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Boundary correlation functions with a hidden quantum group Kalle - - PowerPoint PPT Presentation

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Boundary correlation functions with a hidden quantum group Kalle Kytl kalle.kytola@aalto.fi Department of Mathematics and Systems Analysis, Aalto University joint work with Eveliina Peltola (Univ. Helsinki) Niko Jokela (Univ. Helsinki)

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Boundary correlation functions with a hidden quantum group

Kalle Kytölä

kalle.kytola@aalto.fi

Department of Mathematics and Systems Analysis, Aalto University

joint work with Eveliina Peltola (Univ. Helsinki) Niko Jokela (Univ. Helsinki) & Matti Järvinen (ENS, Paris)

May 22, 2015 — Galileo Galilei Institute Lattice models: exact methods and combinatorics

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Correlation functions with hidden quantum group Outline Kalle Kytölä — Florence, May 2015

This talk

Quantum group construction of boundary correlation fns:

(K. & Peltola [arXiv:1408.1384])

Applications to random conformally invariant curves:

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Correlation functions with hidden quantum group Outline Kalle Kytölä — Florence, May 2015

This talk

Quantum group construction of boundary correlation fns:

(K. & Peltola [arXiv:1408.1384])

◮ seek: n-point correlation functions defined for x1 < · · · < xn

* Möbius covariant, satisfy PDEs, behavior as |xj+1 − xj| → 0

Applications to random conformally invariant curves:

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Correlation functions with hidden quantum group Outline Kalle Kytölä — Florence, May 2015

This talk

Quantum group construction of boundary correlation fns:

(K. & Peltola [arXiv:1408.1384])

◮ seek: n-point correlation functions defined for x1 < · · · < xn

* Möbius covariant, satisfy PDEs, behavior as |xj+1 − xj| → 0

◮ use: quantum group Uq(sl2) and its representations Md

Applications to random conformally invariant curves:

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SLIDE 5

Correlation functions with hidden quantum group Outline Kalle Kytölä — Florence, May 2015

This talk

Quantum group construction of boundary correlation fns:

(K. & Peltola [arXiv:1408.1384])

◮ seek: n-point correlation functions defined for x1 < · · · < xn

* Möbius covariant, satisfy PDEs, behavior as |xj+1 − xj| → 0

◮ use: quantum group Uq(sl2) and its representations Md ◮ correspondence: associate functions to vectors v ∈ n

j=1 MdJ

* representation theoretic properties of v guarantee

desired properties of the function

Applications to random conformally invariant curves:

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SLIDE 6

Correlation functions with hidden quantum group Outline Kalle Kytölä — Florence, May 2015

This talk

Quantum group construction of boundary correlation fns:

(K. & Peltola [arXiv:1408.1384])

◮ seek: n-point correlation functions defined for x1 < · · · < xn

* Möbius covariant, satisfy PDEs, behavior as |xj+1 − xj| → 0

◮ use: quantum group Uq(sl2) and its representations Md ◮ correspondence: associate functions to vectors v ∈ n

j=1 MdJ

* representation theoretic properties of v guarantee

desired properties of the function

Applications to random conformally invariant curves:

◮ Extremal multiple SLEs

(K. & Peltola [arXiv:????.????])

classification of random curves w/ non-trivial conformal moduli effect of bdry conditions on interfaces in lattice models crossing probabilities

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Correlation functions with hidden quantum group Outline Kalle Kytölä — Florence, May 2015

This talk

Quantum group construction of boundary correlation fns:

(K. & Peltola [arXiv:1408.1384])

◮ seek: n-point correlation functions defined for x1 < · · · < xn

* Möbius covariant, satisfy PDEs, behavior as |xj+1 − xj| → 0

◮ use: quantum group Uq(sl2) and its representations Md ◮ correspondence: associate functions to vectors v ∈ n

j=1 MdJ

* representation theoretic properties of v guarantee

desired properties of the function

Applications to random conformally invariant curves:

◮ Extremal multiple SLEs

(K. & Peltola [arXiv:????.????])

classification of random curves w/ non-trivial conformal moduli effect of bdry conditions on interfaces in lattice models crossing probabilities

◮ Chordal SLE bdry visiting probabilities

(Jokela & Järvinen & K. [arXiv:1311.2297])

multi-point boundary Green’s function for SLE [Lawler & . . . ] correlation fn of SLE covariant measure on bdry [Alberts & Sheffield] lattice model probas, e.g. Potts model bdry spin correlation

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Correlation functions with hidden quantum group

  • 1. Conformally invariant random curves

Kalle Kytölä — Florence, May 2015

  • 1. INTRODUCTION: CONFORMALLY INVARIANT

RANDOM CURVES

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Correlation functions with hidden quantum group

  • 1. Conformally invariant random curves

Kalle Kytölä — Florence, May 2015

Lattice model interfaces: loop-erased random walk

SRW: simple random walk

  • started at bottom left corner
  • conditioned to exit the box

through top right corner LERW: loop-erased random walk

  • chronologically erasing the loops
  • f the SRW simple path from

bottom left to top right

Scaling limit result [Lawler & Schramm & Werner 2004, Zhan 2004]

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Correlation functions with hidden quantum group

  • 1. Conformally invariant random curves

Kalle Kytölä — Florence, May 2015

Lattice model interfaces: critical percolation

Critical site percolation

  • color the hexagons black or white

according to independent fair coin tosses The exploration path

  • path from midpoint of the bottom

side to the top corner of the triangle, leaving white hexagons

  • n its immediate left and black

hexagons on its immediate right

Scaling limit result [Smirnov 2001, Camia & Newman 2007]

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Correlation functions with hidden quantum group

  • 1. Conformally invariant random curves

Kalle Kytölä — Florence, May 2015

Lattice model interfaces: random cluster model

FK-model [Fortuin-Kasteleyn 1972]

  • random subset ω of edges

P[{ω}] ∝

  • p

1−p

|ω|Q#conn. comp.(ω)

  • parameters: Q > 0, p ∈ (0, 1)

(pcritical =

√ Q 1+ √ Q)

  • all left and top side edges

conditioned to be in ω The exploration path

  • path from bottom left to top right

closely surrounding the connected component of left and top sides

Scaling limit result for Q = 2 [Smirnov 2010, Chelkak & Duminil-Copin & Hongler & Kemppainen & Smirnov 2013]

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Correlation functions with hidden quantum group

  • 1. Conformally invariant random curves

Kalle Kytölä — Florence, May 2015

Lattice model interfaces: loop-erased random walk

Critical Ising model with Dobrushin boundary conditions

  • spins σz ∈ {±1} at each pixel z,

+1 on top and left bdry, −1 on bottom and right bdry

  • probability ∝ x# disagreeing neighbors

with x = xcritical = √ 2 − 1 interface / domain wall

  • curve between the macroscopic

components of +1, −1 spins

Scaling limit result [Chelkak & Duminil-Copin & Hongler & Kemppainen & Smirnov 2013]

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Correlation functions with hidden quantum group

  • 1. Conformally invariant random curves

Kalle Kytölä — Florence, May 2015

Schramm’s classification result

The chordal Schramm-Loewner evolution (SLEκ):

[Schramm 2000, Lawler & Schramm & Werner, Rohde & Schramm, . . . ] D b a

Random curve γD;a,b in domain D from boundary point a to b

✞ ✝ ☎ ✆

law PD;a,b Classification:

◮ Conformal invariance: ◮ Domain Markov property:

= ⇒ ∃κ > 0 : PD;a,b = ”chordal SLEκ in D from a to b“

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Correlation functions with hidden quantum group

  • 1. Conformally invariant random curves

Kalle Kytölä — Florence, May 2015

Schramm’s classification result

The chordal Schramm-Loewner evolution (SLEκ):

[Schramm 2000, Lawler & Schramm & Werner, Rohde & Schramm, . . . ] D b a

Random curve γD;a,b in domain D from boundary point a to b

✞ ✝ ☎ ✆

law PD;a,b Classification:

◮ Conformal invariance: if f : D → f(D) is conformal then

f(γD;a,b) ∼ γf(D);f(a),f(b).

◮ Domain Markov property:

= ⇒ ∃κ > 0 : PD;a,b = ”chordal SLEκ in D from a to b“

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SLIDE 15

Correlation functions with hidden quantum group

  • 1. Conformally invariant random curves

Kalle Kytölä — Florence, May 2015

Schramm’s classification result

The chordal Schramm-Loewner evolution (SLEκ):

[Schramm 2000, Lawler & Schramm & Werner, Rohde & Schramm, . . . ] D b a

Random curve γD;a,b in domain D from boundary point a to b

✞ ✝ ☎ ✆

law PD;a,b Classification:

◮ Conformal invariance: if f : D → f(D) is conformal then

f(γD;a,b) ∼ γf(D);f(a),f(b).

◮ Domain Markov property: given an initial piece γ′ of γD;a,b

starting from a, the rest has the law of γD\γ′;tip,b. = ⇒ ∃κ > 0 : PD;a,b = ”chordal SLEκ in D from a to b“

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Correlation functions with hidden quantum group

  • 1. Conformally invariant random curves

Kalle Kytölä — Florence, May 2015

Properties of SLE

random fractal curve, dimHausdorff(γ) = 1 + κ

8 for κ ≤ 8 [Beffara 2008]

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Correlation functions with hidden quantum group

  • 1. Conformally invariant random curves

Kalle Kytölä — Florence, May 2015

Properties of SLE

random fractal curve, dimHausdorff(γ) = 1 + κ

8 for κ ≤ 8 [Beffara 2008]

✞ ✝ ☎ ✆

0 < κ ≤ 4 simple curve doesn’t touch boundary

✞ ✝ ☎ ✆

4 < κ < 8 non-self-crossing curve touches boundary on a random Cantor set

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Correlation functions with hidden quantum group

  • 1. Conformally invariant random curves

Kalle Kytölä — Florence, May 2015

Properties of SLE

random fractal curve, dimHausdorff(γ) = 1 + κ

8 for κ ≤ 8 [Beffara 2008]

✞ ✝ ☎ ✆

0 < κ ≤ 4 simple curve doesn’t touch boundary

✞ ✝ ☎ ✆

4 < κ < 8 non-self-crossing curve touches boundary on a random Cantor set

✞ ✝ ☎ ✆

8 ≤ κ random Peano curve

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Correlation functions with hidden quantum group

  • 1. Conformally invariant random curves

Kalle Kytölä — Florence, May 2015

Properties of SLE

random fractal curve, dimHausdorff(γ) = 1 + κ

8 for κ ≤ 8 [Beffara 2008]

✞ ✝ ☎ ✆

0 < κ ≤ 4 simple curve doesn’t touch boundary

✞ ✝ ☎ ✆

4 < κ < 8 non-self-crossing curve touches boundary on a random Cantor set

✞ ✝ ☎ ✆

8 ≤ κ random Peano curve

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Correlation functions with hidden quantum group

  • 1. Conformally invariant random curves

Kalle Kytölä — Florence, May 2015

Properties of SLE

random fractal curve, dimHausdorff(γ) = 1 + κ

8 for κ ≤ 8 [Beffara 2008]

✞ ✝ ☎ ✆

0 < κ ≤ 4 simple curve doesn’t touch boundary

✞ ✝ ☎ ✆

4 < κ < 8 non-self-crossing curve touches boundary on a random Cantor set

✞ ✝ ☎ ✆

8 ≤ κ random Peano curve in this talk κ < 8

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Correlation functions with hidden quantum group

  • 1. Conformally invariant random curves

Kalle Kytölä — Florence, May 2015

Generalizing the Dobrushin boundary conditions

Critical Ising model with Dobrushin boundary conditions

[simulation and picture by Eveliina Peltola]

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Correlation functions with hidden quantum group

  • 1. Conformally invariant random curves

Kalle Kytölä — Florence, May 2015

Generalizing the Dobrushin boundary conditions

Critical Ising model with alternating boundary conditions

[simulation and picture by Eveliina Peltola]

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Correlation functions with hidden quantum group

  • 1. Conformally invariant random curves

Kalle Kytölä — Florence, May 2015

Classification problem of multiple SLEs

D

a2 · · · · · · a3 a1 a2N

[Dubédat 2007] [Bauer & Bernard & K. 2005]

Random curves (γ(i))N

i=1 in

domain D connecting boundary points a1, a2, . . . , a2N

✞ ✝ ☎ ✆

law PD;a1,...,a2N Can we give a classification?

◮ Conformal invariance ◮ Domain Markov property (w.r.t. all initial segments)

  • initial segments absolutely continuous w.r.t. chordal SLEκ

! Convex set of multiple-SLEκ’s (PD;a1,...,a2N)

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Correlation functions with hidden quantum group

  • 2. Local multiple SLEs

Kalle Kytölä — Florence, May 2015

  • 2. MULTIPLE SCHRAMM-LOEWNER

EVOLUTIONS GROWTH PROCESSES

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Correlation functions with hidden quantum group

  • 2. Local multiple SLEs

Kalle Kytölä — Florence, May 2015

Overview of classification of multiple SLEs

local multiple SLEs Möbius covariant with curves starting from ← → positive solutions Z 2N boundary points to a system of PDEs

[pictures by Eveliina Peltola]

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Correlation functions with hidden quantum group

  • 2. Local multiple SLEs

Kalle Kytölä — Florence, May 2015

Overview of classification of multiple SLEs

local multiple SLEs Möbius covariant with curves starting from ← → positive solutions Z 2N boundary points to a system of PDEs

convex set of probability measures (finite dimensional) vector space of solutions (finite dimensional)

[pictures by Eveliina Peltola]

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Correlation functions with hidden quantum group

  • 2. Local multiple SLEs

Kalle Kytölä — Florence, May 2015

Overview of classification of multiple SLEs

local multiple SLEs Möbius covariant with curves starting from ← → positive solutions Z 2N boundary points to a system of PDEs

convex set of probability measures (finite dimensional) vector space of solutions (finite dimensional) extremal points: deterministic connectivity pattern (a planar pair partition α ∈ PPPN)

[pictures by Eveliina Peltola]

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Correlation functions with hidden quantum group

  • 2. Local multiple SLEs

Kalle Kytölä — Florence, May 2015

Overview of classification of multiple SLEs

local multiple SLEs Möbius covariant with curves starting from ← → positive solutions Z 2N boundary points to a system of PDEs

convex set of probability measures (finite dimensional) vector space of solutions (finite dimensional) extremal points: deterministic connectivity pattern (a planar pair partition α ∈ PPPN)

PPP =

N∈N PPPN,

#PPPN = CN =

1 N+1

2N

N

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Correlation functions with hidden quantum group

  • 2. Local multiple SLEs

Kalle Kytölä — Florence, May 2015

Overview of classification of multiple SLEs

local multiple SLEs Möbius covariant with curves starting from ← → positive solutions Z 2N boundary points to a system of PDEs

convex set of probability measures (finite dimensional) vector space of solutions (finite dimensional) dim = CN

[Flores & Kleban 2014]

extremal points: deterministic connectivity pattern (a planar pair partition α ∈ PPPN)

PPP =

N∈N PPPN,

#PPPN = CN =

1 N+1

2N

N

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SLIDE 30

Correlation functions with hidden quantum group

  • 2. Local multiple SLEs

Kalle Kytölä — Florence, May 2015

Overview of classification of multiple SLEs

local multiple SLEs Möbius covariant with curves starting from ← → positive solutions Z 2N boundary points to a system of PDEs

convex set of probability measures (finite dimensional) vector space of solutions (finite dimensional) dim = CN

[Flores & Kleban 2014]

extremal points: deterministic connectivity pattern (a planar pair partition α ∈ PPPN) solutions Zα with particular asymptotic behavior

PPP =

N∈N PPPN,

#PPPN = CN =

1 N+1

2N

N

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Correlation functions with hidden quantum group

  • 2. Local multiple SLEs

Kalle Kytölä — Florence, May 2015

Role of the partition function

H x6 x5 x3 x2 x1 x4

Local multiple SLEκ classification: Z ”partition function“ defined on X2N =

  • x1 < x2 < · · · < x2N
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Correlation functions with hidden quantum group

  • 2. Local multiple SLEs

Kalle Kytölä — Florence, May 2015

Role of the partition function

H x6 x5 x3 x2 x1 x4

Local multiple SLEκ classification: Z ”partition function“ defined on X2N =

  • x1 < x2 < · · · < x2N
  • (h = h1,2 = 6−κ

2κ )

Z specifies Girsanov transforms w.r.t. chordal SLEκ:

d(j:th curve) d(chordal SLEκ) ∝ k=j g′(xk)h × Z

  • g(x1), . . . , g(tip), . . . , g(x2N)).

where g : H \ (j:th curve) → H is conformal s.t. g(z) = z + o(1).

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SLIDE 33

Correlation functions with hidden quantum group

  • 2. Local multiple SLEs

Kalle Kytölä — Florence, May 2015

Role of the partition function

H x6 x5 x3 x2 x1 x4

Local multiple SLEκ classification: Z ”partition function“ defined on X2N =

  • x1 < x2 < · · · < x2N
  • (h = h1,2 = 6−κ

2κ )

(PDE) DjZ = 0 for all j = 1, . . . , 2N, where Dj = κ

2 ∂2 ∂x2

j +

i=j

  • 2

xi−xj ∂ ∂xi − 2h (xi−xj)2

  • .

Z specifies Girsanov transforms w.r.t. chordal SLEκ:

d(j:th curve) d(chordal SLEκ) ∝ k=j g′(xk)h × Z

  • g(x1), . . . , g(tip), . . . , g(x2N)).

where g : H \ (j:th curve) → H is conformal s.t. g(z) = z + o(1).

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Correlation functions with hidden quantum group

  • 2. Local multiple SLEs

Kalle Kytölä — Florence, May 2015

Role of the partition function

H x6 x5 x3 x2 x1 x4

Local multiple SLEκ classification: Z ”partition function“ defined on X2N =

  • x1 < x2 < · · · < x2N
  • (h = h1,2 = 6−κ

2κ )

(PDE) DjZ = 0 for all j = 1, . . . , 2N, where Dj = κ

2 ∂2 ∂x2

j +

i=j

  • 2

xi−xj ∂ ∂xi − 2h (xi−xj)2

  • .

(COV) For µ: H → H Möbius s.t. µ(x1) < · · · < µ(x2N) we have Z

  • x1, . . . , x2N
  • = 2N

j=1 µ′(xj)h × Z

  • µ(x1), . . . , µ(x2N)
  • .

Z specifies Girsanov transforms w.r.t. chordal SLEκ:

d(j:th curve) d(chordal SLEκ) ∝ k=j g′(xk)h × Z

  • g(x1), . . . , g(tip), . . . , g(x2N)).

where g : H \ (j:th curve) → H is conformal s.t. g(z) = z + o(1).

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Correlation functions with hidden quantum group

  • 2. Local multiple SLEs

Kalle Kytölä — Florence, May 2015

Collapsing marked points

Suppose limxj,xj+1→ξ

Z(x1,...,x2N) (xj+1−xj)−2h

= ˆ Z(x1, . . . , xj−1, xj+2, . . . , x2N). Then the law of the curves other than j, j + 1 under local 2N-SLEκ defined by Z tends to the local (2N − 2)-SLEκ defined by ˆ Z as xj, xj+1 → ξ.

1 2 3 4 5 6 7 8 1 2 3 4 5 6

For pure partition functions Zα, α ∈ PPP, thus require (ASY) lim

xj,xj+1→ξ

Zα (xj+1 − xj)−2h =

  • Zα/{j,j+1}

if {j, j + 1} ∈ α if {j, j + 1} / ∈ α

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Correlation functions with hidden quantum group

  • 2. Local multiple SLEs

Kalle Kytölä — Florence, May 2015

Multiple SLEs pure partition function problem

  • Zα
  • α∈PPP

for α ∈ PPPN, function Zα on X2N =

  • x1 < x2 < · · · < x2N
  • s.t.

(PDE) DjZα = 0 where Dj = κ 2 ∂2 ∂x2

j

+

  • i=j
  • 2

xi − xj ∂ ∂xi − 2h (xi − xj)2

  • (COV) Z
  • x1, . . . , x2N
  • =

2N

  • j=1

µ′(xj)h×Z

  • µ(x1), . . . , µ(x2N)
  • (ASY)

lim

xj,xj+1→ξ

Zα (xj+1 − xj)−2h =

  • Zα/{j,j+1}

if {j, j + 1} ∈ α if {j, j + 1} / ∈ α

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Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

  • 3. SOLUTION OF PURE PARTITION FUNCTIONS

BY A HIDDEN QUANTUM GROUP

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SLIDE 38

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

Overview of the quantum group method

Correspondence: vectors in an n-fold tensor product representation ← → functions of n variables

  • f a quantum group

highest weight vectors ← → solutions to partial

  • f subrepresentations

differential equations vectors in the ← → Möbius covariant trivial subrepresentation functions prescribed projections ← → prescribed asymptotic to subrepresentations behavior

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SLIDE 39

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

Idea: Integral solutions to PDEs

How to solve the PDEs? (DjZ)(x1, . . . , x2N) = 0, where Dj = κ

2 ∂2 ∂x2

j +

i=j

  • 2

xi−xj ∂ ∂xi − 2h (xi−xj)2

  • .
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SLIDE 40

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

Idea: Integral solutions to PDEs

How to solve the PDEs? (DjZ)(x1, . . . , x2N) = 0, where Dj = κ

2 ∂2 ∂x2

j +

i=j

  • 2

xi−xj ∂ ∂xi − 2h (xi−xj)2

  • .

Z(x1, . . . , x2N) =

  • Γ

f(x1, . . . , x2N; w1, . . . , wℓ) dw1 · · · dwℓ ?

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SLIDE 41

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

Idea: Integral solutions to PDEs

How to solve the PDEs? (DjZ)(x1, . . . , x2N) = 0, where Dj = κ

2 ∂2 ∂x2

j +

i=j

  • 2

xi−xj ∂ ∂xi − 2h (xi−xj)2

  • .

Z(x1, . . . , x2N) =

  • Γ

f(x1, . . . , x2N; w1, . . . , wℓ) dw1 · · · dwℓ ?

◮ find appropriate f(x1, . . . , x2N; w1, . . . , wℓ)

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SLIDE 42

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

Idea: Integral solutions to PDEs

How to solve the PDEs? (DjZ)(x1, . . . , x2N) = 0, where Dj = κ

2 ∂2 ∂x2

j +

i=j

  • 2

xi−xj ∂ ∂xi − 2h (xi−xj)2

  • .

Z(x1, . . . , x2N) =

  • Γ

f(x1, . . . , x2N; w1, . . . , wℓ) dw1 · · · dwℓ ?

◮ find appropriate f(x1, . . . , x2N; w1, . . . , wℓ) ◮ (Djf)dw1 · · · dwℓ is an exact ℓ-form

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SLIDE 43

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

Idea: Integral solutions to PDEs

How to solve the PDEs? (DjZ)(x1, . . . , x2N) = 0, where Dj = κ

2 ∂2 ∂x2

j +

i=j

  • 2

xi−xj ∂ ∂xi − 2h (xi−xj)2

  • .

Z(x1, . . . , x2N) =

  • Γ

f(x1, . . . , x2N; w1, . . . , wℓ) dw1 · · · dwℓ ?

◮ find appropriate f(x1, . . . , x2N; w1, . . . , wℓ) ◮ (Djf)dw1 · · · dwℓ is an exact ℓ-form ◮ if ∂Γ = ∅, then the integral Z solves (DjZ)(z) = 0

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SLIDE 44

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

Idea: Integral solutions to PDEs

How to solve the PDEs? (DjZ)(x1, . . . , x2N) = 0, where Dj = κ

2 ∂2 ∂x2

j +

i=j

  • 2

xi−xj ∂ ∂xi − 2h (xi−xj)2

  • .

Z(x1, . . . , x2N) =

  • Γ

f(x1, . . . , x2N; w1, . . . , wℓ) dw1 · · · dwℓ ?

◮ find appropriate f(x1, . . . , x2N; w1, . . . , wℓ)

[Dotsenko-Fateev 1984] ◮ f =

i<j(xj − xi)

2 κ ×

i,r(wr − xi)

−4 κ ×

r<s(ws − wr)

8 κ

◮ (Djf)dw1 · · · dwℓ is an exact ℓ-form ◮ if ∂Γ = ∅, then the integral Z solves (DjZ)(z) = 0

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SLIDE 45

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

Idea: Integral solutions to PDEs

How to solve the PDEs? (DjZ)(x1, . . . , x2N) = 0, where Dj = κ

2 ∂2 ∂x2

j +

i=j

  • 2

xi−xj ∂ ∂xi − 2h (xi−xj)2

  • .

Z(x1, . . . , x2N) =

  • Γ

f(x1, . . . , x2N; w1, . . . , wℓ) dw1 · · · dwℓ ?

◮ find appropriate f(x1, . . . , x2N; w1, . . . , wℓ)

[Dotsenko-Fateev 1984] ◮ f =

i<j(xj − xi)

2 κ ×

i,r(wr − xi)

−4 κ ×

r<s(ws − wr)

8 κ

◮ (Djf)dw1 · · · dwℓ is an exact ℓ-form ◮ if ∂Γ = ∅, then the integral Z solves (DjZ)(z) = 0 ◮ find appropriate Γ to solve PDEs with boundary conditions?

slide-46
SLIDE 46

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

Idea: Integral solutions to PDEs

How to solve the PDEs? (DjZ)(x1, . . . , x2N) = 0, where Dj = κ

2 ∂2 ∂x2

j +

i=j

  • 2

xi−xj ∂ ∂xi − 2h (xi−xj)2

  • .

Z(x1, . . . , x2N) =

  • Γ

f(x1, . . . , x2N; w1, . . . , wℓ) dw1 · · · dwℓ ?

◮ find appropriate f(x1, . . . , x2N; w1, . . . , wℓ)

[Dotsenko-Fateev 1984] ◮ f =

i<j(xj − xi)

2 κ ×

i,r(wr − xi)

−4 κ ×

r<s(ws − wr)

8 κ

◮ (Djf)dw1 · · · dwℓ is an exact ℓ-form ◮ if ∂Γ = ∅, then the integral Z solves (DjZ)(z) = 0 ◮ find appropriate Γ to solve PDEs with boundary conditions?

quantum group Uq(sl2) acts on Γ

[Felder & Wieczerkowski 1991, Peltola & K. 2014]

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SLIDE 47

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

The quantum group and its representations

slide-48
SLIDE 48

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

The quantum group and its representations

q = eiπ4/κ (assume κ / ∈ Q)

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SLIDE 49

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

The quantum group and its representations

q = eiπ4/κ (assume κ / ∈ Q)

◮ Algebra Uq(sl2): gen. E, F, K, K −1 and q-Chevalley rel.

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SLIDE 50

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

The quantum group and its representations

q = eiπ4/κ (assume κ / ∈ Q)

◮ Algebra Uq(sl2): gen. E, F, K, K −1 and q-Chevalley rel.

KE = q2EK, KF = q−2FK, KK −1 = K −1K = 1 EF − FE =

1 q−q−1

  • K − K −1
slide-51
SLIDE 51

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

The quantum group and its representations

q = eiπ4/κ (assume κ / ∈ Q)

◮ Algebra Uq(sl2): gen. E, F, K, K −1 and q-Chevalley rel.

KE = q2EK, KF = q−2FK, KK −1 = K −1K = 1 EF − FE =

1 q−q−1

  • K − K −1

◮ Irreducible rep. Md of dimension d: basis e0, e1, . . . , ed−1

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SLIDE 52

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

The quantum group and its representations

q = eiπ4/κ (assume κ / ∈ Q)

◮ Algebra Uq(sl2): gen. E, F, K, K −1 and q-Chevalley rel.

KE = q2EK, KF = q−2FK, KK −1 = K −1K = 1 EF − FE =

1 q−q−1

  • K − K −1

◮ Irreducible rep. Md of dimension d: basis e0, e1, . . . , ed−1

K.ej = qd−1−2jej, F.ej = ej+1, E.ej = [j] [d − j] ej−1 where [n] = qn−q−n

q−q−1 are ”q-integers“

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SLIDE 53

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

The quantum group and its representations

q = eiπ4/κ (assume κ / ∈ Q)

◮ Algebra Uq(sl2): gen. E, F, K, K −1 and q-Chevalley rel.

KE = q2EK, KF = q−2FK, KK −1 = K −1K = 1 EF − FE =

1 q−q−1

  • K − K −1

◮ Irreducible rep. Md of dimension d: basis e0, e1, . . . , ed−1

K.ej = qd−1−2jej, F.ej = ej+1, E.ej = [j] [d − j] ej−1 where [n] = qn−q−n

q−q−1 are ”q-integers“

◮ Tensor products of representations: for v ⊗ w ∈ V ⊗ W set

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SLIDE 54

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

The quantum group and its representations

q = eiπ4/κ (assume κ / ∈ Q)

◮ Algebra Uq(sl2): gen. E, F, K, K −1 and q-Chevalley rel.

KE = q2EK, KF = q−2FK, KK −1 = K −1K = 1 EF − FE =

1 q−q−1

  • K − K −1

◮ Irreducible rep. Md of dimension d: basis e0, e1, . . . , ed−1

K.ej = qd−1−2jej, F.ej = ej+1, E.ej = [j] [d − j] ej−1 where [n] = qn−q−n

q−q−1 are ”q-integers“

◮ Tensor products of representations: for v ⊗ w ∈ V ⊗ W set

K.(v ⊗ w) = K.v ⊗ K.w, E.(v ⊗ w) = E.v ⊗ K.w + v ⊗ E.w, F.(v ⊗ w) = F.v ⊗ w + K −1.v ⊗ F.w

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SLIDE 55

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

The quantum group and its representations

q = eiπ4/κ (assume κ / ∈ Q)

◮ Algebra Uq(sl2): gen. E, F, K, K −1 and q-Chevalley rel.

KE = q2EK, KF = q−2FK, KK −1 = K −1K = 1 EF − FE =

1 q−q−1

  • K − K −1

◮ Irreducible rep. Md of dimension d: basis e0, e1, . . . , ed−1

K.ej = qd−1−2jej, F.ej = ej+1, E.ej = [j] [d − j] ej−1 where [n] = qn−q−n

q−q−1 are ”q-integers“

◮ Tensor products of representations: for v ⊗ w ∈ V ⊗ W set

K.(v ⊗ w) = K.v ⊗ K.w, E.(v ⊗ w) = E.v ⊗ K.w + v ⊗ E.w, F.(v ⊗ w) = F.v ⊗ w + K −1.v ⊗ F.w

◮ Semisimple tensor products of the irreps:

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SLIDE 56

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

The quantum group and its representations

q = eiπ4/κ (assume κ / ∈ Q)

◮ Algebra Uq(sl2): gen. E, F, K, K −1 and q-Chevalley rel.

KE = q2EK, KF = q−2FK, KK −1 = K −1K = 1 EF − FE =

1 q−q−1

  • K − K −1

◮ Irreducible rep. Md of dimension d: basis e0, e1, . . . , ed−1

K.ej = qd−1−2jej, F.ej = ej+1, E.ej = [j] [d − j] ej−1 where [n] = qn−q−n

q−q−1 are ”q-integers“

◮ Tensor products of representations: for v ⊗ w ∈ V ⊗ W set

K.(v ⊗ w) = K.v ⊗ K.w, E.(v ⊗ w) = E.v ⊗ K.w + v ⊗ E.w, F.(v ⊗ w) = F.v ⊗ w + K −1.v ⊗ F.w

◮ Semisimple tensor products of the irreps:

Md2 ⊗ Md1 ∼ = Md1+d2−1 ⊕ Md1+d2−3 ⊕ · · · ⊕ M|d1−d2|+1

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SLIDE 57

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

The correspondence theorem (special case)

κ ∈ (0, 8) \ Q, q = eiπ4/κ

Theorem (K. & Peltola)

F(x0) : M⊗2N

2

− → {functions on X(x0)

2N }

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SLIDE 58

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

The correspondence theorem (special case)

◮ X(x0) 2N = {(xj)2N j=1

  • x0 < x1 < · · · < x2N},

X2N =

x0 X(x0) 2N

κ ∈ (0, 8) \ Q, q = eiπ4/κ

Theorem (K. & Peltola)

F(x0) : M⊗2N

2

− → {functions on X(x0)

2N }

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SLIDE 59

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

The correspondence theorem (special case)

◮ X(x0) 2N = {(xj)2N j=1

  • x0 < x1 < · · · < x2N},

X2N =

x0 X(x0) 2N

κ ∈ (0, 8) \ Q, q = eiπ4/κ

Theorem (K. & Peltola)

F(x0) : M⊗2N

2

− → {functions on X(x0)

2N }

(X) If E.v = 0, then F(x0)[v]: X(x0)

n

→ C is independent of x0, thus defines a function F[v]: X2N → C.

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SLIDE 60

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

The correspondence theorem (special case)

◮ X(x0) 2N = {(xj)2N j=1

  • x0 < x1 < · · · < x2N},

X2N =

x0 X(x0) 2N

κ ∈ (0, 8) \ Q, q = eiπ4/κ

Theorem (K. & Peltola)

F(x0) : M⊗2N

2

− → {functions on X(x0)

2N }

(X) If E.v = 0, then F(x0)[v]: X(x0)

n

→ C is independent of x0, thus defines a function F[v]: X2N → C.

(PDE) If E.v = 0, then Z = F[v] satisfies (PDE).

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SLIDE 61

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

The correspondence theorem (special case)

◮ X(x0) 2N = {(xj)2N j=1

  • x0 < x1 < · · · < x2N},

X2N =

x0 X(x0) 2N

κ ∈ (0, 8) \ Q, q = eiπ4/κ

Theorem (K. & Peltola)

F(x0) : M⊗2N

2

− → {functions on X(x0)

2N }

(X) If E.v = 0, then F(x0)[v]: X(x0)

n

→ C is independent of x0, thus defines a function F[v]: X2N → C.

(PDE) If E.v = 0, then Z = F[v] satisfies (PDE). (COV) If E.v = 0 and K.v = v, then Z = F[v] satisfies (COV).

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SLIDE 62

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

The correspondence theorem (special case)

◮ X(x0) 2N = {(xj)2N j=1

  • x0 < x1 < · · · < x2N},

X2N =

x0 X(x0) 2N

κ ∈ (0, 8) \ Q, q = eiπ4/κ

Theorem (K. & Peltola)

F(x0) : M⊗2N

2

− → {functions on X(x0)

2N }

(X) If E.v = 0, then F(x0)[v]: X(x0)

n

→ C is independent of x0, thus defines a function F[v]: X2N → C.

(PDE) If E.v = 0, then Z = F[v] satisfies (PDE). (COV) If E.v = 0 and K.v = v, then Z = F[v] satisfies (COV). (ASY) limxj,xj+1→ξ

F(x0)[v] (xj+1−xj)−2h = F(x0)[ˆ

πj(v)]

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SLIDE 63

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

The correspondence theorem (special case)

◮ M2 ⊗ M2 ∼

= M1 ⊕ M3

◮ X(x0) 2N = {(xj)2N j=1

  • x0 < x1 < · · · < x2N},

X2N =

x0 X(x0) 2N

κ ∈ (0, 8) \ Q, q = eiπ4/κ

Theorem (K. & Peltola)

F(x0) : M⊗2N

2

− → {functions on X(x0)

2N }

(X) If E.v = 0, then F(x0)[v]: X(x0)

n

→ C is independent of x0, thus defines a function F[v]: X2N → C.

(PDE) If E.v = 0, then Z = F[v] satisfies (PDE). (COV) If E.v = 0 and K.v = v, then Z = F[v] satisfies (COV). (ASY) limxj,xj+1→ξ

F(x0)[v] (xj+1−xj)−2h = F(x0)[ˆ

πj(v)]

slide-64
SLIDE 64

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

The correspondence theorem (special case)

◮ M2 ⊗ M2 ∼

= M1 ⊕ M3, proj. to M1 ∼ = C is ˆ π: M2 ⊗ M2 → C.

◮ X(x0) 2N = {(xj)2N j=1

  • x0 < x1 < · · · < x2N},

X2N =

x0 X(x0) 2N

κ ∈ (0, 8) \ Q, q = eiπ4/κ

Theorem (K. & Peltola)

F(x0) : M⊗2N

2

− → {functions on X(x0)

2N }

(X) If E.v = 0, then F(x0)[v]: X(x0)

n

→ C is independent of x0, thus defines a function F[v]: X2N → C.

(PDE) If E.v = 0, then Z = F[v] satisfies (PDE). (COV) If E.v = 0 and K.v = v, then Z = F[v] satisfies (COV). (ASY) limxj,xj+1→ξ

F(x0)[v] (xj+1−xj)−2h = F(x0)[ˆ

πj(v)]

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SLIDE 65

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

The correspondence theorem (special case)

◮ M2 ⊗ M2 ∼

= M1 ⊕ M3, proj. to M1 ∼ = C is ˆ π: M2 ⊗ M2 → C.

◮ ˆ

πj : M⊗2N

2

→ M⊗2(N−1)

2

, projection ˆ π in factors j and j + 1

◮ X(x0) 2N = {(xj)2N j=1

  • x0 < x1 < · · · < x2N},

X2N =

x0 X(x0) 2N

κ ∈ (0, 8) \ Q, q = eiπ4/κ

Theorem (K. & Peltola)

F(x0) : M⊗2N

2

− → {functions on X(x0)

2N }

(X) If E.v = 0, then F(x0)[v]: X(x0)

n

→ C is independent of x0, thus defines a function F[v]: X2N → C.

(PDE) If E.v = 0, then Z = F[v] satisfies (PDE). (COV) If E.v = 0 and K.v = v, then Z = F[v] satisfies (COV). (ASY) limxj,xj+1→ξ

F(x0)[v] (xj+1−xj)−2h = F(x0)[ˆ

πj(v)]

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SLIDE 66

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

Translation of the multiple SLE problem

◮ M2 ⊗ M2 ∼

= M1 ⊕ M3, proj. to M1 ∼ = C is ˆ π: M2 ⊗ M2 → C.

◮ ˆ

πj : M⊗2N

2

→ M⊗2(N−1)

2

, projection ˆ π in factors j and j + 1 The translation: If (vα)α∈PPPN satisfies (SING) K.vα = vα, E.vα = 0, (F.vα = 0) (PROJ) ˆ πj(vα) =

  • vα/{j,j+1}

if {j, j + 1} ∈ α if {j, j + 1} / ∈ α ∀j then the functions Zα = F[vα] satisfy (PDE), (COV), (ASY).

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SLIDE 67

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

Translation of the multiple SLE problem

◮ M2 ⊗ M2 ∼

= M1 ⊕ M3, proj. to M1 ∼ = C is ˆ π: M2 ⊗ M2 → C.

◮ ˆ

πj : M⊗2N

2

→ M⊗2(N−1)

2

, projection ˆ π in factors j and j + 1 The translation: If (vα)α∈PPPN satisfies (SING) K.vα = vα, E.vα = 0, (F.vα = 0) (PROJ) ˆ πj(vα) =

  • vα/{j,j+1}

if {j, j + 1} ∈ α if {j, j + 1} / ∈ α ∀j then the functions Zα = F[vα] satisfy (PDE), (COV), (ASY). Trivial subrepresentation: dim{v ∈ M⊗2N

2

  • (SING)} = CN =

1 N+1

2N

N

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SLIDE 68

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

Translation of the multiple SLE problem

◮ M2 ⊗ M2 ∼

= M1 ⊕ M3, proj. to M1 ∼ = C is ˆ π: M2 ⊗ M2 → C.

◮ ˆ

πj : M⊗2N

2

→ M⊗2(N−1)

2

, projection ˆ π in factors j and j + 1 The translation: If (vα)α∈PPPN satisfies (SING) K.vα = vα, E.vα = 0, (F.vα = 0) (PROJ) ˆ πj(vα) =

  • vα/{j,j+1}

if {j, j + 1} ∈ α if {j, j + 1} / ∈ α ∀j then the functions Zα = F[vα] satisfy (PDE), (COV), (ASY). Trivial subrepresentation: dim{v ∈ M⊗2N

2

  • (SING)} = CN =

1 N+1

2N

N

  • Uniqueness of solutions: The only solution of the

homogeneous problem, ˆ πj(v) = 0 ∀j & (SING), is v = 0.

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SLIDE 69

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

Explicit solution for the maximally nested case

Rainbow configuration: ⋓N = {{1, 2N}, {2, 2N − 1}, . . . , {N, N + 1}}

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SLIDE 70

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

Explicit solution for the maximally nested case

Rainbow configuration: ⋓N = {{1, 2N}, {2, 2N − 1}, . . . , {N, N + 1}}

Note: for rainbow configuration ⋓N ∈ PPPN, (PROJ) becomes ˆ πN(v⋓N ) = v⋓N−1 and ˆ πj(v⋓N ) = 0 for j = N

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SLIDE 71

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

Explicit solution for the maximally nested case

Rainbow configuration: ⋓N = {{1, 2N}, {2, 2N − 1}, . . . , {N, N + 1}}

Note: for rainbow configuration ⋓N ∈ PPPN, (PROJ) becomes ˆ πN(v⋓N ) = v⋓N−1 and ˆ πj(v⋓N ) = 0 for j = N

Explicit formula: The solution for rainbow configurations is v⋓N = const.×

N

  • k=0

(−1)kqk(N−k−1) ×(F k.(e⊗N ))⊗(F N−k.(e⊗N )).

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SLIDE 72

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

Recursive solution on the poset of configurations

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SLIDE 73

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

Recursive solution on the poset of configurations

  • Tying operation

℘j : PPPN → PPPN:

  • connect j and j + 1
  • connect the points to which

j and j + 1 were previously connected

l2 l1 j j + 1

α ℘j

l2 l1 j j + 1

α℘j

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SLIDE 74

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

Recursive solution on the poset of configurations

  • Tying operation

℘j : PPPN → PPPN:

  • connect j and j + 1
  • connect the points to which

j and j + 1 were previously connected Recursion based on formula: if {j, j + 1} ∈ ̺ ∈ PPPN, then (id − πj) (v̺) = −1 [2]

  • β∈℘−1

j

(̺)\{̺}

vβ

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SLIDE 75

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

Summary: solution of pure partition functions

Theorem (K. & Peltola)

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SLIDE 76

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

Summary: solution of pure partition functions

Theorem (K. & Peltola)

◮ With v∅ = 1, there is a unique collection (vα)α∈PPP solving

the system (SING) & (PROJ).

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SLIDE 77

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

Summary: solution of pure partition functions

Theorem (K. & Peltola)

◮ With v∅ = 1, there is a unique collection (vα)α∈PPP solving

the system (SING) & (PROJ).

◮ The vectors (vα)α∈PPPN span the CN-dimensional trivial

subrepresentation {v ∈ W ⊗2N

2

  • (SING)}
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SLIDE 78

Correlation functions with hidden quantum group

  • 3. Quantum group solutions for multiple SLEs

Kalle Kytölä — Florence, May 2015

Summary: solution of pure partition functions

Theorem (K. & Peltola)

◮ With v∅ = 1, there is a unique collection (vα)α∈PPP solving

the system (SING) & (PROJ).

◮ The vectors (vα)α∈PPPN span the CN-dimensional trivial

subrepresentation {v ∈ W ⊗2N

2

  • (SING)}

◮ The functions Zα = F[vα], span cN-dimensional solution

spaces of the system

(PDE) DjZα = 0, Dj = κ

2 ∂2 ∂x2

j +

i=j

  • 2

xi−xj ∂ ∂xi − 2h (xi−xj)2

  • (COV) Z
  • x1, . . .
  • = 2N

j=1 µ′(xj)h × Z

  • µ(x1), . . .
  • and their asymptotic behavior as xj, xj+1 → ξ is

(ASY) lim

Zα (xj+1−xj)−2h =

  • Zα/{j,j+1}

if {j, j + 1} ∈ α if {j, j + 1} / ∈ α .

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SLIDE 79

Correlation functions with hidden quantum group

  • 4. Details about the quantum group method

Kalle Kytölä — Florence, May 2015

  • 4. GENERAL QUANTUM GROUP METHOD AND

SOME DETAILS

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SLIDE 80

Correlation functions with hidden quantum group

  • 4. Details about the quantum group method

Kalle Kytölä — Florence, May 2015

Overview of the quantum group method (again)

Correspondence: vectors in an n-fold tensor product representation ← → functions of n variables

  • f a quantum group

highest weight vectors ← → solutions to partial

  • f subrepresentations

differential equations vectors in the ← → Möbius covariant trivial subrepresentation functions prescribed projections ← → prescribed asymptotic to subrepresentations behavior

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SLIDE 81

Correlation functions with hidden quantum group

  • 4. Details about the quantum group method

Kalle Kytölä — Florence, May 2015

Integral solutions to PDEs of CFTs

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SLIDE 82

Correlation functions with hidden quantum group

  • 4. Details about the quantum group method

Kalle Kytölä — Florence, May 2015

The correspondence theorem (general case)

F (x0)

d1,...,dn : n j=1 Mdj −

→ {functions on X(x0)

n

}

Theorem (K. & Peltola)

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SLIDE 83

Correlation functions with hidden quantum group

  • 4. Details about the quantum group method

Kalle Kytölä — Florence, May 2015

The correspondence theorem (general case)

(v highest weight vector ⇔ E.v = 0) (v in trivial subrepresentation ⇔ E.v = 0 and K.v = v)

F (x0)

d1,...,dn : n j=1 Mdj −

→ {functions on X(x0)

n

}

Theorem (K. & Peltola)

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SLIDE 84

Correlation functions with hidden quantum group

  • 4. Details about the quantum group method

Kalle Kytölä — Florence, May 2015

The correspondence theorem (general case)

(v highest weight vector ⇔ E.v = 0) (v in trivial subrepresentation ⇔ E.v = 0 and K.v = v)

F (x0)

d1,...,dn : n j=1 Mdj −

→ {functions on X(x0)

n

}

Theorem (K. & Peltola)

(Xn) If v is a highest weight vector, then F(x0)[v]: X(x0)

n

→ C is independent of x0, thus defines a function F[v]: Xn → C.

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SLIDE 85

Correlation functions with hidden quantum group

  • 4. Details about the quantum group method

Kalle Kytölä — Florence, May 2015

The correspondence theorem (general case)

(v highest weight vector ⇔ E.v = 0) (v in trivial subrepresentation ⇔ E.v = 0 and K.v = v)

F (x0)

d1,...,dn : n j=1 Mdj −

→ {functions on X(x0)

n

}

Theorem (K. & Peltola)

(Xn) If v is a highest weight vector, then F(x0)[v]: X(x0)

n

→ C is independent of x0, thus defines a function F[v]: Xn → C.

(PDE) If v is a highest weight vector, then F[v]: Xn → C satisfies

n linear homogeneous PDEs of orders d1, . . . , dn.

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SLIDE 86

Correlation functions with hidden quantum group

  • 4. Details about the quantum group method

Kalle Kytölä — Florence, May 2015

The correspondence theorem (general case)

(v highest weight vector ⇔ E.v = 0) (v in trivial subrepresentation ⇔ E.v = 0 and K.v = v)

F (x0)

d1,...,dn : n j=1 Mdj −

→ {functions on X(x0)

n

}

Theorem (K. & Peltola)

(Xn) If v is a highest weight vector, then F(x0)[v]: X(x0)

n

→ C is independent of x0, thus defines a function F[v]: Xn → C.

(PDE) If v is a highest weight vector, then F[v]: Xn → C satisfies

n linear homogeneous PDEs of orders d1, . . . , dn.

(COV) F(x0)[v]: X(x0)

n

→ C is

  • translation invariant
  • homogeneous, if v is K-eigenvector
  • Möbius covariant, if v is in trivial subrepresentation
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SLIDE 87

Correlation functions with hidden quantum group

  • 4. Details about the quantum group method

Kalle Kytölä — Florence, May 2015

The correspondence theorem (general case)

(v highest weight vector ⇔ E.v = 0) (v in trivial subrepresentation ⇔ E.v = 0 and K.v = v)

F (x0)

d1,...,dn : n j=1 Mdj −

→ {functions on X(x0)

n

}

Theorem (K. & Peltola)

(Xn) If v is a highest weight vector, then F(x0)[v]: X(x0)

n

→ C is independent of x0, thus defines a function F[v]: Xn → C.

(PDE) If v is a highest weight vector, then F[v]: Xn → C satisfies

n linear homogeneous PDEs of orders d1, . . . , dn.

(COV) F(x0)[v]: X(x0)

n

→ C is

  • translation invariant
  • homogeneous, if v is K-eigenvector
  • Möbius covariant, if v is in trivial subrepresentation

(ASY) Mdj+1 ⊗ Mdj ∼

=

d Md induces a decomp. of n j=1 Mdj.

If v ∈

i>j+1 Mdi

  • ⊗ Md ⊗

i<j Mdi

  • , then

F(x0)

...,dj,dj+1,...[v] ∼ (xj+1 − xj)∆d × F(x0) ...,d,...[v].

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SLIDE 88

Correlation functions with hidden quantum group

  • 4. Details about the quantum group method

Kalle Kytölä — Florence, May 2015

Sketch: definition of the correspondence

* anchor x0, chamber X(x0)

n

= {x0 < x1 < x2 < · · · < xn} ⊂ Rn

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SLIDE 89

Correlation functions with hidden quantum group

  • 4. Details about the quantum group method

Kalle Kytölä — Florence, May 2015

Sketch: definition of the correspondence

* anchor x0, chamber X(x0)

n

= {x0 < x1 < x2 < · · · < xn} ⊂ Rn * parameters d1, d2, . . . , dn ∈ N

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SLIDE 90

Correlation functions with hidden quantum group

  • 4. Details about the quantum group method

Kalle Kytölä — Florence, May 2015

Sketch: definition of the correspondence

* anchor x0, chamber X(x0)

n

= {x0 < x1 < x2 < · · · < xn} ⊂ Rn * parameters d1, d2, . . . , dn ∈ N

F(x0) :

n

  • j=1

Mdj − → {functions on X(x0)

n

}

Informally, F(x0)[v](x) = “

  • Γ[v] f(x; w)dw”,

where the integration surface Γ[v] depends on v ∈ n

j=1 Mdj .

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SLIDE 91

Correlation functions with hidden quantum group

  • 4. Details about the quantum group method

Kalle Kytölä — Florence, May 2015

Sketch: definition of the correspondence

* anchor x0, chamber X(x0)

n

= {x0 < x1 < x2 < · · · < xn} ⊂ Rn * parameters d1, d2, . . . , dn ∈ N

F(x0) :

n

  • j=1

Mdj − → {functions on X(x0)

n

}

Informally, F(x0)[v](x) = “

  • Γ[v] f(x; w)dw”,

where the integration surface Γ[v] depends on v ∈ n

j=1 Mdj .

F(x0)[eln ⊗ · · · ⊗ el1] = ϕ(x0)

l1,...,ln (below), extend linearly

ϕ(x0)

l1,...,ln(x) =

l1 l2 ln

x0 x1 x2 xn . . .

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SLIDE 92

Correlation functions with hidden quantum group

  • 4. Details about the quantum group method

Kalle Kytölä — Florence, May 2015

Sketch: definition of the correspondence

* anchor x0, chamber X(x0)

n

= {x0 < x1 < x2 < · · · < xn} ⊂ Rn * parameters d1, d2, . . . , dn ∈ N

F(x0) :

n

  • j=1

Mdj − → {functions on X(x0)

n

}

Informally, F(x0)[v](x) = “

  • Γ[v] f(x; w)dw”,

where the integration surface Γ[v] depends on v ∈ n

j=1 Mdj .

F(x0)[eln ⊗ · · · ⊗ el1] = ϕ(x0)

l1,...,ln (below), extend linearly

ϕ(x0)

l1,...,ln(x) =

l1 l2 ln

x0 x1 x2 xn . . .

f ∝ (xj − xi)

2 κ (di −1)(dj −1) × (ws − wr) 8 κ × (wr − xi)− 4 κ (di −1)

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SLIDE 93

Correlation functions with hidden quantum group

  • 4. Details about the quantum group method

Kalle Kytölä — Florence, May 2015

Sketch: asymptotics with subrepresentations

Md ֒ → Mdj+1 ⊗ Mdj d = dj + dj+1 − 1 − 2m

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SLIDE 94

Correlation functions with hidden quantum group

  • 4. Details about the quantum group method

Kalle Kytölä — Florence, May 2015

Sketch: asymptotics with subrepresentations

Md ֒ → Mdj+1 ⊗ Mdj, e0 → τ

(d;dj,dj+1)

, d = dj + dj+1 − 1 − 2m

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SLIDE 95

Correlation functions with hidden quantum group

  • 4. Details about the quantum group method

Kalle Kytölä — Florence, May 2015

Sketch: asymptotics with subrepresentations

Md ֒ → Mdj+1 ⊗ Mdj, e0 → τ

(d;dj,dj+1)

, d = dj + dj+1 − 1 − 2m τ

(d;dj,dj+1)

∝

k(−1)k [dj−1−k]![dj+1−1−m+k]! [k]![dj−1]![m−k]![d2−1]! qk(d1−k) (q−q−1)m (ek ⊗em−k)

slide-96
SLIDE 96

Correlation functions with hidden quantum group

  • 4. Details about the quantum group method

Kalle Kytölä — Florence, May 2015

Sketch: asymptotics with subrepresentations

Md ֒ → Mdj+1 ⊗ Mdj, e0 → τ

(d;dj,dj+1)

, d = dj + dj+1 − 1 − 2m τ

(d;dj,dj+1)

∝

k(−1)k [dj−1−k]![dj+1−1−m+k]! [k]![dj−1]![m−k]![d2−1]! qk(d1−k) (q−q−1)m (ek ⊗em−k)

Calculation for v = eln ⊗ · · · ⊗ elj+2 ⊗ (F l.τ0) ⊗ elj−1 ⊗ · · · ⊗ el1

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SLIDE 97

Correlation functions with hidden quantum group

  • 4. Details about the quantum group method

Kalle Kytölä — Florence, May 2015

Sketch: asymptotics with subrepresentations

Md ֒ → Mdj+1 ⊗ Mdj, e0 → τ

(d;dj,dj+1)

, d = dj + dj+1 − 1 − 2m τ

(d;dj,dj+1)

∝

k(−1)k [dj−1−k]![dj+1−1−m+k]! [k]![dj−1]![m−k]![d2−1]! qk(d1−k) (q−q−1)m (ek ⊗em−k)

Calculation for v = eln ⊗ · · · ⊗ elj+2 ⊗ (F l.τ0) ⊗ elj−1 ⊗ · · · ⊗ el1 F(x0)[v](x) =

l1 l ln m

x0 x1 xj xj+1 xn . . . . . .

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SLIDE 98

Correlation functions with hidden quantum group

  • 4. Details about the quantum group method

Kalle Kytölä — Florence, May 2015

Sketch: asymptotics with subrepresentations

Md ֒ → Mdj+1 ⊗ Mdj, e0 → τ

(d;dj,dj+1)

, d = dj + dj+1 − 1 − 2m τ

(d;dj,dj+1)

∝

k(−1)k [dj−1−k]![dj+1−1−m+k]! [k]![dj−1]![m−k]![d2−1]! qk(d1−k) (q−q−1)m (ek ⊗em−k)

Calculation for v = eln ⊗ · · · ⊗ elj+2 ⊗ (F l.τ0) ⊗ elj−1 ⊗ · · · ⊗ el1 F(x0)[v](x) =

l1 l ln m

x0 x1 xj xj+1 xn . . . . . .

dominated convergence:

F

(x0) ...,dj ,dj+1,...[v](...)

(xj+1−xj)∆

dj ,dj +1 d

− →

xj,xj+1→ξ F(x0) ...,d,...[v](. . . , ξ, . . .)

where ∆

dj,dj+1 d

=

2(1+d2−d2

j −d2 j+1)+κ(dj+dj+1−d−1)

2κ

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SLIDE 99

Correlation functions with hidden quantum group

  • 4. Details about the quantum group method

Kalle Kytölä — Florence, May 2015

Sketch: anchor point independence

Write ϕ(x0)

l1,...,ln(x) in terms of α(x0) m1,...,mn(x)

ϕ(x0)

l1,...,ln(x) =

l1 l2 ln

x0 x1 x2 xn . . .

α(x0)

m1,...,mn(x) = x0 x1 x2 xn−1 xn . . .

m1 m2 mn

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SLIDE 100

Correlation functions with hidden quantum group

  • 4. Details about the quantum group method

Kalle Kytölä — Florence, May 2015

Sketch: anchor point independence

Write ϕ(x0)

l1,...,ln(x) in terms of α(x0) m1,...,mn(x)

ϕ(x0)

l1,...,ln(x) =

l1 l2 ln

x0 x1 x2 xn . . .

α(x0)

m1,...,mn(x) = x0 x1 x2 xn−1 xn . . .

m1 m2 mn

Highest weight vectors: If E.v = 0, then in F(x0)[v](x), the coefficient of α(x0)

m1,...,mn(x)

vanishes whenever m1 = 0.

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SLIDE 101

Correlation functions with hidden quantum group

  • 4. Details about the quantum group method

Kalle Kytölä — Florence, May 2015

Sketch: anchor point independence

Write ϕ(x0)

l1,...,ln(x) in terms of α(x0) m1,...,mn(x)

ϕ(x0)

l1,...,ln(x) =

l1 l2 ln

x0 x1 x2 xn . . .

α(x0)

m1,...,mn(x) = x0 x1 x2 xn−1 xn . . .

m1 m2 mn

Highest weight vectors: If E.v = 0, then in F(x0)[v](x), the coefficient of α(x0)

m1,...,mn(x)

vanishes whenever m1 = 0. F[v](x) well defined for x ∈ Xn

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SLIDE 102

Correlation functions with hidden quantum group

  • 4. Details about the quantum group method

Kalle Kytölä — Florence, May 2015

Sketch: Stokes thm and highest weight vectors

◮ ⋑l1,....ln the ℓ-dimensional integration surface of ϕ(x0) l1,...,ln ◮ g(w1; w2, . . . , wℓ) single valued, symmetric in last ℓ − 1 vars

slide-103
SLIDE 103

Correlation functions with hidden quantum group

  • 4. Details about the quantum group method

Kalle Kytölä — Florence, May 2015

Sketch: Stokes thm and highest weight vectors

◮ ⋑l1,....ln the ℓ-dimensional integration surface of ϕ(x0) l1,...,ln ◮ g(w1; w2, . . . , wℓ) single valued, symmetric in last ℓ − 1 vars

Stokes formula / integration by parts:

  • ⋑l1,....ln

ℓ

r=1 ∂ ∂wr

  • g(wr; w1, . . . , wr

X , . . . , wℓ) f(x; w)

  • dw1 · · · dwℓ

= n

j=1

  • (q−1 − q)
  • lj

dj − lj

  • q
  • i<j(di−1−2li)

×

  • ⋑...,lj −1,...
  • γ(w1, . . . , wℓ−1) f(x; w1, . . . , wℓ−1)
  • dw1 · · · dwℓ−1
slide-104
SLIDE 104

Correlation functions with hidden quantum group

  • 4. Details about the quantum group method

Kalle Kytölä — Florence, May 2015

Sketch: Stokes thm and highest weight vectors

◮ ⋑l1,....ln the ℓ-dimensional integration surface of ϕ(x0) l1,...,ln ◮ g(w1; w2, . . . , wℓ) single valued, symmetric in last ℓ − 1 vars

Stokes formula / integration by parts:

  • ⋑l1,....ln

ℓ

r=1 ∂ ∂wr

  • g(wr; w1, . . . , wr

X , . . . , wℓ) f(x; w)

  • dw1 · · · dwℓ

= n

j=1

  • (q−1 − q)
  • lj

dj − lj

  • q
  • i<j(di−1−2li)

×

  • ⋑...,lj −1,...
  • γ(w1, . . . , wℓ−1) f(x; w1, . . . , wℓ−1)
  • dw1 · · · dwℓ−1
  • where γ(w1, . . . , wℓ−1)

= n

i=1 |x0−xi|− 4

κ (di−1) ℓ−1

r=1 |x0−wr|

8 κ g(x0; w1, . . . , wℓ−1).

slide-105
SLIDE 105

Correlation functions with hidden quantum group

  • 4. Details about the quantum group method

Kalle Kytölä — Florence, May 2015

Sketch: Stokes thm and highest weight vectors

◮ ⋑l1,....ln the ℓ-dimensional integration surface of ϕ(x0) l1,...,ln ◮ g(w1; w2, . . . , wℓ) single valued, symmetric in last ℓ − 1 vars

Stokes formula / integration by parts:

  • ⋑l1,....ln

ℓ

r=1 ∂ ∂wr

  • g(wr; w1, . . . , wr

X , . . . , wℓ) f(x; w)

  • dw1 · · · dwℓ

= n

j=1

  • (q−1 − q)
  • lj

dj − lj

  • q
  • i<j(di−1−2li)

×

  • ⋑...,lj −1,...
  • γ(w1, . . . , wℓ−1) f(x; w1, . . . , wℓ−1)
  • dw1 · · · dwℓ−1
  • where γ(w1, . . . , wℓ−1)

= n

i=1 |x0−xi|− 4

κ (di−1) ℓ−1

r=1 |x0−wr|

8 κ g(x0; w1, . . . , wℓ−1).

Highest weight vect.: v = Cl1,...,ln (eln ⊗ · · · ⊗ el1) s.t. E.v = 0

Cl1,...,ln

  • ⋑l1,....ln

ℓ

r=1 ∂ ∂wr

  • g(wr; . . .) f(x; w)
  • dw = 0.
slide-106
SLIDE 106

Correlation functions with hidden quantum group

  • 4. Details about the quantum group method

Kalle Kytölä — Florence, May 2015

Sketch: PDEs by Stokes

Benoit & Saint-Aubin differential operators: D(j) =

dj

  • k=1
  • n1,...,nk≥1

n1+...+nk=dj

(κ/4)dj−k (dj − 1)!2 k−1

j=1 (j i=1 ni)(k i=j+1 ni)

×L(j)

−n1 · · · L(j) −nk

where L(j)

p (j = 1, . . . , n and p ∈ Z) are 1st order diff. operators

L(j)

p = − i=j(xi − xj)p

(1 + p) (di−1)(2(di+1)−κ)

2κ

+ (xi − xj) ∂

∂xi

slide-107
SLIDE 107

Correlation functions with hidden quantum group

  • 4. Details about the quantum group method

Kalle Kytölä — Florence, May 2015

Sketch: PDEs by Stokes

Benoit & Saint-Aubin differential operators: D(j) =

dj

  • k=1
  • n1,...,nk≥1

n1+...+nk=dj

(κ/4)dj−k (dj − 1)!2 k−1

j=1 (j i=1 ni)(k i=j+1 ni)

×L(j)

−n1 · · · L(j) −nk

where L(j)

p (j = 1, . . . , n and p ∈ Z) are 1st order diff. operators

L(j)

p = − i=j(xi − xj)p

(1 + p) (di−1)(2(di+1)−κ)

2κ

+ (xi − xj) ∂

∂xi

  • The integrand f(x; w) satisfies
  • D(j)f
  • (x; w) = ℓ

r=1 ∂ ∂wr

  • g(wr; w1, . . . , wr

X , . . . , wℓ) × f(x; w)

  • .
slide-108
SLIDE 108

Correlation functions with hidden quantum group

  • 4. Details about the quantum group method

Kalle Kytölä — Florence, May 2015

Sketch: PDEs by Stokes

Benoit & Saint-Aubin differential operators: D(j) =

dj

  • k=1
  • n1,...,nk≥1

n1+...+nk=dj

(κ/4)dj−k (dj − 1)!2 k−1

j=1 (j i=1 ni)(k i=j+1 ni)

×L(j)

−n1 · · · L(j) −nk

where L(j)

p (j = 1, . . . , n and p ∈ Z) are 1st order diff. operators

L(j)

p = − i=j(xi − xj)p

(1 + p) (di−1)(2(di+1)−κ)

2κ

+ (xi − xj) ∂

∂xi

  • The integrand f(x; w) satisfies
  • D(j)f
  • (x; w) = ℓ

r=1 ∂ ∂wr

  • g(wr; w1, . . . , wr

X , . . . , wℓ) × f(x; w)

  • .

Highest weight vectors: If E.v = 0, Stokes formula gives D(j) F[v](x) = 0.

slide-109
SLIDE 109

Correlation functions with hidden quantum group

  • 4. Details about the quantum group method

Kalle Kytölä — Florence, May 2015

Sketch: Möbius covariance

ϕ(x0)

l1,...,ln(x1, . . . , xn) =

  • ⋑l1,...,ln

f(x1, . . . , xn; w1, . . . , wℓ) dw1 · · · dwℓ Möbius covariance: if ν(x1) < · · · < ν(xn) for ν(z) = az+b

cz+d , want

F[v]

  • ν(x1), . . . , ν(xn)
  • ×

n

  • j=1

ν′(xj)

(di −1)(2(di +1)−κ) 2κ

= F[v](x1, . . . , xn)

◮ translation invariance, z → z + ξ:

ϕ(x0+ξ)

l1,...,ln (x1 + ξ, . . . , xn + ξ) = ϕ(x0) l1,...,ln(x1, . . . , xn)

* make changes of variables w′

r = wr + ξ

◮ homogeneity, z → λz:

ϕ(λx0)

l1,...,ln(λx1, . . . , λxn) = λ∆ ϕ(x0) l1,...,ln(x1, . . . , xn)

* make changes of variables w′

r = λwr

◮ special conformal transformations, z → z 1+az :

* vary a infinitesimally * use a property of the integrand f * apply Stokes formula

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SLIDE 110

Correlation functions with hidden quantum group

  • 5. Chordal SLE boundary visit problem

Kalle Kytölä — Florence, May 2015

  • 5. CHORDAL SLE BOUNDARY VISIT PROBLEM
slide-111
SLIDE 111

Correlation functions with hidden quantum group

  • 5. Chordal SLE boundary visit problem

Kalle Kytölä — Florence, May 2015

Chordal SLE boundary visit amplitude

PH;x,∞

  • SLEκ visits Bε(y1), then Bε(y2), then . . . then Bε(yN)
  • H

x y2 y1 y··· yN

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SLIDE 112

Correlation functions with hidden quantum group

  • 5. Chordal SLE boundary visit problem

Kalle Kytölä — Florence, May 2015

Chordal SLE boundary visit amplitude

PH;x,∞

  • SLEκ visits Bε(y1), then Bε(y2), then . . . then Bε(yN)
  • ∼

εN 8−κ

κ

H x y2 y1 y··· yN

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SLIDE 113

Correlation functions with hidden quantum group

  • 5. Chordal SLE boundary visit problem

Kalle Kytölä — Florence, May 2015

Chordal SLE boundary visit amplitude

PH;x,∞

  • SLEκ visits Bε(y1), then Bε(y2), then . . . then Bε(yN)
  • ∼ const. × εN 8−κ

κ × ζN(x; y1, y2, . . . , yN)

H x y2 y1 y··· yN

slide-114
SLIDE 114

Correlation functions with hidden quantum group

  • 5. Chordal SLE boundary visit problem

Kalle Kytölä — Florence, May 2015

Boundary visit amplitude problem

(COV) ζN(x; y1, . . . , yN) = λNh1,3 × ζN(λx + ξ; λy1 + ξ, . . . , λyN + ξ)

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SLIDE 115

Correlation functions with hidden quantum group

  • 5. Chordal SLE boundary visit problem

Kalle Kytölä — Florence, May 2015

Boundary visit amplitude problem

(COV) ζN(x; y1, . . . , yN) = λNh1,3 × ζN(λx + ξ; λy1 + ξ, . . . , λyN + ξ) (PDE)

  • κ

2 ∂2 ∂x2 + N j=1

  • 2

yj−x ∂ ∂yj + 2 κ−8

κ

(yj−x)2

  • ζN(x; y1, . . . , yN) = 0

* Itô for martingale N

j=1 g′ t(yj)

8−κ κ

× ζN(Xt; gt(y1), . . . , gt(yN))

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SLIDE 116

Correlation functions with hidden quantum group

  • 5. Chordal SLE boundary visit problem

Kalle Kytölä — Florence, May 2015

Boundary visit amplitude problem

(COV) ζN(x; y1, . . . , yN) = λNh1,3 × ζN(λx + ξ; λy1 + ξ, . . . , λyN + ξ) (PDE)

  • κ

2 ∂2 ∂x2 + N j=1

  • 2

yj−x ∂ ∂yj + 2 κ−8

κ

(yj−x)2

  • ζN(x; y1, . . . , yN) = 0

* Itô for martingale N

j=1 g′ t(yj)

8−κ κ

× ζN(Xt; gt(y1), . . . , gt(yN))

(ASY) As yj → x, asymptotics are |yj − x|

8−κ κ × ζN(. . .) →

  • ζN−1(x; y2, . . . , yN)

if j = 1

  • therwise .
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SLIDE 117

Correlation functions with hidden quantum group

  • 5. Chordal SLE boundary visit problem

Kalle Kytölä — Florence, May 2015

Boundary visit amplitude problem

(COV) ζN(x; y1, . . . , yN) = λNh1,3 × ζN(λx + ξ; λy1 + ξ, . . . , λyN + ξ) (PDE)

  • κ

2 ∂2 ∂x2 + N j=1

  • 2

yj−x ∂ ∂yj + 2 κ−8

κ

(yj−x)2

  • ζN(x; y1, . . . , yN) = 0

* Itô for martingale N

j=1 g′ t(yj)

8−κ κ

× ζN(Xt; gt(y1), . . . , gt(yN))

(ASY) As yj → x, asymptotics are |yj − x|

8−κ κ × ζN(. . .) →

  • ζN−1(x; y2, . . . , yN)

if j = 1

  • therwise .

(ASY) As yj, yk → y, asymptotics are |yj − yk|

8−κ κ × ζN(. . .) →

  • ζN−1(. . . , y, . . .)

if |j − k| = 1

  • therwise

.

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SLIDE 118

Correlation functions with hidden quantum group

  • 5. Chordal SLE boundary visit problem

Kalle Kytölä — Florence, May 2015

Boundary visit amplitude problem

(COV) ζN(x; y1, . . . , yN) = λNh1,3 × ζN(λx + ξ; λy1 + ξ, . . . , λyN + ξ) (PDE)

  • κ

2 ∂2 ∂x2 + N j=1

  • 2

yj−x ∂ ∂yj + 2 κ−8

κ

(yj−x)2

  • ζN(x; y1, . . . , yN) = 0

* Itô for martingale N

j=1 g′ t(yj)

8−κ κ

× ζN(Xt; gt(y1), . . . , gt(yN))

(PDE) moreover N third order linear homogeneous PDEs for ζN (ASY) As yj → x, asymptotics are |yj − x|

8−κ κ × ζN(. . .) →

  • ζN−1(x; y2, . . . , yN)

if j = 1

  • therwise .

(ASY) As yj, yk → y, asymptotics are |yj − yk|

8−κ κ × ζN(. . .) →

  • ζN−1(. . . , y, . . .)

if |j − k| = 1

  • therwise

.

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SLIDE 119

Correlation functions with hidden quantum group

  • 5. Chordal SLE boundary visit problem

Kalle Kytölä — Florence, May 2015

Translation of the bdry visit amplitude problem

L points on the left and R on the right, L + R = N, visit order ω: vω ∈ M⊗R

3

⊗ M2 ⊗ M⊗L

3

H x y2 y1 y··· yN

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SLIDE 120

Correlation functions with hidden quantum group

  • 5. Chordal SLE boundary visit problem

Kalle Kytölä — Florence, May 2015

Translation of the bdry visit amplitude problem

L points on the left and R on the right, L + R = N, visit order ω: vω ∈ M⊗R

3

⊗ M2 ⊗ M⊗L

3

(K − q).vω = 0, E.vω = 0

H x y2 y1 y··· yN

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SLIDE 121

Correlation functions with hidden quantum group

  • 5. Chordal SLE boundary visit problem

Kalle Kytölä — Florence, May 2015

Translation of the bdry visit amplitude problem

L points on the left and R on the right, L + R = N, visit order ω: vω ∈ M⊗R

3

⊗ M2 ⊗ M⊗L

3

(K − q).vω = 0, E.vω = 0 ˆ π(1)

(pos)(vω) = 0,

ˆ π(3)

j

(vω) =

  • vω\(pos)

if consecutive if not consecutive ˆ π(2)

(middle)(vω) =

  • vω\(1:st visit)

if first visit if not first visit

H x y2 y1 y··· yN

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SLIDE 122

Correlation functions with hidden quantum group

  • 5. Chordal SLE boundary visit problem

Kalle Kytölä — Florence, May 2015

Translation of the bdry visit amplitude problem

L points on the left and R on the right, L + R = N, visit order ω: vω ∈ M⊗R

3

⊗ M2 ⊗ M⊗L

3

(K − q).vω = 0, E.vω = 0 ˆ π(1)

(pos)(vω) = 0,

ˆ π(3)

j

(vω) =

  • vω\(pos)

if consecutive if not consecutive ˆ π(2)

(middle)(vω) =

  • vω\(1:st visit)

if first visit if not first visit

H x y2 y1 y··· yN

Thm (Jokela & Järvinen & K., K. & Peltola)

If vω satisfies this, then the function ζω = F[vω] satisfies the PDEs and asymptotics for the zig-zag problem. Solutions exist and are unique.

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SLIDE 123

Correlation functions with hidden quantum group

  • 5. Chordal SLE boundary visit problem

Kalle Kytölä — Florence, May 2015

Boundary visits of interfaces in lattice models

LERW − → chordal SLEκ=2 Percolation − → chordal SLEκ=6 Q-FK model

?

− → chordal SLEκ=κ(Q) as lattice mesh δ ց 0

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SLIDE 124

Correlation functions with hidden quantum group

  • 5. Chordal SLE boundary visit problem

Kalle Kytölä — Florence, May 2015

Boundary visits of interfaces in lattice models

LERW − → chordal SLEκ=2 Percolation − → chordal SLEκ=6 Q-FK model

?

− → chordal SLEκ=κ(Q) as lattice mesh δ ց 0

◮ sample configuration and find the curve (interface)

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SLIDE 125

Correlation functions with hidden quantum group

  • 5. Chordal SLE boundary visit problem

Kalle Kytölä — Florence, May 2015

Boundary visits of interfaces in lattice models

LERW − → chordal SLEκ=2 Percolation − → chordal SLEκ=6 Q-FK model

?

− → chordal SLEκ=κ(Q) as lattice mesh δ ց 0

◮ sample configuration and find the curve (interface) ◮ collect frequencies of boundary visits from the samples

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SLIDE 126

Correlation functions with hidden quantum group

  • 5. Chordal SLE boundary visit problem

Kalle Kytölä — Florence, May 2015

Boundary visits of interfaces in lattice models

LERW − → chordal SLEκ=2 Percolation − → chordal SLEκ=6 Q-FK model

?

− → chordal SLEκ=κ(Q) as lattice mesh δ ց 0

◮ sample configuration and find the curve (interface) ◮ collect frequencies of boundary visits from the samples ◮ P[γ visits x1, . . . , xN] ≈ const.× j(δ f ′(xj))

8−κ κ ζN(f(x1), . . .),

where f = conformal map to (H; 0, ∞)

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SLIDE 127

Correlation functions with hidden quantum group

  • 5. Chordal SLE boundary visit problem

Kalle Kytölä — Florence, May 2015

Lattice model simulations vs. solutions

N = 1, one-point visit frequencies, log-log-scale ζ1(x; y1) ∝ |y1 − x|

κ−8 κ (set x = 0)

0.1 10 1000 logy1

blue: percolation red: Q = 2 FK model green: Q = 3 FK model

0.01 0.1 1 10 100 logy1

magenta: LERW

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SLIDE 128

Correlation functions with hidden quantum group

  • 5. Chordal SLE boundary visit problem

Kalle Kytölä — Florence, May 2015

Lattice model simulations vs. solutions

N = 2, two-point visit frequencies, log-scale the 4 pieces of ζ2(x; y1, y2) are hypergeometric functions

(set x = 0, y1 = 1)

5 5 y2 0.02 0.05 0.1 0.2 0.5 1. 2.

blue: percolation red: Q = 2 FK model green: Q = 3 FK model

5 5 y2 0.001 0.1 10 1000

magenta: LERW

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SLIDE 129

Correlation functions with hidden quantum group

  • 5. Chordal SLE boundary visit problem

Kalle Kytölä — Florence, May 2015

Lattice model simulations vs. solutions

N = 3, three-point visit frequencies, log-scale solving for the 8 pieces of ζ3(x; y1, y2, y3) not reducible to ODE percolation

2 1 1 2 3 y2 0.01 0.05 0.1 0.5 1. 5. 2 1 1 2 3 y2 0.1 0.2 0.3 0.15

(set x = 0, y1 = 1, y3 = 2) (set x = 0, y1 = 1, y3 = −1)

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SLIDE 130

Correlation functions with hidden quantum group

  • 5. Chordal SLE boundary visit problem

Kalle Kytölä — Florence, May 2015

Lattice model simulations vs. solutions

N = 3, three-point visit frequencies, log-scale solving for the 8 pieces of ζ3(x; y1, y2, y3) not reducible to ODE Q = 3 FK model

2 1 1 2 3 y2 0.01 0.1 1 10

(set x = 0, y1 = 1, y3 = 2)

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SLIDE 131

Correlation functions with hidden quantum group

  • 5. Chordal SLE boundary visit problem

Kalle Kytölä — Florence, May 2015

Lattice model simulations vs. solutions

N = 4, four-point visit frequencies, log-scale

solving for the 16 pieces of ζ4(x; y1, y2, y3, y4) not reducible to ODE

percolation

  • 2

2 4 y 3 0.005 0.01 0.02 0.05

  • 0.5

1.0 1.5 2.0 2.5 3.0 y 3 0.5 0.3

(set x = 0, y1 = 1, y2 = −1, y4 = 2) (set x = 0, y1 = −1, y2 = 1, y4 = 2)

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SLIDE 132

Correlation functions with hidden quantum group The end Kalle Kytölä — Florence, May 2015

THANK YOU!