t v v Or ( := dscrete dfferentaton) G d = s t F s,t, where F s,t counts mn(f ) s f mn(f )= t Γ mn Γ = d st method corresponds to t = f f mn(f ) t Γ mn Γ =

45 nother way of computng the -pont functon Then by the lnk between cyclc and sequental excursons: F s,t = log(x s,t ) }{{} counts s t Equaton for X s,t : X s,t = + gr s R t X s,t ( + gr s+ R t+ X s+,t+ ) soluton (guessng/checkng): X s,t = [3] x[s+] x [t+] x [s+t+3] x [] x [s+3] x [t+3] x [s+t+] ) x recover G d = log ( [s+t] x [s+t+3] x [s+t ] x [s+t+] x

46 A frst covered case for the 3-pont functon [Boutter, Gutter 8] mn(f )= s f mn(f )= t Ths solves the case of 3 algned vertces s t Γ v 3 v v 3 v tr-ponted quadrangulatons wth d = s + t, d 3 = s, d 3 = t f mn Γ =.e., v 3 s on a geodesc path from v to v at respectve dstances s, t from v, v

47 A frst covered case for the 3-pont functon [Boutter, Gutter 8] mn(f )= s f mn(f )= t Ths solves the case of 3 algned vertces s t Γ v 3 v v 3 v tr-ponted quadrangulatons wth d = s + t, d 3 = s, d 3 = t f mn Γ = Hence G s+t,s,t (g) = s t X s,t.e., v 3 s on a geodesc path from v to v at respectve dstances s, t from v, v where X s,t = [3] x[s+] x [t+] x [s+t+3] x [] x [s+3] x [t+3] x [s+t+] x mn(f ) s X s,t counts f mn(f ) t Γ v 3 f mn Γ =

48 The dfferent cases for the 3-pont functon [Boutter, Gutter 8] D = (d, d 3, d 3 ) can be acheved only f d d 3 + d 3 d 3 d + d 3 d 3 d + d 3

49 The dfferent cases for the 3-pont functon [Boutter, Gutter 8] D = (d, d 3, d 3 ) can be acheved only f d d 3 + d 3 d 3 d + d 3 d 3 d + d 3 parametrze d = s + t d 3 = s + u d 3 = t + u wth s, t, u

50 The dfferent cases for the 3-pont functon [Boutter, Gutter 8] D = (d, d 3, d 3 ) can be acheved only f d d 3 + d 3 d 3 d + d 3 d 3 d + d 3 parametrze d = s + t d 3 = s + u d 3 = t + u wth s, t, u 3 ponts are dstnct at most one of s, t, u s zero One of s, t, u (say u) s zero algned ponts (precedng slde) Generc case: s, t, u > (non-algned ponts)

51 The generc case Boutter, Gutter 8] wrte D as d = s + t d 3 = s + u d 3 = t + u wth s, t, u > Endow Q wth unque very-well labellng wth 3 local mn at v, v, v 3 and where l(v )= s, l(v )= t, l(v 3 )= u Apply the Mermont bjecton f obtan a 3-face well-labelled map where mn(f )= s mn(f )= t mn(f 3 )= u mn Γ = mn Γ3 = mn Γ3 = f f 3 Γ Γ 3 Γ 3

52 The generc case Boutter, Gutter 8] wrte D as d = s + t d 3 = s + u d 3 = t + u wth s, t, u > Endow Q wth unque very-well labellng wth 3 local mn at v, v, v 3 and where l(v )= s, l(v )= t, l(v 3 )= u -s Apply the Mermont bjecton obtan a 3-face well-labelled map where mn(f )= s mn(f )= t mn(f 3 )= u mn Γ = mn Γ3 = mn Γ3 = f -u f 3 -t Γ Γ 3 Γ 3 f

53 The generc case Boutter, Gutter 8] wrte D as d = s + t d 3 = s + u d 3 = t + u wth s, t, u > Endow Q wth unque very-well labellng wth 3 local mn at v, v, v 3 and where l(v )= s, l(v )= t, l(v 3 )= u -s Apply the Mermont bjecton obtan a 3-face well-labelled map where mn(f )= s mn(f )= t mn(f 3 )= u mn Γ = mn Γ3 = mn Γ3 = f -u f 3 -t Γ Γ 3 Γ 3 expresson of G d,d 3,d 3 (g) as s t u F s,t,u, wth F s,t,u (g) explct f

54 Computng the two-pont functon of general maps usng the Ambjørn-Budd bjecton

55 The Ambjørn-Budd bjecton Λ [Ambjørn-Budd 3] Recall the Mermont bjecton Φ (reformulated by Ambjørn-Budd) Q local mn of Q face f of W - - Φ W mn(f) = +

56 The Ambjørn-Budd bjecton Λ [Ambjørn-Budd 3] Recall the Mermont bjecton Φ (reformulated by Ambjørn-Budd) Q local mn of Q face f of W local max of Q local max of W - - Φ W mn(f) = +

57 The Ambjørn-Budd bjecton Λ Recall the Mermont bjecton Φ (reformulated by Ambjørn-Budd) Q local mn of Q face f of W Let op : Z Z - local max of Q local max of W - - Φ = op Φ op Φ W W [Ambjørn-Budd 3] mn(f) = + - -

58 The Ambjørn-Budd bjecton Λ Recall the Mermont bjecton Φ (reformulated by Ambjørn-Budd) Q local mn of Q face f of W Let op : Z Z mn(f) = + local mn of W - local max of Q local max of W - - face f of W Φ = op Φ op Φ max(f) = W W [Ambjørn-Budd 3] - -

59 The Ambjørn-Budd bjecton Λ Recall the Mermont bjecton Φ (reformulated by Ambjørn-Budd) Q local mn of Q face f of W Let op : Z Z mn(f) = + local mn of W - local max of Q local max of W - - face f of W Φ = op Φ op Φ max(f) = W W Λ + Λ s a new dualty relaton for well-labelled map [Ambjørn-Budd 3] - -

60 The bjecton Λ appled to ponted maps Rk: ponted maps+geodesc labellng well-labelled maps wth one local mn, of label v M T ponted maps n edges well-labelled trees mn-label= and n edges (as for quadrang., but ths tme vertex of M v non-local max of T )

61 The bjecton Λ appled to ponted maps Rk: ponted maps+geodesc labellng well-labelled maps wth one local mn, of label v M T ponted maps n edges well-labelled trees mn-label= and n edges (as for quadrang., but ths tme vertex of M v non-local max of T ) k: In that case, Φ gves a new bjecton from ponted quadrangulatons wth n faces to ponted maps wth n edges that preserves the dstances to the ponted vertex (not the case wth the easy local bjecton)

62 The two-pont functon of general maps Let G d (g) the -pont functon of general maps d = v AB bjecton G d (g) s the seres of well-labelled trees wth mn-label wth a marked non local max of label d 3

63 The two-pont functon of general maps Let G d (g) the -pont functon of general maps d = v AB bjecton G d (g) s the seres of well-labelled trees wth mn-label wth a marked non local max of label d G d = F d F d, wth F d (g) := the seres of well-labelled trees wth postve labels and a marked non local max of label d 3

64 The two-pont functon of general maps Let G d (g) the -pont functon of general maps d = v AB bjecton G d (g) s the seres of well-labelled trees wth mn-label wth a marked non local max of label d G d = F d F d, wth F d (g) := the seres of well-labelled trees wth postve labels and a marked non local max of label d F = log g(r +R +R + ) log g(r +R ) = log( + gr R + ) 3 G d = log ( [d+] 3 x [d+3] [d] x [d+] 3 x ) for general maps

65 The two-pont functon of general maps Let G d (g) the -pont functon of general maps d = v AB bjecton G d (g) s the seres of well-labelled trees wth mn-label wth a marked non local max of label d G d = F d F d, wth F d (g) := the seres of well-labelled trees wth postve labels and a marked non local max of label d F = log g(r +R +R + ) log g(r +R ) = log( + gr R + ) G d = log recall G d = log ( [d+] 3 x [d+3] [d] x [d+] 3 x ) ( [d] x [d+3] x [d ] x [d+] x 3 for general maps ) for quadrang. (same asymptotc laws)

66 The case of two local mn Let M a well-labelled map wth two local mn v, v Let M = Λ(M), let f, f the two faces of M Let Γ the (cycle) boundary of M, := mn Γ Two cases: A): no edge of labels on Γ dst M (v, v ) = l(v ) l(v ) B): an edge of labels on Γ dst M (v, v ) = l(v ) l(v ) l(v ) l(v ) v + v l(v ) l(v ) - - v v

67 other ways to compute the -pont functon [F, Gutter 4] For d, let M a b-ponted map wth d = d A) Wrte d as s + t wth s, t. Endow M wth unque well-labellng where v, v are unque local mn and l(v ) = s, l(v ) = t - v v - - f - - B) Wrte d as s + t wth s, t. Endow M wth unque well-labellng where v, v are unque local mn and l(v ) = s, l(v ) = t v - - v f Γ f f Γ mn Γ = no edge - on Γ - mn Γ = edge - on Γ

68 other ways to compute the -pont functon ase (A): X s,t = G s+t (g) = s t log(n s,t ) }{{} N s,t gr s R t N s,t exact expresson for N s,t recover G s+t = log ( [s+t] x [s+t+3] x [s+t ] x [s+t+] x counts Rk: s t N s,t gves GF of tr-ponted maps wth algned ponts: d, d 3, d 3 = (s + t, s, t) ) mn(f ) s f f mn(f ) t Γ mn Γ = no edge on Γ v 3 Case (B): G s+t (g) = s t log( gr s R t N s,t ) mn(f ) s recover G s+t = log ( [s+t ] x [s+t+] x [s+t ] x [s+t+] x ) counts f f mn(f ) t Γ mn Γ = edges -

69 3-pont functon: generc (non-algned) case Case A: d + d 3 + d 3 even parametrze as: d = s + t d 3 = s + u d 3 = t + u wth s, t, u > endow tr-ponted map wth unque ( s, t, u)-well-labellng and apply the AB bjecton Λ mn(f )= s mn(f )= t mn(f 3 )= u mn Γ = mn Γ3 = mn Γ3 = and no edge - on Γ f f 3 Γ Γ 3 Γ 3 f

70 3-pont functon: generc (non-algned) case Case A: d + d 3 + d 3 even parametrze as: d = s + t d 3 = s + u d 3 = t + u wth s, t, u > endow tr-ponted map wth unque ( s, t, u)-well-labellng and apply the AB bjecton Λ -s mn(f )= s mn(f )= t mn(f 3 )= u mn Γ = mn Γ3 = mn Γ3 = and no edge - on Γ f -u f 3 -t Γ Γ 3 Γ 3 f

71 3-pont functon: generc (non-algned) case Case A: d + d 3 + d 3 even parametrze as: d = s + t d 3 = s + u d 3 = t + u wth s, t, u > endow tr-ponted map wth unque ( s, t, u)-well-labellng and apply the AB bjecton Λ -s mn(f )= s mn(f )= t mn(f 3 )= u mn Γ = mn Γ3 = mn Γ3 = and no edge - on Γ f -u f 3 -t Γ Γ 3 Γ 3 f expresson of G d,d 3,d 3 (g) as s t u F even s,t,u, wth F even s,t,u(g) explct

72 3-pont functon: generc (non-algned) case Case B: d + d 3 + d 3 odd (dd not exst for quadrang.) parametrze as: d = s + t d 3 = s + u d 3 = t + u wth s, t, u > endow tr-ponted map wth unque ( s, t, u)-well-labellng and apply the AB bjecton Λ mn(f )= s mn(f )= t mn(f 3 )= u mn Γ = mn Γ3 = mn Γ3 = and there s an edge - on each of Γ, Γ 3, Γ 3 f f 3 Γ Γ 3 Γ 3 f

73 3-pont functon: generc (non-algned) case Case B: d + d 3 + d 3 odd (dd not exst for quadrang.) parametrze as: d = s + t d 3 = s + u d 3 = t + u wth s, t, u > endow tr-ponted map wth unque ( s, t, u)-well-labellng and apply the AB bjecton Λ -s mn(f )= s mn(f )= t mn(f 3 )= u mn Γ = mn Γ3 = mn Γ3 = and there s an edge - on each of Γ, Γ 3, Γ f -u f 3 -t - - Γ Γ 3 Γ 3 - f

74 3-pont functon: generc (non-algned) case Case B: d + d 3 + d 3 odd (dd not exst for quadrang.) parametrze as: d = s + t d 3 = s + u d 3 = t + u wth s, t, u > endow tr-ponted map wth unque ( s, t, u)-well-labellng and apply the AB bjecton Λ -s mn(f )= s mn(f )= t mn(f 3 )= u mn Γ = mn Γ3 = mn Γ3 = and there s an edge - on each of Γ, Γ 3, Γ f -u f 3 -t - - Γ Γ 3 Γ 3 expresson of G d,d 3,d 3 (g) as s t u F odd s,t,u, wth F odd s,t,u(g) explct - - f

75 Examples Case A: - v - f v - - f f 3 - v 3 - Case B: - v f v f v f 3

76 Concluson and remarks There are exact expressons for the -pont and 3-pont functons of quadrangulatons and general maps (bjectons + GF calculatons) Asymptotcally the lmt laws (rescalng by n /4 ) are the same for the random quad. Q n of sze n as for the random map M n of sze n Rk: also follows from [Bettnell, Jacob, Mermont 3] (Q n, dst/n /4 ) and (M n, dst/n /4 ) are close as metrc spaces, when couplng (M n, Q n ) by the AB bjecton We can also obtan smlar expressons for bpartte maps (assocated well-labelled maps are restrcted to have no edge ) The GF expressons G D (g) for maps/bpartte maps can be extended to expressons G D (g, z) where z marks the number of faces