Liouville theory and log-correlated processes Xiangyu Cao (LPTMS, - PowerPoint PPT Presentation

Liouville & log-REM Liouville theory and log-correlated processes Xiangyu Cao (LPTMS, Orsay) Random Geometry & Physics, 10 / 2016

Liouville & log-REM c ≥ 25 pure Liouville theory and multi-fractality in log-correlated processes Collaborators: Outline: ◮ Pierre Le Doussal ENS ◮ c ≥ 25 pure Liouville & 2d GFF disordered stat. ϕ ◮ Alberto Rosso Orsay ◮ discrete terms in Liouville. ◮ Raoul Santachiara Orsay ⇔ Termination point transition Thanks: in disordered systems. ◮ Yan Fyodorov ◮ R´ emi Rhodes ◮ Sylvain Ribault ◮ Vincent Vargas

Liouville & log-REM Liouville and 2D GFF c ≥ 25 pure Liouville in disordered stat. ϕ Connection to pure Liouville (96) Freezing & 2D GFF (96, earlier: Derrida-Spohn, David) Full-fledged freezing (98-00) Rigorous renewal (see also Duplantier & She ffi eld, . . . )

Liouville & log-REM Liouville and 2D GFF Gibbs measure of 2D GFF + log potential p β ( z ) = 1 Ze − β ( φ ( z ) + U ( z )) , β = 1 / T φ ( z ) + U ( z ) 24 φ ( z ) φ ( w ) = − 4 ln | z − w | (2d GFF) 3 18 2 k 12 � 1 U ( z ) = 4 a j ln | z − w j | , 6 Im ( z ) 0 j = 1 0 − 1 − 6  β β < 1  Q = b + b − 1 , b = min(1 , β ) =  − 12 − 2   1 β ≥ 1  − 18  − 3 − 24 − 4 − 3 − 2 − 1 0 1 2 3 4 Re ( z ) k = 2 , w 1 , 2 = 0 , 1 , a 1 , 2 / Q = . 2 , . 4

Liouville & log-REM Liouville and 2D GFF correlation of p ( z ) = pure Liouville c ≥ 25 � ℓ ℓ k � β< 1 � � � p q i β ( z i ) V q i β ( z i ) V a j ( w j ) × V a ∞ ( ∞ ) ∝ i = 1 i = 1 j = 1 LFT( c = 1 + 6 Q 2 )  a ∞ = Q − � k j = 1 a j ⇔ Q − a ∞ = � k j = 1 a j     β< 1  if Q = b + b − 1 , b = min(1 , β )  = β      a j , q i meet Seiberg bounds (see next slide)  β> 1 β > 1 ⇒ freezing ( b = 1 ) + 1RSB (spin-glass physics, UV origin) β> 1 β> 1 = Tp 1 ( z 1 ) p 1 ( z 2 ) + (1 − T ) δ 2 ( z 1 − z 2 ) . . . = p 1 ( z ) , p β ( z 1 ) p β ( z 2 ) p β ( z )

Liouville & log-REM Liouville and 2D GFF Seiberg bounds � ℓ k � � � = µ − s / b Z − s / b . . . V q i β ( z i ) V a j ( w j ) × V a ∞ ( ∞ ) (1) i = 1 j = 1 LFT √ s = � ℓ i = 1 q i b + � k j = 1 a j + a ∞ − Q = � ℓ i = 1 q i b > 0 ( q i > 0); ◮ (1) by conformal bootstrap. [Zamolodchikov, Zamolodchikov, Belavin, . . . ] √ a j < Q / 2 ⇔ no binding, a ∞ < Q / 2 ⇔ confinement ◮ q i β < Q / 2 ⇔ away from termination point transition .

Liouville & log-REM Discrete terms Liouville by bootstrap: continuous & discrete U ( z ) = 4 a 1 ln | z | + 4 a 2 ln | z − 1 | 10 3 LFT a 1 b LFT(t) 10 2 � only � p β ( z ) ∝ d α α num. Q 2 + i R P 10 1 a 2 a ∞ 10 0 � 2 π i Res[ . . . ; α ] + β , a 1 Q , a 2 Q = 0.4, 0.1, 0.45 α ∈ D 0.0 0.2 0.4 0.6 0.8 1.0 x a ∞ = b + b − 1 − a 1 − a 2 , b = min(1 , β ) Discrete terms ⇔ s channel = t channel = numerics [A.A. Belavin, A.B. Zamolodchikov, Theor.Math.Phys. 147 (2006) 729-754] See also Konstantin Aleshkin’s talk tomorrow

Liouville & log-REM Discrete terms Liouville fusion / OPE: continuous & discrete Q Q Q α 1 + α 2 < Q /2 α 1 + α 2 = Q /2 α 1 + α 2 > Q /2 2 + i R 2 + i R 2 + i R [ α 1 ] + [ α 2 ] [ α 1 ] + [ α 2 ] [ α 1 ] + [ α 2 ] [ α 1 + α 2 ] + . . . � � [ Q /2 + i P ] P 2 dP [ Q /2 + i P ] dP Goal: interpret as termination point transition [A.A. Belavin, A.B. Zamolodchikov, Theor.Math.Phys. 147 (2006) 729-754] See also : Ribault, Santachiara arXiv:1503.02067, Ribault arXiv:1406.4290, Ex. 3.3 Konstantin Aleshkin’s talk tomorrow

Liouville & log-REM Multi-fractality Multi-fractal spectrum 10 − 1 10 − 2 Set U ( z ) = 0, cut domain into M boxes: 10 − 3 10 − 4 � 10 − 5 p β ( z ) d 2 z , i = 1 , . . . , M → + ∞ . p i = 10 − 6 10 − 7 box i 10 − 8 def Num. of p i ∈ [ M − α , M − ( α + d α ) ] = M f ( α ) d α 10 − 9 10 − 10 For 2D GFF (general log-correlated) , f ( α ) b = β < 1 f ( α ) = 4( α + − α )( α − α − ) + o (1) ( α + − α − ) 2 α − = (1 − b ) 2 , b = min(1 , β ) , α α + α − = ( 1 − b ) 2 α + = α − + 4 β .

Liouville & log-REM Multi-fractality Inverse participation ratios (IPR) M � α q − f ( α ) � , q > 0 . def � i ∼ M − τ ( q ) , τ ( q ) = min p q P q = α ∈ I i = 1 Annealed: P q averaged over all samples, Quenched: I = [0 , + ∞ ) ⇒ termination point transition . one big sample, I = [ α − , α + ] f ( α ) f ( α ) α α q c = f ′ ( α − ) termination point  ∆ q β − 1 q β < 1   τ ( q ) =  q (1 − b ) 2  q β ≥ 1   ∆ q β − 1 q β < Q / 2    τ ( q ) =   q β ≥ Q / 2 ∆ Q / 2 − 1   ∆ α = α ( Q − α ) , Q = b + b − 1 , b = min(1 , β )

Liouville & log-REM Multi-fractality Log corrections in annealed IPR b = min(1 , β ) , Q = b + b − 1 , ∆ α = α ( Q − α ) . M 1 − ∆ q β  q β < Q / 2    β< 1  M 1 − ∆ Q / 2 ln − 1  2 M P q ∼ q β = Q / 2 (2)     M 1 − ∆ Q / 2 ln − 3  2 M  q β > Q / 2  ◮ Nb : ln − 1 + x � M − P 2 P x d P comes from continuous spectrum 2 M ∼ integral. No log-CFT. ◮ Fyodorov, arxiv / 0903.2502, uncorrelated potential: same exponent, di ff erent log corrections (eq. 9). Nb. He used the term pre-freezing. ◮ β > 1 , q β > Q / 2 = 1 ⇒ P q ∼ O (1), no more log.

Liouville & log-REM Other applications Joint occupation probability 2 particles ( z 1 , 2 ) in 1 potential, distribution of z = z 1 − z 2 ◮ averaged over all potential samples ◮ | z 1 − z 2 | ≪ | z 1 − w j | ⇒ U ∼ const ⇒ local translation invariance | z | − 4 β 2 β < 3 − 1  − exponent 2     | z | − 4 / 3 ln − 1 β = 3 − 1  β = 1 2 | 1 / z |   2   P β ( z ) ∼  | z | − 3 + β 2 + β − 2  ln − 3 β ∈ (3 − 1 2 | 1 / z | 2 , 1]   2   β = 3 − 1   2 c 1 | z | − 2 ln − 3  2 | 1 / z | + c 0 δ 2 ( z )  β > 1  note: β = 3 − 1 2 ⇔ q β = Q / 2 for q = 2 β At zero- T , distribution of 1st & 2nd minima positions P ( ξ = ξ 1 − ξ 2 ) ξ → 0 1 | ξ | − 2 ln − 3 ∼ c ′ 2 | 1 /ξ | + c ′ 0 δ 2 ( ξ ) δ comes from freezing / 1RSB in β > 1 Again: log’s come from continuous spectrum. No log-CFT.

Liouville & log-REM Other applications Directed polymers on disordered Cayley tree Hierarchical cousin of 2D GFF:  ǫ ( 3, 2 ) i.i.d. random i = 1 ǫ ( x i , i ) , ǫ ( x , i ) � e − β � t   Z =  ǫ ( 2, 1 ) s.t. β c = 1    ( x i ) t ǫ ( 1, 2 ) i = 1 distance � overlap q ∈ [0 , 1] . ǫ ( − 1, 2 ) ǫ ( − 2, 1 ) e (2 β 2 − 1) tq t β < 3 − 1  ǫ ( − 3, 2 ) 2     e − qt / 3 q − 1 1 β = 3 − 1 2 t  t = 0 qt = 3 t = 5  2 2   P β ( q ) ∼  [Derrida-Spohn, Arguin et.al ] : e − ( β − β − 1 ) 2 tq  4 t − 1 2 q − 3 β ∈ (3 − 1 2 , 1)  2    t →∞  = m δ ( q ) + 1 − m δ (1 − q ) .  P β ( q ) t − 1 2 q − 3  β ≥ 1 , q ≪ 1  2 m = min(1 , 1 /β ) translation: qt = − 2 log 2 | z | , t < ∞ , β ≥ 1: [Derrida etal 1607.06610] t = 2 log 2 ( R / a )

Liouville & log-REM Other applications Conclusion ◮ Discrete terms ⇔ termination point transition. ◮ Continuous spectrum ⇒ 3 2 log corrections in β < 1 phase. Similar applications: ◮ p β ( z → 0) with U ( z ) = a ln | z | + . . . . 10 − 1 10 − 2 ◮ The case a ≥ Q / 2 (bound phase). 10 − 3 10 − 4 Puzzles: 10 − 5 ◮ Some issues in β ≥ 1 phase. 10 − 6 10 − 7 ◮ Avoid freezing ? 10 − 8 ◮ Extend to c ≤ 1 ? 10 − 9 10 − 10

Liouville & log-REM Appendices Charge at z = ∞ ⇔ charges at z � ∞ → − �� � ∇ U V ... ( . . . ) × V a ∞ ( ∞ ) i , j LFT = µ ∗ Z ∗ � e ∗ φ ( ∗ ) × e ( Q − a ∞ ) φ ( ∞ ) , z ≫∀ w j φ ( z ) + � k j = 1 4 a j ln | z − w j | −→ � k j = 1 4 a j ln | z | = 4( Q − a ∞ ) ln | z | n . − → � � ∇ U = 4 π ( Q − a ∞ ). Here U ( z ) = � k | z | = R d � In general | z | < R ∆ U = j = 1 4 a j ln | z − w j | , ∆ U = � 4 π a j δ z , w j point charges. Dilute charges will not give n < ∞ -point Liouville.

Liouville & log-REM Appendices � � V a 1 (0) V a 2 (1) V b ( z ) V a 3 = Q − a 1 − a 2 ( ∞ ) LFT , b = min( β, 1) � Q / 2 + i R + | F ( z | a i , α, b ) | 2 C DOZZ α, a 1 , a 2 C DOZZ = − α, a 3 , b d α � | F ( z | a i , α 0 , b ) | 2 2 π i Res � � C DOZZ α, a 1 , a 2 C DOZZ α, a 3 , b ; α → α 0 + α 0 ∈ D

Liouville & log-REM Appendices Liouville fusion rule: continous and / or discrete See also Konstantin Aleshkin’s talk tomorrow ◮ Discrete terms come from fusion results outside the Liouville spectrum Q / 2 + i R . ◮ When they are present, they dominate OPE s. Sylvain Ribault, arXiv:1406.4290 [A.A. Belavin, A.B. Zamolodchikov, Theor.Math.Phys. 147 (2006) 729-754]