Geodesics in large planar quadrangulations J er emie Bouttier, - PowerPoint PPT Presentation

Geodesics in large planar quadrangulations J´ er´ emie Bouttier, Emmanuel Guitter arXiv:0712.2160 , arXiv:0805.2355 and work in progress Institut de Physique Th´ eorique, CEA Saclay INRIA, 29 September 2008

Outline Statistics of geodesics Geodesic points Geodesic loops Confluence of geodesics

Introduction Reminder : geodesic = shortest path between two points v 1 v 2

Outline Statistics of geodesics Geodesic points Geodesic loops Confluence of geodesics

Quadrangulations with geodesic boundary geodesic geodesic boundary boundary

Quadrangulations with geodesic boundary 0 simply pointed 1 1 2 2 3 i − 2 i − 1 i − 1 i

Quadrangulations with geodesic boundary Schaeffer 1 well − labeled 2 tree min ℓ ( v ) = 1 i − 1 i

Quadrangulations with geodesic boundary 1 2 min ℓ ( v ) = 1 i − 1 i i Π R m m= 1

Quadrangulations with geodesic boundary The generating function for quadrangulations with geodesic boundary is therefore: i R j = R i (1 − x )(1 − x i +3 ) � Z i ( g ) = (1 − x 3 )(1 − x i +1 ) j =1 Reminder: g weight per square, R ( g ) = 1 − √ 1 − 12 g 1 1 x ( g ) + x ( g ) + 1 = gR ( g ) 2 6 g

Quadrangulations with a marked geodesic

Quadrangulations with a marked geodesic Almost the same as quadrangulations with geodesic boundary... 0 0 0 1 1 1 1 1 2 2 2 2 2 3 3 3 3 i− 2 i− 1 i− 1 i− 1 i− 1 i− 1 i i i

Quadrangulations with a marked geodesic Arbitrary geodesic boundaries may have “pinch points”. Marked geodesics correspond to irreducible boundaries. 0 0 0 1 1 1 1 1 2 2 2 2 3 3 3 i− 2 i− 1 2 i− i− 1 i− 1 1 i− i− 1 i i i

Quadrangulations with a marked geodesic An arbitrary geodesic boundary may be decomposed into irreducible components. 0 0 0 1 1 U 1 1 1 1 j j i− 1 i Σ Z = U = = Z − i i j= 1 Z i−j i i i

Quadrangulations with a marked geodesic i − 1 ˆ Z ( g ; t ) � ˆ U i ( g ) = Z i ( g ) − U j ( g ) Z i − j ( g ) i . e . U ( g ; t ) = 1 + ˆ Z ( g ; t ) j =1 From the exact formula for Z i we can perform asymptotic analysis: 12 n U i ( g ) | g n ∼ 2 √ π n 5 / 2 δ i as n → ∞ where: 3 t ( 2 t ( 3+177 t − 412 t 2 +708 t 3 − 624 t 4 +224 t 5 ) +3(1 − 2 t ) 6 log(1 − 2 t ) ) ˆ δ ( t ) = 70(1 − 2 t ) 4 ( t − (1 − 2 t ) log(1 − 2 t )) 2

Quadrangulations with a marked geodesic i − 1 ˆ Z ( g ; t ) � ˆ U i ( g ) = Z i ( g ) − U j ( g ) Z i − j ( g ) i . e . U ( g ; t ) = 1 + ˆ Z ( g ; t ) j =1 From the exact formula for Z i we can perform asymptotic analysis: 12 n U i ( g ) | g n ∼ 2 √ π n 5 / 2 δ i as n → ∞ where: δ i ∼ 9 72 i i 3 as i → ∞

Quadrangulations with a marked geodesic In the local limit: 12 n 2 √ π n 5 / 2 × 3 7 · i 3 × 3 · 2 i U i ( g ) | g n ∼

Quadrangulations with a marked geodesic In the local limit: 12 n 2 √ π n 5 / 2 × 3 7 · i 3 × 3 · 2 i U i ( g ) | g n ∼ 12 n 2 √ π n 5 / 2 : asymptotic number of pointed quadrangulations ◮

Quadrangulations with a marked geodesic In the local limit: 12 n 2 √ π n 5 / 2 × 3 7 · i 3 × 3 · 2 i U i ( g ) | g n ∼ 12 n 2 √ π n 5 / 2 : asymptotic number of pointed quadrangulations ◮ ◮ 3 7 · i 3 : average number of vertices at distance i ≫ 1 from the origin

Quadrangulations with a marked geodesic In the local limit: 12 n 2 √ π n 5 / 2 × 3 7 · i 3 × 3 · 2 i U i ( g ) | g n ∼ 12 n 2 √ π n 5 / 2 : asymptotic number of pointed quadrangulations ◮ ◮ 3 7 · i 3 : average number of vertices at distance i ≫ 1 from the origin ◮ 3 · 2 i : mean number of geodesics between two given points at distance i ≫ 1

Quadrangulations with a marked geodesic In the local limit: 12 n 2 √ π n 5 / 2 × 3 7 · i 3 × 3 · 2 i U i ( g ) | g n ∼ 12 n 2 √ π n 5 / 2 : asymptotic number of pointed quadrangulations ◮ ◮ 3 7 · i 3 : average number of vertices at distance i ≫ 1 from the origin ◮ 3 · 2 i : mean number of geodesics between two given points at distance i ≫ 1 A similar result holds in the scaling limit i = r · n 1 / 4 : 12 n 2 √ π n 7 / 4 × ρ ( r ) × 3 · 2 i U i ( g ) | g n ∼ ρ ( r ): canonical two-point function

Geodesic watermelons Our method does not easily give access to higher moments for the number of geodesics. We shall consider quadrangulations with several marked geodesics, which might have complicated crossings.

Geodesic watermelons Our method does not easily give access to higher moments for the number of geodesics. We shall consider quadrangulations with several marked geodesics, which might have complicated crossings. However one can consider “geodesic watermelons”: sets of k non-crossing geodesics with common endpoints. These correspond to k quadrangulations with geodesic boundary placed side-by-side.

Geodesic watermelons 0 0 0 i i i

Geodesic watermelons Our method does not easily give access to higher moments for the number of geodesics. We shall consider quadrangulations with several marked geodesics, which might have complicated crossings. However one can consider “geodesic watermelons”: sets of k non-crossing geodesics with common endpoints. These correspond to k quadrangulations with geodesic boundary placed side-by-side.

Geodesic watermelons Our method does not easily give access to higher moments for the number of geodesics. We shall consider quadrangulations with several marked geodesics, which might have complicated crossings. However one can consider “geodesic watermelons”: sets of k non-crossing geodesics with common endpoints. These correspond to k quadrangulations with geodesic boundary placed side-by-side. ◮ Weakly avoiding case: the whole must be irreducible i − 1 U ( k ) = ( Z i ) k − U ( k ) � ( Z i − j ) k i j j =1

Geodesic watermelons Our method does not easily give access to higher moments for the number of geodesics. We shall consider quadrangulations with several marked geodesics, which might have complicated crossings. However one can consider “geodesic watermelons”: sets of k non-crossing geodesics with common endpoints. These correspond to k quadrangulations with geodesic boundary placed side-by-side. ◮ Weakly avoiding case: the whole must be irreducible i − 1 U ( k ) = ( Z i ) k − U ( k ) � ( Z i − j ) k i j j =1 ◮ Strongly avoiding case: each part must be irreducible U ( k ) ˜ = ( U i ) k i

Geodesic watermelons In the weakly avoiding case, in the local limit: 12 n 2 √ π n 5 / 2 × 3 � 7 · i 3 × k · 3 · 2 i � k U ( k ) � ( g ) g n ∼ � i � 3 · 2 i � k : mean number of k -watermelons � k ·

Geodesic watermelons In the weakly avoiding case, in the local limit: 12 n 2 √ π n 5 / 2 × 3 � 7 · i 3 × k · 3 · 2 i � k U ( k ) � ( g ) g n ∼ � i � 3 · 2 i � k : mean number of k -watermelons � k · The k factor corresponds to symmetry breaking: among the k delimited regions, only one has macroscopic ( ∝ n ) size.

Geodesic watermelons In the weakly avoiding case, in the local limit: 12 n 2 √ π n 5 / 2 × 3 � 7 · i 3 × k · 3 · 2 i � k U ( k ) � ( g ) g n ∼ � i � 3 · 2 i � k : mean number of k -watermelons � k · The k factor corresponds to symmetry breaking: among the k delimited regions, only one has macroscopic ( ∝ n ) size. Further computations ( k = 2): ◮ two weakly avoiding geodesics of length i ≫ 1 have in average i / 3 common vertices ◮ they delimit two regions with respective areas n vs O ( i 3 )

Geodesic watermelons In the weakly avoiding case, in the local limit: 12 n 2 √ π n 5 / 2 × 3 � 7 · i 3 × k · 3 · 2 i � k U ( k ) � ( g ) g n ∼ � i � 3 · 2 i � k : mean number of k -watermelons � k · The k factor corresponds to symmetry breaking: among the k delimited regions, only one has macroscopic ( ∝ n ) size. Further computations ( k = 2): ◮ two weakly avoiding geodesics of length i ≫ 1 have in average i / 3 common vertices ◮ they delimit two regions with respective areas n vs O ( i 3 ) Similar results hold in the scaling limit: 12 n � 3 · 2 i � k U ( k ) � ( g ) g n ∼ 2 √ π n 7 / 4 × ρ ( r ) × k · � i �

Geodesic watermelons In the strongly avoiding case, in the local limit: 2 √ π n 5 / 2 × 3 · 4 k − 1 12 n � i 6 − 3 k × k · (3 · 2 i ) k U ( k ) ˜ ( g ) g n ∼ � i 7 �

Geodesic watermelons In the strongly avoiding case, in the local limit: 2 √ π n 5 / 2 × 3 · 4 k − 1 12 n � i 6 − 3 k × k · (3 · 2 i ) k U ( k ) ˜ ( g ) g n ∼ � i 7 � The constraint of strong avoidance is relevant. In the scaling limit: 12 n � 3 · 2 i � k U ( k ) ˜ 2 √ π n 3 k / 4+1 × σ ( k ) ( r ) × k · � ( g ) g n ∼ � i � σ ( k ) ( r ): new scaling functions