The local limit of uniform triangulations in high genus Thomas Budzinski (joint work with Baptiste Louf) ENS Paris and Université Paris Saclay 2019, March 6th UBC probability seminar Thomas Budzinski High genus triangulations
Motivations We would like to define a discrete "random two-dimensional geometry", in a way that is as uniform as possible. Regular prototypes: the 6-regular and the 7-regular triangular lattices. For physicists, discrete model of "2d quantum gravity". Basic idea: use finite random objects, and take the limit. Thomas Budzinski High genus triangulations
Finite triangulations A triangulation with 2 n faces is a set of 2 n triangles whose sides have been glued two by two, in such a way that we obtain a connected, orientable surface. The genus g of the triangulation is the number of holes of this surface ( g = 0 on the figure). Our triangulations are of type I (we may glue two sides of the same triangle), and rooted (oriented root edge). Thomas Budzinski High genus triangulations
Finite triangulations A triangulation with 2 n faces is a set of 2 n triangles whose sides have been glued two by two, in such a way that we obtain a connected, orientable surface. The genus g of the triangulation is the number of holes of this surface ( g = 0 on the figure). Our triangulations are of type I (we may glue two sides of the same triangle), and rooted (oriented root edge). Thomas Budzinski High genus triangulations
Finite triangulations − → A triangulation with 2 n faces is a set of 2 n triangles whose sides have been glued two by two, in such a way that we obtain a connected, orientable surface. The genus g of the triangulation is the number of holes of this surface ( g = 0 on the figure). Our triangulations are of type I (we may glue two sides of the same triangle), and rooted (oriented root edge). Thomas Budzinski High genus triangulations
Some combinatorics Let T n , g be the set of triangulations of genus g with 2 n faces, and τ ( n , g ) its size. Let also τ p ( n , g ) be the number of triangulations of size n and genus g , where the face on the right of the root has perimeter p . Can we compute those numbers? In the planar case, exact formulas [Tutte, 60s] : √ 4 n ( 3 n )!! � 6 3 ) n n − 5 / 2 , τ ( n , 0 ) = 2 ∼ π ( 12 ( n + 1 )!( n + 2 )!! n → + ∞ where n !! = n ( n − 2 )( n − 4 ) ... . We also know τ p ( n , 0 ) explicitely. In general, double recurrence relations [Goulden–Jackson, 2008] , but no close formula. Known asymptotics when n → + ∞ with g fixed, but not when both n , g → + ∞ . Thomas Budzinski High genus triangulations
Local convergence For any (finite or infinite) rooted triangulation t and r ≥ 0, let B r ( t ) be the ball of radius r around the root vertex in t . For any two triangulations t and t ′ , set � − 1 . d loc ( t , t ′ ) = � 1 + max { r ≥ 0 | B r ( t ) = B r ( t ′ ) } This is the local distance : we focus on the neighbourhood of the root. Let T n , g be uniform in T n , g . We want to study local limits of T n , g when both n and g go to infinity. Thomas Budzinski High genus triangulations
The planar case Theorem (Angel–Schramm, 2003) We have the convergence ( d ) − − − − → T n , 0 n → + ∞ T in distribution for the local topology, where T is an infinite triangulation of the plane called the UIPT (Uniform Infinite Planar Triangulation). Quick sketch of the proof: if t has size v and perimeter p , then P ( t ⊂ T n , 0 ) = τ p ( n − v , 0 ) , τ ( n , 0 ) and the limit is given by the results of Tutte. Thomas Budzinski High genus triangulations
A sample of T 32400 , 0 Thomas Budzinski High genus triangulations
The UIPT Thomas Budzinski High genus triangulations
The spatial Markov property of T Let t be a small triangulation with perimeter p and v vertices in total. ⊂ t ( p = 6 , v = 9 ) T Then P ( t ⊂ T ) = C p × λ v 1 c , where λ c = 3 and the C p are √ 12 explicit. Consequence: conditionally on t ⊂ T , the law of T \ t only depends on p . Allows to explore T in a Markovian way: peeling explorations are one of the most important tools in the study of T [Angel, 2004...] . Thomas Budzinski High genus triangulations
The non-planar case: what is going on? Euler formula: T n , g has # E = 3 n edges and # V = n + 2 − 2 g vertices. In particular g ≤ n 2 . Hence, the average degree in T n , g is 2 # E 6 n 6 # V = n + 2 − 2 g ≈ 1 − 2 g / n . Interesting regime: g 0 , 1 � � n → θ ∈ . The average degree in the 2 limit is strictly between 6 and + ∞ , so we expect a hyperbolic behaviour. Thomas Budzinski High genus triangulations
The Planar Stochastic Hyperbolic Triangulations 1 The PSHT ( T λ ) 0 <λ ≤ λ c , where λ c = 3 , have been √ 12 introduced in [Curien, 2014] , following similar works on half-planar maps [Angel–Ray, 2013] . For every triangulation t with perimeter p and volume v , we have P ( t ⊂ T λ ) = C p ( λ ) λ v , where the number C p ( λ ) are explicit [B. 2016] . T λ c is the UIPT. For λ < λ c , they have a hyperbolic behaviour : exponential volume growth [Curien, 2014] , transience and positive speed of the simple random walk [Curien, 2014] , existence of infinite geodesics in many different directions [B., 2018] ... Thomas Budzinski High genus triangulations
A sample of a PSHT Thomas Budzinski High genus triangulations
The local limit of T n , g Theorem (B.–Louf, 2019) Let g n 0 , 1 � � n → θ ∈ . Then we have the convergence 2 ( d ) T n , g n − − − − → n → + ∞ T λ ( θ ) in distribution for the local topology, where λ ( θ ) and θ are linked by an explicit equation. In particular, if g n = o ( n ) , then the limit is the UIPT. It may seem surprising that highly non-planar objects become planar in the limit, but this is already the case in other contexts (ex: random regular graphs). The case θ = 1 2 is degenerate (vertices with "infinite degrees"). Thomas Budzinski High genus triangulations
Back to combinatorics Natural idea to prove the theorem: as in the planar case, use asymptotic results on the number τ p ( n , g n ) of triangulations of size n with genus g and a boundary of length p . Unfortunately, this seems very hard to obtain directly asymptotics, so new ideas are needed. On the other hand, our local convergence result gives the limit value of the ratio τ ( n + 1 , g n ) when g n n → θ , and allows to obtain τ ( n , g n ) asymptotic enumeration results up to sub-exponential factors. Theorem (B.–Louf, 2019) When g n 0 , 1 � � n → θ ∈ , we have 2 τ ( n , g n ) = n 2 g n exp ( f ( θ ) n + o ( n )) , � 1 where f ( θ ) = 2 θ log 12 θ 1 e + θ 2 θ log λ ( θ/ t ) d t , and λ ( θ ) is the same as in the previous theorem. Thomas Budzinski High genus triangulations
Steps of the proof Tightness result, plus planarity and one-endedness of the limits. Any subsequential limit T is weakly Markovian : for any finite t , the probability P ( t ⊂ T ) only depends on the perimeter and volume of t . Any weakly Markovian random triangulation of the plane is a mixture of PSHT (i.e. T Λ for some random Λ ). Ergodicity: Λ is deterministic, characterized by the fact that 6 the average degree must be 1 − 2 θ . Thomas Budzinski High genus triangulations
Tightness: the bounded ratio lemma Lemma Fix ε > 0. There is a constant C ε such that, for every p , n and for � 1 � every g ≤ 2 − ε n , we have τ p ( n , g ) τ p ( n − 1 , g ) ≤ C ε . This is the "minimal combinatorial input" needed to adapt the Angel–Schramm argument for tightness. n + 2 − 2 g n ≤ 3 6 n Proof: the average degree is ε , so there are ε n good vertices with degree ≤ 6 ε . Consider a good vertex v and remember its degree d ≤ 6 ε . Choose an edge e joining v to another vertex v ′ . We will contract e . Thomas Budzinski High genus triangulations
Proof of the bounded ratio lemma e’ e → → v v ′ d = 4 From a triangulation with size n and a good vertex v , we obtain a triangulation with size n − 1 with a marked (oriented) edge e ′ , and a degree d ≤ 6 ε . Given d , we can find the other blue edge and reverse the operation, so the operation is injective. At least τ ( n , g n ) × ε n inputs, and at most τ ( n , g n ) τ ( n − 1 , g n ) × 6 n × 6 τ ( n − 1 , g n ) ≤ 36 ε outputs, so ε 2 . Thomas Budzinski High genus triangulations
Tightness As in [Angel–Schramm, 2003] , we first prove that the degree of the root in T n , g n is tight. We explore the neighbours of the root vertex ρ step by step. t + t ρ ρ We have P ( t + ⊂ T n , g n | t ⊂ T n , g n ) = τ p ( n − v − 1 , g n ) 1 ≥ C ε . τ p ( n − v , g n ) Hence, the number of steps needed to finish the exploration of the root has exponential tail uniformly in n , so the root degree is tight. The root vertex degree is tight and T n , g n is stationary for the simple random walk, so the degrees in all the neighbourhood of the root are tight, which is enough to ensure tightness for the local topology. Thomas Budzinski High genus triangulations
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