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:

◮ Pierre Le Doussal ENS ◮ Alberto Rosso Orsay ◮ Raoul Santachiara Orsay

Thanks:

◮ Yan Fyodorov ◮ R´

emi Rhodes

◮ Sylvain Ribault ◮ Vincent Vargas

Outline:

◮ c ≥ 25 pure Liouville & 2d

GFF disordered stat. ϕ

◮ discrete terms in Liouville.

⇔ Termination point transition in disordered systems.

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 & Sheffield, . . . )

Liouville & log-REM Liouville and 2D GFF

Gibbs measure of 2D GFF + log potential

pβ(z) = 1 Ze−β(φ(z)+U(z)) , β = 1/T φ(z)φ(w) = −4 ln |z − w| (2d GFF) U(z) =

k

• j=1

4aj ln |z − wj| , Q = b + b−1, b = min(1, β) =        β β < 1 1 β ≥ 1 φ(z) + U(z)

−3 −2 −1 1 2 3 4 Re(z) −4 −3 −2 −1 1 2 3 Im(z) −24 −18 −12 −6 6 12 18 24

k = 2, w1,2 = 0, 1, a1,2/Q = .2, .4

Liouville & log-REM Liouville and 2D GFF

correlation of p(z) = pure Liouville c ≥ 25

ℓ

• i=1

pqi

β (zi) β<1

∝ ℓ

• i=1

Vqiβ(zi)

k

• j=1

Vaj(wj) × Va∞(∞)

• LFT(c=1+6Q2)

if              a∞ = Q − k

j=1 aj ⇔ Q − a∞ = k j=1 aj

Q = b + b−1, b = min(1, β)

β<1

= β

aj, qi meet Seiberg bounds (see next slide)

β > 1 ⇒ freezing (b

β>1

= 1) + 1RSB (spin-glass physics, UV origin) pβ(z)

β>1

= p1(z) , pβ(z1)pβ(z2)

β>1

= Tp1(z1)p1(z2) + (1 − T)δ2(z1 − z2) . . .

Liouville & log-REM Liouville and 2D GFF

Seiberg bounds

ℓ

• i=1

Vqiβ(zi)

k

• j=1

Vaj(wj) × Va∞(∞)

• LFT

= µ−s/bZ−s/b . . . (1) √ s = ℓ

i=1 qib + k j=1 aj + a∞ − Q = ℓ i=1 qib > 0 (qi > 0); ◮ (1) by conformal bootstrap. [Zamolodchikov, Zamolodchikov, Belavin, . . . ]

√ aj < Q/2 ⇔ no binding, a∞ < Q/2 ⇔ confinement

◮ qiβ < Q/2 ⇔ away from termination point transition.

Liouville & log-REM Discrete terms

Liouville by bootstrap: continuous & discrete

U(z) = 4a1 ln |z| + 4a2 ln |z − 1| pβ(z) ∝

• Q

2 +iR

dα

a1 a2 b a∞ α

+

• α∈D

2πi Res[. . . ; α] a∞ = b+b−1−a1−a2, b = min(1, β)

0.0 0.2 0.4 0.6 0.8 1.0 x 100 101 102 103 P β, a1

Q, a2 Q = 0.4, 0.1, 0.45

LFT LFT(t)

• nly

num.

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 2 + iR

α1 + α2 < Q/2 [α1] + [α2] [α1 + α2] + . . .

Q 2 + iR

α1 + α2 = Q/2 [α1] + [α2]

• [Q/2 + iP]dP

Q 2 + iR

α1 + α2 > Q/2 [α1] + [α2]

• [Q/2 + iP]P2dP

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

Set U(z) = 0, cut domain into M boxes: pi =

• box i

pβ(z) d2z , i = 1, . . . , M → +∞.

• Num. of pi ∈ [M−α, M−(α+dα)]

def

= M f(α)dα For 2D GFF (general log-correlated), f(α) = 4(α+ − α)(α − α−) (α+ − α−)2 + o(1) α− = (1 − b)2 , b = min(1, β) , α+ = α− + 4β .

10−10 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1

α f(α) α+ b = β < 1 α− = (1 − b)2

Liouville & log-REM Multi-fractality

Inverse participation ratios (IPR)

Pq

def

=

M

• i=1

pq

i ∼ M−τ(q) , τ(q) = min α∈I

αq − f(α) , q > 0 . Annealed: Pq averaged over all samples, I = [0, +∞) ⇒ termination point transition.

α f(α) termination point

τ(q) =        ∆qβ − 1 qβ < Q/2 ∆Q/2 − 1 qβ ≥ Q/2

Quenched:

• ne big sample,

I = [α−, α+]

α f(α) qc = f ′(α−)

τ(q) =        ∆qβ − 1 qβ < 1 q(1 − b)2 qβ ≥ 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). Pq

β<1

∼              M1−∆qβ qβ < Q/2 M1−∆Q/2ln− 1

2 M

qβ = Q/2 M1−∆Q/2ln− 3

2 M

qβ > Q/2 (2)

◮ Nb: ln− 1+x

2 M ∼

• M−P2PxdP comes from continuous spectrum
• integral. No log-CFT.

◮ Fyodorov, arxiv/0903.2502, uncorrelated potential: same

exponent, different log corrections (eq. 9). Nb. He used the term pre-freezing.

◮ β > 1, qβ > Q/2 = 1 ⇒ Pq ∼ O(1), no more log.

Liouville & log-REM Other applications

Joint occupation probability

2 particles (z1,2) in 1 potential, distribution of z = z1 − z2

◮ averaged over all potential samples ◮ |z1 − z2| ≪ |z1 − wj| ⇒ U ∼ const ⇒ local translation invariance

Pβ(z) ∼                      |z|−4β2 β < 3− 1

2

|z|−4/3 ln− 1

2 |1/z|

β = 3− 1

2

|z|−3+ β2+β−2

2

ln− 3

2 |1/z|

β ∈ (3− 1

2 , 1]

c1 |z|−2 ln− 3

2 |1/z| + c0δ2(z)

β > 1 note: β = 3− 1

2 ⇔ q β = Q/2 for q = 2

β − exponent β = 3−1

2

β = 1

At zero-T, distribution of 1st & 2nd minima positions P(ξ = ξ1 − ξ2) ξ→0 ∼ c′

1 |ξ|−2 ln− 3

2 |1/ξ| + c′

0δ2(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: Z =

• (xi)t

i=1

e−β t

i=1 ǫ(xi,i) , ǫ(x, i)

       i.i.d. random s.t. βc = 1 distance overlap q ∈ [0, 1].

Pβ(q) ∼                    e(2β2−1)tqt β < 3− 1

2

e−qt/3q− 1

2 t 1 2

β = 3− 1

2

e−(β−β−1)2 tq

4 t− 1 2 q− 3 2

β ∈ (3− 1

2 , 1)

t− 1

2 q− 3 2

β ≥ 1, q ≪ 1

translation: qt = −2 log2 |z|, t = 2 log2(R/a)

ǫ(2, 1) ǫ(−2, 1) ǫ(3, 2) ǫ(1, 2) ǫ(−1, 2) ǫ(−3, 2)

t = 5 t = 0 qt = 3

[Derrida-Spohn, Arguin et.al]:

Pβ(q)

t→∞

= mδ(q) + 1 − mδ(1 − q). m = min(1, 1/β) t < ∞, β ≥ 1: [Derrida etal 1607.06610]

Liouville & log-REM Other applications

Conclusion

◮ Discrete terms ⇔ termination point transition. ◮ Continuous spectrum ⇒ 3 2 log corrections in β < 1 phase.

Similar applications:

◮ pβ(z → 0) with U(z) = a ln |z| + . . . . ◮ The case a ≥ Q/2 (bound phase).

Puzzles:

◮ Some issues in β ≥ 1 phase. ◮ Avoid freezing ? ◮ Extend to c ≤ 1 ?

10−10 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1

Liouville & log-REM Appendices

Charge at z = ∞ ⇔ charges at z ∞

• i,j

V...(. . . ) × Va∞(∞)

• LFT

=µ∗Z∗ e∗φ(∗) × e(Q−a∞)φ(∞) , φ(z) + k

j=1 4aj ln |z − wj| z≫∀wj

−→ k

j=1 4aj ln |z| = 4(Q − a∞) ln |z|

− → ∇U

In general

• |z|<R ∆U =
• |z|=R d

n.− → ∇U = 4π(Q − a∞). Here U(z) = k

j=1 4aj ln |z − wj|,

∆U = 4πajδz,wj point charges. Dilute charges will not give n < ∞-point Liouville.

Liouville & log-REM Appendices

• Va1(0)Va2(1)Vb(z)Va3=Q−a1−a2(∞)
• LFT,b=min(β,1)

=

• Q/2+iR+ |F (z|ai, α, b)|2 CDOZZ

α,a1,a2 CDOZZ −α,a3,b dα

+

• α0∈D

|F (z|ai, α0, b)|2 2πiRes

• CDOZZ

α,a1,a2 CDOZZ α,a3,b ; α → α0

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 + iR.

◮ When they are present, they dominate OPEs.

Sylvain Ribault, arXiv:1406.4290

[A.A. Belavin, A.B. Zamolodchikov, Theor.Math.Phys. 147 (2006) 729-754]

Liouville & log-REM Appendices

Liouville OPE: continuous and/or discrete

See also Konstantin Aleshkin’s talk tomorrow

Suppose β1, β2 ∈ (0, Q/2), β1 + β2 Vβ1 × Vβ2 Vα1(z)Vα2(z′ → z) × . . .

LFT

< Q

2

Vβ1+β2 + . . . |z − z′|−2δ0 = Q

2

• |p|≪1 V Q

2 +ipdp + . . .

|z − z′|−2δ0 ln− 1

2 (1/ |z − z′|)

> Q

2

• |p|≪1 V Q

2 +ipp2dp + . . .

|z − z′|−2δ ln− 3

2 (1/ |z − z′|)

δ0 = ∆β1 + ∆β2 − ∆β1+β2 = 2β1β2 , δ = ∆β1 + ∆β2 − ∆Q/2 , ∆x = x(Q − x)

• Nb. The log’s come from continue spectrum, not log-CFT.

Goal: Extract physics of pβ(z) and extend to log-correlated processes in general.

[Ribault & Santachiara arXiv:1503.02067, Rhodes & Vargas private Email.]