On multiple SLEs Eveliina Peltola Universit de Genve; Section de - PowerPoint PPT Presentation

On multiple SLEs Eveliina Peltola Université de Genève; Section de Mathématiques < eveliina.peltola@unige.ch > July 25th 2018 Based on joint works with Vincent Beffara (Université Grenoble Alpes, Institut Fourier) , and Hao Wu (Yau Mathematical Sciences Center, Tsinghua University) ICMP, Equilibrium Statistical Mechanics Session

P Motivation 1 critical models in statistical physics, scaling limits conformal invariance of interfaces & correlations Multiple SLEs: classification 2 “global” multiple SLEs “local” multiple SLEs (i.e., commuting SLEs) Partition functions of multiple SLEs 3 marginals of global multiple SLEs local ⇔ global ??? Relation to connection probabilities 4 multichordal loop-erased random walks / UST branches level lines of the Gaussian free field double-dimer pairings Ising model crossing probabilities 1

M x 1 x 6 ⊕ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊕ ⊕ ⊖ ⊕ ⊖ ⊕ ⊕ ⊖ ⊕ ⊖ ⊕ ⊕ ⊕ ⊖ ⊕ ⊖ ⊖ ⊖ ⊖ ⊖ ⊕ ⊖ ⊕ ⊕ ⊕ ⊕ ⊖ ⊖ ⊖ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊖ ⊖ ⊕ ⊕ ⊖ ⊕ ⊕ ⊕ ⊖ ⊕ x 5 x 2 ⊖ ⊖ ⊕ ⊕ ⊖ ⊖ ⊕ ⊖ ⊕ ⊕ ⊖ ⊖ ⊕ ⊖ ⊖ ⊕ ⊖ ⊕ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊕ ⊕ ⊖ ⊖ ⊕ ⊖ ⊕ ⊖ ⊖ ⊖ ⊖ ⊕ ⊖ ⊖ ⊕ ⊖ ⊖ ⊖ ⊕ ⊖ ⊕ ⊕ ⊖ ⊕ ⊕ ⊕ ⊕ ⊖ ⊖ ⊖ ⊖ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊖ x 3 x 4 1

E:  I  put random spins σ x = ± 1 at vertices x of a graph � 1 � nearest neighbor interaction: P [ config. ] ∝ exp � x ∼ y σ x σ y T phase transition at critical temperature T = T c look at correlation of a pair of spins at x and y C ( x , y ) = E [ σ x σ y ] − E [ σ x ] E [ σ y ] when | x − y | >> 1: T < T c T = T c T c < T C ( x , y ) ∼ e − 1 ξ | x − y | C ( x , y ) ∼ | x − y | − β C ( x , y ) ∼ const . scaling limit at critical temperature T c : conformal invariance 2

C    I  Dobrushin boundary conditions: ∂ Ω δ � {⊕ segment } � {⊖ segment } δ → 0 interface of Ising model −→ Schramm-Loewner evolution, SLE 3 [ Chelkak, Duminil-Copin, Hongler, Kemppainen, Smirnov (2014) ] 3

C    I  fix discrete domain data ( Ω δ ; x δ 1 , . . . , x δ x 5 2 N ) consider critical Ising model in Ω δ ⊂ δ Z 2 with alternating ⊕ / ⊖ b.c. x 6 x 4 Izyurov (2015): δ → 0 interfaces −→ (local) multiple SLE 3 x 1 x 3 Proof: multi-point holomorphic observable x 2 4

C    I  fix discrete domain data ( Ω δ ; x δ 1 , . . . , x δ x 5 2 N ) consider critical Ising model in Ω δ ⊂ δ Z 2 with alternating ⊕ / ⊖ b.c. x 6 x 4 Izyurov (2015): δ → 0 interfaces −→ (local) multiple SLE 3 x 1 x 3 Proof: multi-point holomorphic observable If we condition on the event that the x 2 interfaces connect the boundary points according to a given connectivity ... ... then we have: Theorem The law of the N macroscopic interfaces of the critical Ising model converges in the scaling limit δ → 0 to the N - SLE κ with κ = 3. Wu [arXiv:1703.02022] Proof: convergence for N = 1 and Beffara, P. & Wu [arXiv:1801.07699] classification of multiple SLE 3 4

C    FK-I  fix discrete domain data ( Ω δ ; x δ 1 , . . . , x δ x 6 2 N ) x 6 consider critical FK-Ising model in x 5 Ω δ ⊂ δ Z 2 with alternating free/wir b.c. x 5 x 4 Kemppainen & Smirnov (2018): x 4 x 1 x 1 δ → 0 2 interfaces −→ hSLE 16 / 3 Proof: holomorphic observable x 2 x 2 3 x x 3 5

C    FK-I  fix discrete domain data ( Ω δ ; x δ 1 , . . . , x δ x 6 2 N ) x 6 consider critical FK-Ising model in x 5 Ω δ ⊂ δ Z 2 with alternating free/wir b.c. x 5 x 4 Kemppainen & Smirnov (2018): x 4 x 1 x 1 δ → 0 2 interfaces −→ hSLE 16 / 3 Proof: holomorphic observable x 2 x 2 If we condition on the event that the 3 x x 3 interfaces connect the boundary points according to a given connectivity ... ... then we have: Theorem The law of the N macroscopic interfaces of the critical FK-Ising model converges in the scaling limit δ → 0 to the N - SLE κ with κ = 16 / 3. Beffara, P. & Wu [arXiv:1801.07699] Proof: convergence for N = 1 and classification of multiple SLE 16 / 3 5

S-L  SLE κ 5

S’    SLE κ Theorem [Schramm ∼ 2000] ∃ ! one-parameter family ( SLE κ ) κ ≥ 0 of probability measures on chordal curves with conformal invariance and domain Markov property encode SLE κ random curves in random conformal maps ( g t ) t ≥ 0 driving process � image of the tip : g t z → γ ( t ) g t ( z ) = √ κ B t X t : = lim g t : H \ γ [ 0 , t ] → H solutions to Loewner equation: d 2 X t = g t ( γ ( t )) d tg t ( z ) = , g 0 ( z ) = z g t ( z ) − X t 6

C    SLE κ x b j family of random curves in ( Ω ; x 1 , . . . , x 2 N ) C L j various connectivities encoded in C R j planar pair partitions α ∈ LP N x a j Theorem Let κ ∈ ( 0 , 4 ] ∪ { 16 / 3 , 6 } . For any fixed connectivity α of 2 N points, there exists a unique probability measure on N curves such that conditionally on N − 1 of the curves, the remaining one is the chordal SLE κ in the random domain where it can live . Dubédat (2006); Kozdron & Lawler (2007–2009); Miller & Sheffield (2016); Miller, Sheffield & Werner (2018); P. & Wu (2017); Beffara, P. & Wu (2018) 7

U: ∃     N - SLE κ Theorem Let κ ∈ ( 0 , 4 ] ∪ { 16 / 3 , 6 } . For any fixed connectivity α of 2 N points, there exists at most one probability measure on N curves such that conditionally on N − 1 of the curves, the remaining one is the chordal SLE κ in the random domain where it can live . Idea of proof: [ Beffara, P. & Wu [arXiv:1801.07699] ] sample curves according to conditional law ⇒ Markov chain on space of curves prove that there is a coupling of two such Markov chains, started from any two initial configurations, such that they have a uniformly positive chance to agree after a few steps ⇒ there is at most one stationary measure 8

U: ∃     N - SLE κ Theorem Let κ ∈ ( 0 , 4 ] ∪ { 16 / 3 , 6 } . For any fixed connectivity α of 2 N points, there exists at most one probability measure on N curves such that conditionally on N − 1 of the curves, the remaining one is the chordal SLE κ in the random domain where it can live . Remarks: uniformly exponential mixing: ∃ coupling s.t. P [ X 4 n � ˜ X 4 n ] ≤ ( 1 − p 0 ) n 9

U: ∃     N - SLE κ Theorem Let κ ∈ ( 0 , 4 ] ∪ { 16 / 3 , 6 } . For any fixed connectivity α of 2 N points, there exists at most one probability measure on N curves such that conditionally on N − 1 of the curves, the remaining one is the chordal SLE κ in the random domain where it can live . Remarks: uniformly exponential mixing: ∃ coupling s.t. P [ X 4 n � ˜ X 4 n ] ≤ ( 1 − p 0 ) n the case of N = 2 was proved in Miller & Sheffield: IG II using the coupling of SLE with the Gaussian free field (GFF) but N ≥ 3 curves cannot be coupled with the GFF our proof does not use the GFF in any way 9

U: ∃     N - SLE κ Theorem Let κ ∈ ( 0 , 4 ] ∪ { 16 / 3 , 6 } . For any fixed connectivity α of 2 N points, there exists at most one probability measure on N curves such that conditionally on N − 1 of the curves, the remaining one is the chordal SLE κ in the random domain where it can live . Remarks: uniformly exponential mixing: ∃ coupling s.t. P [ X 4 n � ˜ X 4 n ] ≤ ( 1 − p 0 ) n the case of N = 2 was proved in Miller & Sheffield: IG II using the coupling of SLE with the Gaussian free field (GFF) but N ≥ 3 curves cannot be coupled with the GFF our proof does not use the GFF in any way update: N = 2, κ ∈ ( 0 , 8 ) : Miller, Sheffield, Werner (2018) 9

C   N - SLE κ Theorem For any fixed connectivity α of 2 N points, there exists at least one probability measure on N curves such that conditionally on N − 1 of the curves, the remaining one is the chordal SLE κ in the random domain where it can live . 1. From scaling limits of multiple interfaces in critical models works for κ ∈ { 2 , 3 , 4 , 16 / 3 , 6 } [ Schramm & Sheffield (2013); Izyurov (2015); Beffara, P. & Wu (2018) ] 10