Eulerian orientations and the six-vertex model on planar maps - PowerPoint PPT Presentation

Eulerian orientations and the six-vertex model on planar maps Andrew Elvey Price Joint work with Mireille Bousquet-Mélou and Paul Zinn-Justin Université de Bordeaux, France 02/07/2019 Eulerian orientations and the six-vertex model on planar maps Andrew Elvey Price

P LANAR MAPS = � = Eulerian orientations and the six-vertex model on planar maps Andrew Elvey Price

R OOTED PLANAR MAPS = � = Eulerian orientations and the six-vertex model on planar maps Andrew Elvey Price

A CHRONOLOGY OF PLANAR MAPS Random maps Bijections (enumeration) Matrix integrals (enumeration) Recursive approach (enumeration) 1960 1978 1981 1995 2000 • Recursive approach: Tutte, Brown, Bender, Canfield, Richmond, Goulden, Jackson, Wormald, Walsh, Lehman, Gao, Wanless... • Matrix integrals: Brézin, Itzykson, Parisi, Zuber, Bessis, Ginsparg, Kostov, Zinn-Justin, Boulatov, Kazakov, Mehta, Bouttier, Di Francesco, Guitter, Eynard... • Bijections: Cori & Vauquelin, Schaeffer, Bouttier, Di Francesco & Guitter (BDG), Bernardi, Fusy, Poulalhon, Bousquet-Mélou, Chapuy... • Geometric properties of random maps: Chassaing & Schaeffer, BDG, Marckert & Mokkadem, Jean-François Le Gall, Miermont, Curien, Albenque, Bettinelli, Ménard, Angel, Sheffield, Miller, Gwynne... Eulerian orientations and the six-vertex model on planar maps Andrew Elvey Price

M APS EQUIPPED WITH AN ADDITIONAL STRUCTURE • How many maps equipped with... a spanning tree [Mullin 67, Bernardi] a spanning forest? [Bouttier et al., Sportiello et al., Bousquet-Mélou & Courtiel] a self-avoiding walk? [Duplantier & Kostov; Gwynne & Miller] a proper q -colouring? [Tutte 74-83, Bouttier et al.] a bipolar orientation? [Kenyon, Miller, Sheffield, Wilson, Fusy, Bousquet-Mélou...] • What is the expected partition function of... the Ising model? [Boulatov, Kazakov, Bousquet-Mélou, Schaeffer, Chen, Turunen, Bouttier et al., Albenque, Ménard...] the hard-particle model? [Bousquet-Mélou, Schaeffer, Jehanne, Bouttier et al.] the Potts model? [Eynard-Bonnet, Baxter, Bousquet-Mélou & Bernardi, Guionnet et al., Borot et al., ...] Eulerian orientations and the six-vertex model on planar maps Andrew Elvey Price

E ULERIAN ORIENTATIONS GENERATING FUNCTIONS The 6-vertex model Q ( t , γ ) Non-alternating G ( t ) = 2 Q ( t , 0 ) (weight t ) The 4-valent case: the ice model Q ( t , 1 ) Alternating (weight tγ ) Each vertex has equally many incoming as outgoing edges. Eulerian orientations and the six-vertex model on planar maps Andrew Elvey Price

Part 1: Counting Eulerian orientations Eulerian orientations and the six-vertex model on planar maps Andrew Elvey Price

E ULERIAN ORIENTATIONS Aim: Determine the number g n of (rooted planar) Eulerian orientations with n edges ∞ g n t n = t + 5 t 2 + . . . � The generating function G ( t ) = t = 1 Eulerian orientations and the six-vertex model on planar maps Andrew Elvey Price

E NUMERATING E ULERIAN ORIENTATIONS Problem posed by Bonichon, Bousquet-Mélou, Dorbec and Pennarun in 2016. In 2017, E.P. and Guttmann: Computed the number g n of Eulerian orientations for n < 100. Predicted that ( 4 π ) n g n ∼ κ g n 2 (log n ) 2 . This led us to conjecture the exact solution. Eulerian orientations and the six-vertex model on planar maps Andrew Elvey Price

P REVIEW : E XACT SOLUTION Let R 0 ( t ) be the unique power series with constant term 0 satisfying ∞ � 2 � 2 n 1 � R 0 ( t ) n + 1 . t = n + 1 n n = 0 ∞ g n t n of rooted planar Eulerian � The generating function G ( t ) = n = 0 orientations counted by edges is given by G ( t ) = 1 4 t 2 ( t − 2 t 2 − R 0 ( t )) . Eulerian orientations and the six-vertex model on planar maps Andrew Elvey Price

E ULERIAN ORIENTATIONS OUTLINE Bijections Functional equations Guess and check solution Eulerian orientations and the six-vertex model on planar maps Andrew Elvey Price

Step 1: Bijection to labelled maps (EP and Guttmann, 2017) Eulerian orientations and the six-vertex model on planar maps Andrew Elvey Price

B IJECTION TO LABELLED MAPS − 1 0 1 +1 1 0 +1 1 0 1 2 +1 − 1 − 1 − 1 0 1 − 1 3 2 1 0 ℓ + 1 ℓ 1 Eulerian orientations and the six-vertex model on planar maps Andrew Elvey Price

L ABELLED MAPS Labelled maps are rooted planar maps with labelled vertices such that: The root edge is labelled from 0 to 1. Adjacent labels differ by 1. By the bijection, G ( t ) counts labelled maps by edges. 1 0 1 2 − 1 1 3 2 1 0 1 Eulerian orientations and the six-vertex model on planar maps Andrew Elvey Price

L ABELLED QUADRANGULATIONS By our bijection, Q ( t , γ ) counts labelled quadrangulations by faces ( t ) and alternating faces ( γ ). ℓ + 2 ℓ + 1 ℓ ℓ + 1 ℓ + 1 ℓ + 1 ℓ ℓ Non-alternating Alternating (weight t ) (weight tγ ) Eulerian orientations and the six-vertex model on planar maps Andrew Elvey Price

L ABELLED QUADRANGULATIONS By our bijection, Q ( t , γ ) counts labelled quadrangulations by faces ( t ) and alternating faces ( γ ). Q ( t , 0 ) counts labelled quadrangulations with no alternating faces. 1 0 1 2 3 − 1 2 1 1 Eulerian orientations and the six-vertex model on planar maps Andrew Elvey Price

E ULERIAN ORIENTATIONS Step 2: Bijection between labelled quadrangulations with no alternating faces and labelled maps (Miermont (2009)/Ambjørn and Budd (2013)). Eulerian orientations and the six-vertex model on planar maps Andrew Elvey Price

L ABELLED QUADRANGULATIONS TO LABELLED MAPS Highlight edges according to the rule. The red edges (sometimes) form a labelled map. The bijection implies that Q ( t , 0 ) = 2 G ( t ) . 1 0 ℓ + 2 0 − 1 − 1 − 1 1 − 2 ℓ + 1 ℓ + 1 0 ℓ − 1 − 1 Eulerian orientations and the six-vertex model on planar maps Andrew Elvey Price

Exact solution using labelled quadrangulations at γ = 0 (Bousquet-Mélou and E.P.) Eulerian orientations and the six-vertex model on planar maps Andrew Elvey Price

D ECOMPOSITION OF LABELLED QUADRANGULATIONS 0 1 0 1 0 0 − 1 0 1 − 1 1 2 0 − 1 − 1 1 1 1 0 C 1 0 1 2 0 1 1 2 1 2 1 0 P D Eulerian orientations and the six-vertex model on planar maps Andrew Elvey Price

E QUATIONS FOR PLANAR E ULERIAN ORIENTATIONS The series 2 G ( t ) = Q ( t , 0 ) is given by Q ( t , 0 ) = [ y 1 ] P ( t , y ) − 1 , where the series P ( t , y ) , C ( t , x , y ) and D ( t , x , y ) are characterised by the equations P ( t , 0 ) = 1 P ( t , y ) = 1 y [ x 1 ] C ( t , x , y ) , 1 D ( t , x , y ) = � , � 1 1 − C t , 1 − x , y � � �� t , 1 C ( t , x , y ) = xy [ x ≥ 0 ] P ( t , tx ) D x , y , We solve these using a guess and check method. Eulerian orientations and the six-vertex model on planar maps Andrew Elvey Price

S OLUTION FOR PLANAR E ULERIAN ORIENTATIONS Let R 0 ( t ) be the unique power series with constant term 0 satisfying ∞ � 2 � 2 n 1 � R 0 ( t ) n + 1 . t = n + 1 n n = 0 Then the series P ( t , y ) , C ( t , x , y ) and D ( t , x , y ) are given by: n � 2 n �� 2 n − j � 1 � � y j R n + 1 t P ( t , ty ) = , 0 n + 1 n n n ≥ 0 j = 0   n n � 2 n − j �� 2 n − i � 1 � � � x i + 1 y j + 1 R n + 1  , C ( t , x , ty ) = 1 − exp  − 0 n + 1 n n n ≥ 0 j = 0 i = 0   n � 2 n − j �� 2 n + i + 1 � 1 � � �  . x i y j + 1 R n + 1 D ( t , x , ty ) = exp 0 n + 1 n n n ≥ 0 j = 0 i ≥ 0 Eulerian orientations and the six-vertex model on planar maps Andrew Elvey Price

S OLUTION FOR PLANAR E ULERIAN ORIENTATIONS Let R 0 ( t ) be the unique power series with constant term 0 satisfying ∞ � 2 � 2 n 1 � R 0 ( t ) n + 1 , t = n + 1 n n = 0 Then the generating function of rooted planar Eulerian orientations counted by edges is G ( t ) = 1 2 Q ( t , 0 ) = 1 4 t 2 ( t − 2 t 2 − R 0 ( t )) . Asymptotically, the coefficients behave as µ n + 2 g n ∼ κ n 2 (log n ) 2 , where κ = 1 / 16 and µ = 4 π . Eulerian orientations and the six-vertex model on planar maps Andrew Elvey Price

S OLUTION FOR QUARTIC E ULERIAN ORIENTATIONS Let R 1 ( t ) be the unique power series with constant term 0 satisfying ∞ � 2 n �� 3 n � 1 � R 1 ( t ) n + 1 , t = n + 1 n n n = 0 Then the generating function of quartic rooted planar Eulerian orientations counted by edges is Q ( t , 1 ) = 1 3 t 2 ( t − 3 t 2 − R 1 ( t )) . Asymptotically, the coefficients behave as µ n + 2 q n ∼ κ n 2 (log n ) 2 , √ where κ = 1 / 18 and µ = 4 3 π . Eulerian orientations and the six-vertex model on planar maps Andrew Elvey Price

Part 2: Solution for general γ (Kostov/ E.P. and Zinn-Justin) ℓ + 2 ℓ + 1 ℓ ℓ + 1 ℓ + 1 ℓ ℓ Non-alternating Alternating (weight t ) (weight tγ ) Eulerian orientations and the six-vertex model on planar maps Andrew Elvey Price

B ACKGROUND ( FROM PHYSICS ) Solved at criticality by Zinn-Justin in 2000. Exactly solved by Kostov later in 2000 (to the satisfaction of physicists). Solution was not completely rigorous. We corrected a mistake and simplified the form of the solutions. Eulerian orientations and the six-vertex model on planar maps Andrew Elvey Price