Random planar maps, alternating knots and links Gilles Sc - PowerPoint PPT Presentation

Random planar maps, alternating knots and links Gilles Sc hae�er CNRS � S�bastien Kunz-Jacques Join w ork with (LIX � Corps des t�l�coms) 1

An overview of the talk The enumeration of maps examples of algebraic functions Random plana r maps almost sure prop erties Enumerative knot theo ry ? prime alternating links Asymptotic enumeration of links as an application of random maps 2

Ro oted plana r maps. De�nition a plana r map = an em b edding of a connected graph in the plane. planar map = planar graph + cyclic order around v ertices. W e consider r o ote d planar maps: a r o ot edge is c hosen around the in�nite face and orien ted coun terclo c kwise. 3

Ro oted plana r maps. Examples The smallest maps: A planar map with only one face is a plane tr e e . A planar map with only one v ertex is a cycle of lo ops. 4

Ro oted plana r maps. On the sphere ? Sometimes I lik e to replace the plane b y a sphere : : : This is equiv alen t but lo oks more symmetric: all faces are simply connected (=disc). ! nicer pictures but that are more di�cult to do : : : 5

Ro oted plana r maps. Example of subfamilies T riangulations 4-regular maps Suc h lo cal restrictions should b e irrelev an t in the large size limit. Compare to simple trees : m -ary trees, plane trees, 1-2 trees ) they usually are all the same. 6

Enumeration of maps in combinato rics as opp osed to physics & enumerative top ology � T utte (1962): a c ensus of triangulations Originally to attac k the four color theorem via en umeration. � Counting plana r maps (70's): T utte, Bro wn, Mullin, Cori, Liu : : : Results for more than t w en t y subfamilies of planar maps. : : : Gao-W ormald (2001) 5-c onne cte d triangulations � Maps on surfaces (80's), random plana r maps (90's): Bender, Can�eld, Arqu�s, Gao, Ric hmond, W ormald, : : : F or instance, Bender-Compton-Ric hmond (1999): 0-1 laws for F O lo gic pr op erties of r andom maps on surfac es. 7

Enumeration via generating functions just one step a w a y from plane trees... 8

6 T utte's ro ot deletion metho d. (i) plane trees Usual plane tr e es are exactly maps with one face. T’’ T’ T’ T’’ 0 00 X X j T j 1+ j T j + j T j z = z 0 00 T = � T ;T 2 Th us the equation t ( z ) � 1 = z t ( z ) . Plane tr e es (and in gener al simple tr e es) have algebr aic GF. 9

6 T utte's ro ot deletion metho d. (ii) lo ops A map with only one v ertex is a cycle of lo ops . 1 1 5 2 2 4 3 3 X X j w j d ( w ) 1+ j w j 2 d ( w )+1 z u = z ( u + u + � � � + u ) w = � w 2 z u z u L ( z ; u ) � 1 = L ( z ; u ) � ` ( z ) : u � 1 u � 1 A line ar e quation in L ( z ; u ) with p olynomial c o e�s in z , u and ` ( z ) : 2 ( u � 1 + z u ) L ( z ; u ) = u � 1 � z u ` ( z ) : 10

T utte's ro ot deletion metho d. (iii) all maps The t w o previous cases generalize: 1 2 6 M’ M’’ M’ M’’ 3 5 4 2 2 z u F ( z ; u ) � 1 = z u F ( z ; u ) + ( uF ( z ; u ) � f ( z )) u � 1 or equiv alen tly (dep endenc es in z hidden) 2 2 2 ( u � 1) z u F ( u ) + ( u � 1 + z u ) F ( u ) + ( u � 1 � z uf ) = 0 a quadr atic equation in F ( u ) with p olynomial co e�s in z , u and f . 11

Linea r equations with a catalytic va riable. The k ernel metho d. The k ernel metho d for K ( u ) L ( u ) = q ( z ; u; ` ) : � L o ok for a r o ot u of K ( u ) such that L ( u ) makes sense . 0 0 Here L ( u ) [ u ][[ z ]] and the ro ots of K ( u ) are C 2 p p 1+ 1 � 4 z 1 � 1 � 4 z 2 u = = 1 =z + O (1) and u = = z + O ( z ) : 1 2 2 z 2 z L ( u ) is not ok but L ( u ) con v erges as a formal p o w er series. 1 2 The substitution u u in the linear equation giv es 0 p 1 � 1 � 4 z 0 = u � 1 + z u `; so that ` ( z ) = 2 2 2 z (See also Cyril Banderier's talk) 12

P olynomial equations with a catalytic va riable. Bousquet-Melou's metho d � (extends k ernel & T utte's quadratic metho ds) F ( z ; u ) and f ( z ) such that ther e is a p olynomial P ( a; b; c ) with P ( F ( u ) ; u; f ) = 0 (dep endenc e in z hidden) Di�eren tiate with resp ect to u : � 0 0 0 F ( u ) P ( F ( u ) ; u; f ) + P ( F ( u ) ; u; f ) = 0 u a b Supp ose w e �nd u = u ( z ) suc h that F ( u ) is w ell de�ned and � 0 0 0 0 P ( F ( u ) ; u ; f ) = 0 : 0 0 b 0 Then P ( F ( u ) ; u ; f ) = 0 and P ( F ( u ) ; u ; f ) = 0 : 0 0 0 0 a A p olynomial system in F ( u ) , u , f : algebr aic solutions ! 0 0 13

Ro oted plana r maps. The solution � W e obtain an algebraic generating function f ( z ) : 3 = 2 1 � 18 z � (1 � 12 z ) X j M j f ( z ) = z = 1 � 2 54 z M (remark that F ( z ; u ) is also algebraic ! face degree) � T ransfert theorems ( e.g. ) yield an asymptotic expansion: � 5 = 2 n # f ro oted maps with n edges g c � n � 12 : The exp onent 5 = 2 is char acteristic of planar map emuner ations (c omp ar e to 3 = 2 for various simple tr e es). 14

A �rst summa ry . � P olynomial equations with one catalytic v ariable should ha v e algebraic solutions (cf. Mireille Bousquet-M�lou): P ( F ( z ; u ) ; z ; u; f ( z ) ; : : : ; f ( z )) = 0 1 k (if y ou kno w examples, w e are in terested in collecting them !) � Ro ot deletion applies to man y families of maps and yields �univ ersal� asymptotic b eha vior: � n � 5 = 2 # f ro oted F -maps of size n g = c� n where c and � dep end on the family F . � In some cases the explicit form ulas are nice. 15

Nice fo rmulas, random maps and why plana r maps a re almost Galton-W atson trees 16

T utte's fo rmulas fo r ro oted plana r maps. (60's) The ro ot deletion metho d pro vides surprisingly nice form ulas in sev eral cases, among whic h: n � � 2 2 3 n c 1 n # f triangulations with 2 n faces g = � (27 = 2) 5 = 2 2 n + 2 2 n + 1 n n n � � 2 3 2 n c 2 n # f 4-regular maps with n v ert. g = � 12 5 = 2 n + 2 n + 1 n n All families should b eha v e the same ) concen trate on those simpler mo dels ! (lik e binary trees in tree en umeration, b ernoulli w alks, : : : ) 17

T utte's fo rmulae. A bijectiv e pro of (i). n 2 n 2 3 � � # f 4-regular maps with n v ertices g is � . n +2 n +1 n There are � � 1 2 n n + 1 n binary trees with n no des. Suc h trees ha v e n (in ternal) no des and n + 2 lea v es (ro ot included). 18

T utte's fo rmulae. A bijectiv e pro of (ii). n 2 n 2 3 � � # f 4-regular maps with n v ertices g is � . n +2 n +1 n On eac h no de, a bud can b e added in three w a ys, giving rise to n 3 � 2 n � n + 1 n blossom tr e es with n no des. Blossom trees ha v e n buds and n + 2 lea v es around the tree. Up on matc hing them coun terclo c kwise, t w o lea v es remain unmatche d . 19

T utte's fo rmulae. A bijectiv e pro of (ii). n 2 n 2 3 � � # f 4-regular maps with n v ertices g is � . n +2 n +1 n On eac h no de, a bud can b e added in three w a ys, giving rise to n 3 � 2 n � n + 1 n blossom tr e es with n no des. Blossom trees ha v e n buds and n + 2 lea v es around the tree. Up on matc hing them coun terclo c kwise, t w o lea v es remain unmatche d . 20

T utte's fo rmulae. A bijectiv e pro of (ii). n 2 n 2 3 � � # f 4-regular maps with n v ertices g is � . n +2 n +1 n On eac h no de, a bud can b e added in three w a ys, giving rise to n 3 � 2 n � n + 1 n blossom tr e es with n no des. Blossom trees ha v e n buds and n + 2 lea v es around the tree. Up on matc hing them coun terclo c kwise, t w o lea v es remain unmatche d . 21

T utte's fo rmulae. A bijectiv e pro of (ii). n 2 n 2 3 � � # f 4-regular maps with n v ertices g is � . n +2 n +1 n On eac h no de, a bud can b e added in three w a ys, giving rise to n 3 � 2 n � n + 1 n blossom tr e es with n no des. Blossom trees ha v e n buds and n + 2 lea v es around the tree. Up on matc hing them coun terclo c kwise, t w o lea v es remain unmatche d . 22