Sheffield’s freshburgers q c ( S ) − 1+ n ( S ) = q # marked fresh . � � ( M,S ) , n edges W walk on N 2 , n steps subgraph-decotated map + some marked fresh edges Critical FK-cluster model on maps ( q determines the central charge) Rk: q ր ⇐ ⇒ fresh steps ր ⇐ ⇒ correlation of coordinates ր . Brownian excursion Brownian excursion ⇐ ⇒ with correlated in a cone α ( q ) coordinates Theorem: [Sheffield 16] For q ∈ [0 , 4] the scaling limit of fresh-weighted walks is a Brownian motion in a cone of angle α ( q ) . Example: α (0) = π/ 2 for tree-decorated, α (1) = 2 π/ 3 for percolation, α (2) = 3 π/ 8 for Ising.
Mating of trees [Duplantier, Miller, Sheffield] There is a parallel story in the context of Liouville quantum gravity:
Mating of trees [Duplantier, Miller, Sheffield] There is a parallel story in the context of Liouville quantum gravity: Theorem: [Duplantier, Miller, Sheffield 14] Grow a space-filling SLE κ ( q ) on LQG γ ( q ) and record length of sides. This process is equal to the “correlated” 2D-Brownian motion which is the limit of q -fresh-weighted walks. Moreover, the above is a σ -algebra preserving correspondence.
Mating of trees [Duplantier, Miller, Sheffield] There is a parallel story in the context of Liouville quantum gravity: Theorem: [Duplantier, Miller, Sheffield 14] Grow a space-filling SLE κ ( q ) on LQG γ ( q ) and record length of sides. This process is equal to the “correlated” 2D-Brownian motion which is the limit of q -fresh-weighted walks. Moreover, the above is a σ -algebra preserving correspondence. Gives hope of connecting (FK-weighted) maps to LQG and SLE via the walk encoding.
B.II Percolated triangulations and a bijective path to LQG
Percolation on triangulation CLE on Liouville Quantum Gravity
Percolation on triangulation “Random curves on a random surface” CLE on Liouville Quantum Gravity
Kreweras excursion Percolation on triangulation 2D Brownian excursion CLE on Liouville Quantum Gravity
Percolation on triangulations
Percolation on a regular lattice Triangular lattice
Percolation on a regular lattice Site percolation: Color vertices black or white with probability 1 / 2 .
Percolation on a regular lattice A n × B n box Site percolation: Color vertices black or white with probability 1 / 2 . Questions: • Crossing probabilities ?
Percolation on a regular lattice Site percolation: Color vertices black or white with probability 1 / 2 . Questions: • Crossing probabilities ? • Crossing probabilities ? • Law of interfaces ?
Percolation on a regular lattice Site percolation: Color vertices black or white with probability 1 / 2 . Questions: • Crossing probabilities ? • Crossing probabilities ? • Law of interfaces ? • Mixing properties ?
Triangulations (of the disk) Def. A triangulation of the disk is a decomposition into triangles.
Triangulations (of the disk) Def. A triangulation of the disk is a decomposition into triangles (considered up to deformation). =
Triangulations (of the disk) Def. A triangulation of the disk is a decomposition into triangles (considered up to deformation). (multiple edges allowed, loops forbidden)
Triangulations (of the disk) Def. A triangulation of the disk is a decomposition into triangles (considered up to deformation). Def. A triangulation is rooted by marking an edge on the boundary.
Percolation on triangulations We can consider percolation on random triangulations of the disk. ( k exterior vertices, n interior vertices; uniform probability)
Percolation on triangulations We can consider percolation on random triangulations of the disk. ( k exterior vertices, n interior vertices; uniform probability) Same questions: • Crossing probabilities ? • Law of interfaces ? • Mixing properties ?
Percolation on triangulations We can consider percolation on random triangulations of the disk. ( k exterior vertices, n interior vertices; uniform probability) Same questions: • Crossing probabilities ? • Law of interfaces ? • Mixing properties ? Goal 1: Answer these questions. (as n → ∞ , k ∼ √ n )
Percolation on triangulations We can consider percolation on random triangulations of the disk. ( k exterior vertices, n interior vertices; uniform probability) Same questions: • Crossing probabilities ? • Law of interfaces ? • Mixing properties ? Goal 1: Answer these questions. (as n → ∞ , k ∼ √ n ) We can also consider infinite triangulations . Uniform Infinite Planar Triangulation [Angel,Schramm 04]
Regular lattices Vs random lattices Is it interesting to study statistical mechanics on random lattices ? Vs regular lattice random lattice
Regular lattices Vs random lattices Is it interesting to study statistical mechanics on random lattices ? Vs regular lattice random lattice Yes! New tools: random matrices , generating functions , bijections .
Regular lattices Vs random lattices Is it interesting to study statistical mechanics on random lattices ? Vs regular lattice random lattice Yes! New tools: random matrices , generating functions , bijections . Yes! The “critical exponents” on regular Vs random lattices are related by the KPZ formula [Knizhnik, Polyakov, Zamolodchikov].
Regular lattices Vs random lattices Is it interesting to study statistical mechanics on random lattices ? Vs regular lattice random lattice Yes! New tools: random matrices , generating functions , bijections . Yes! The “critical exponents” on regular Vs random lattices are related by the KPZ formula [Knizhnik, Polyakov, Zamolodchikov]. Yes! Critically weighted random lattices � random surfaces .
Liouville Quantum Gravity (LQG) and Schramm–Loewner Evolution (SLE)
What is . . . Liouville Quantum Gravity?
What is . . . Liouville Quantum Gravity? (image by J. Miller) LQG is a random area measure µ on a C -domain which is related to the Gaussian free field.
What is . . . Liouville Quantum Gravity? 1D LQG µ = e γ h dx h 0 1 0 1 Brownian motion 1D LQG
What is . . . Liouville Quantum Gravity? 1D LQG h n : [ n ] → R h = lim h n µ = e γ h dx 0 n 0 1 0 1 Random function Brownian motion 1D LQG chosen with probability proportional to n ( h ( i ) − h ( i − 1)) 2 � − 2 i =1 e
What is . . . Liouville Quantum Gravity? µ = e γ h dxdy h n : [ n ] 2 → R h = lim h n Random function Gaussian Free Field LQG chosen with probability (a distribution) (area measure) proportional to ( h ( u ) − h ( v )) 2 � − 2 u ∼ v e
What is . . . Liouville Quantum Gravity? µ = e γ h dxdy h n : [ n ] 2 → R h = lim h n γ ∈ [0 , 2] controls how wild LQG measure is. � Today: γ = 8 / 3 . “pure gravity”
What is... a SLE (Schramm–Loewner evolution)?
What is... a SLE (Schramm–Loewner evolution)? SLE κ is a random (non-crossing, parametrized) curve in a C -domain.
What is... a SLE (Schramm–Loewner evolution)? SLE κ is a random (non-crossing, parametrized) curve in a C -domain. The parameter κ determines how much the curve “wiggles”. SLE κ were introduced to describe the scaling limit of curves from statistical mechanics.
What is... a SLE (Schramm–Loewner evolution)? SLE are characterized by: • Conformal invariance property • Markov domain property γ 0 1
What is... a SLE (Schramm–Loewner evolution)? SLE are characterized by: • Conformal invariance property • Markov domain property φ conformal 0 0 1 1 γ ( t )
What is... a SLE (Schramm–Loewner evolution)? SLE are characterized by: • Conformal invariance property • Markov domain property ˜ φ conformal 0 1 0 1 e i √ κW ( t ) γ ( t ) Brownian
What is... a SLE (Schramm–Loewner evolution)? Today: κ = 6 (percolation)
What is... a SLE (Schramm–Loewner evolution)? Today: κ = 6 (percolation) Theorem [Smirnov 01]: Convergence. SLE 6
What is... a SLE (Schramm–Loewner evolution)? Today: κ = 6 (percolation) Theorem [Smirnov 01]: Convergence.
What is... a SLE (Schramm–Loewner evolution)? Today: κ = 6 (percolation) Theorem [Smirnov 01]: Convergence. Theorem [Camia, Newman 09]: Convergence. Conformal Loop Ensemble CLE 6
The big conjecture Nice embedding Riemann mapping ?
The big conjecture under nice embedding LQG √ 8 / 3
The big conjecture under nice embedding LQG √ 8 / 3 + CLE 6
Strategy of proof bijection [Bernardi 2007] [Bernardi, Holden, Sun 2019] under nice embedding σ -algebra preserving coupling [Duplantier, Miller, Sheffield 2014] “mating of trees” LQG √ 8 / 3 + CLE 6
B.III Bijection: Percolated triangulations ← → Kreweras walks
Kreweras walks A Kreweras walk is a lattice walk on Z 2 using the steps Def. a = (1 , 0) , b = (0 , 1) and c = ( − 1 , − 1) . b a c
Thm [Bernardi 07/ Bernardi, Holden, Sun 18]: There is a bijection between: • K = set of Kreweras walks starting and ending at (0 , 0) and staying in N 2 . • T = set of percolated triangulations of the disk with 2 exterior vertices: one white and one black. K T Φ 3 n steps n interior vertices
Example: w = baabbcacc b Φ a c
Example: w = baabbcacc b a c a c a b c b
Example: w = baabbcacc b b a a b b Definition: b a
Example: w = baabbcacc c a c c Definition: c
Well defined? b # b − # c # a − # c a c
Well defined? b # b − # c # a − # c a c Bijective? c a b a a b c c b
Well defined? b # b − # c # a − # c a c Bijective? c a b a a b c c b inverse bijection
Variants of the bijection Spherical case Disk case UIPT case
B.IV Dictionary between triangulations and walks
Dictionary percolated triangulation walk walk edges steps left-boundary length x-coordinate of walk vertices c -steps black vertices c steps of type a perco-interface toward t walk of excursions envelope intervals clusters cluster’s bubbles cone intervals
b Basic observations a c b c a Dictionary: ← → triangles a, b steps ← → inner vertices c steps inner edges +root-edge ← → a, b, c steps
b Basic observations a c b c a Remark. Subwalk made of a, c steps form a parenthesis system . Subwalk made of b, c steps form a parenthesis system . Example: w = b a a b b c a c c .
b Basic observations a c b c a Remark. Subwalk made of a, c steps form a parenthesis system . Subwalk made of b, c steps form a parenthesis system . Example: w = b a a b b c a c c . Dictionary. ← → unmatched a -steps. triangles incident to left boundary triangles incident to right boundary ← → unmatched b -steps.
b Basic observations a c b c a Def: c -steps are of two types : xxxxx a xxxx b xxxx c xxxx xxxxx b xxxx a xxxx c xxxx type a type b
b Basic observations a c b c a Def: c -steps are of two types : xxxxx a xxxx b xxxx c xxxx xxxxx b xxxx a xxxx c xxxx type a type b Dictionary. white inner vertex ← → c -step of type a black outer vertex ← → c -step of type b
b Basic observations a c b c a Def: c -steps are of two types : xxxxx a xxxx b xxxx c xxxx xxxxx b xxxx a xxxx c xxxx type a type b Dictionary. white inner vertex ← → c -step of type a black outer vertex ← → c -step of type b Math puzzle: Prove directly that the number of c -steps of type a follows a binomial B ( n, 1 / 2) distribution.
Percolation interface
Dictionary: percolation-interface to t ← → walk of excursions
Dictionary: percolation-interface to t ← → walk of excursions Flatten each cone-excursion into a single step empty the bubbles Shuffle of 2 looptrees Flattened walk
Dictionary: percolation-interface to t ← → walk of excursions
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