graphical representations of some statistical...
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Graphical Representations of Some
Statistical Models
Youjin Deng
New York University
– Introduction to critical phenomena
– High-temperature Graph
– Low-temperature Graph
– Fortuin-Kasteleyn Representation
– Baxter-Kelland-Wu Representation and
Stochastic Lowner Evolution
Cooperate with
Henk W.J. Blote (Leiden Univ., Delft Univ. of Tech.)
Alan D. Sokal (New York Univ.)
Wenan Guo (Beijing Normal. Univ.)
Timothy M. Garoni (New York Univ.)
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Introduction
• Examples
P cTc
ice
water
273K 647K
218atm ,( )
T
P
vapor
0
h
T
Tc
1044K
first−order
critical
The liquid-gas critical The ferromagnetic
point of H2O: Tc = 647K, point of Fe: hc = 0,
pc = 218 atm. Tc = 1044K.
• Statistical Models
– Ising model on Graph (V, E).
1, Configuration Space �s ∈ {+1,−1}|V |.
2, A priori measure 12(δs,+1 + δs,−1).
3, Hamiltonian
H/kBT = −K∑
〈i,j〉sisj − H
∑
k
sk
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– Partition function Z =∑
e−H/kbT
or free energy F = − lnZ
– Physical quantities as derivatives of F .
First derivative: ρH2O, m
Second derivative: C, χ
� nth-order transition:
nth derivative of F is singular,
but (n − 1)th derivative is analytic.
0
m
h
m
h0
First-order Second-order
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• Critical phenomena
Specific-heat : C ∝ |T − Tc|−α
Susceptibility : χ ∝ |T − Tc|−γ
Magnetization: m ∝ |T − Tc|β
At Tc, correlation: g(r) ∝ r−2X
� Universality: critical exponents, α, γ, · · ·,
take same values in different systems.
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Four-state Potts at criticality; the Hamiltonian reads
H/kBT = −J∑〈i,j〉
δσi,σj
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High-temperature expansion of thezero-field Ising model
Parition function:
Z =∑
{±1}|V |e−H
=∑
{±1}|V |
∏
〈i,j〉eKsisj (1)
Identity
eKsisj = coshK + sisj sinhK
= coshK(1 + vsisj) (v = tanhK)
(2)
Insert (2) into (1), and expand the product∏. Then, label each term in the expansion by
a graph Λ:
• If factor 1 is taken, do nothing;
• If vsisj, put a bond 〈i, j〉.
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A closed graph An non-closed graph
Parition function Z reads
Z = (coshK)|E| ∑
G∈G0
vb(G) ∑
{±1}|V |
⎛⎝∏
i
smi(G)i
⎞⎠
– mi(G): number of bonds incident on vertex i.
�: Any graph with at least one odd number
mi contributes to Z by zero.
Z becomes
Z = 2|V |(coshK)|E| ∑
G∈G0 ,∂G=0
vb(G).
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� High-T graph of 〈sisj〉: Almost closed graphs
with ∂G = (i, j).
� High-T graph of the Potts model: Flow
polynomial
Application:
• 1D Ising model: Z = 2L(coshK)L (free)
Z = 2L(coshK)L[1+(tanhK)L] (periodic)
• Exact solution of the 2D Ising model
• Worm Algorithm (has considerable appli-
cations in quantum systems).
• O(n) loop model and its cluster simula-
tions.
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Low-temperature expansion of thezero-field Ising model
Dual Lattice: G → G∗
G: solid and black lines G∗: dashed and blue lines
Partition function Z:
Z =∑
{±1}|V |e−H
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1) All spins DOWN: e|E|K
2) One spin UP: e|E|K ∗ e−8K = e|E|K ∗ u4
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2) Two spins UP:
e|E|K ∗ u6 e|E|K ∗ u8
...
Z becomes
Z = e|E|K ∑
G∗∈G∗0 ,∂G∗=0
ub(G∗).
Duality Relation:
Z(G; v) = constant × Z(G∗;u) if
v = u. Namely, tanhK = e−2K∗
Criticality on Square Lattice:2Kc = ln(1 +√2).
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Fortuin-Kasteleyn Representation ofthe zero-field Ising model
Hamiltonian H
H/ = −K∑
〈i,j〉sisj = −2K
∑
〈i,j〉δsi,sj + constant
Identity:
e2Kδsi,sj = 1 + μδsi,sj (μ = e2K − 1)
Partition function Z:
Z =∑
{±1}|V |e−H
=∑
{±1}|V |
∏
〈i,j〉(1 + μδsi,sj) (3)
Expand the product∏, label each term by
the graph: if the factor 1 is taken, do nothing;
if μδsi,sj is taken, put a bond on 〈i, j〉.
δsi,sj ⇒ Each component is of a unique color
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Z becomes
Z =∑
G∈G0
μb(G)2k(G)
–b: bond number, k: component number
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Random-cluster model:
Z(q, μ) =∑
G∈G0
μb(G)qk(G)
Some special cases:
• q → 0: spanning tree or spanning forests
• μ = −1: Chromatic polynomial (such as
The Four Color Prolem)
Monte Carlo Methods:
• Sweeny algorithm
• Swendsen-Wang-Chayes-Machta Algorithm
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Baxter-Kelland-Wu (BKW) mappingof the random-cluster model
Medial Graph : G → G
G–Water: blue area; Island: white
Operation:
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Random-Cluster (RC) Model:
Z(q, μ) =∑
G∈G0
μb(G)(√
q)c(G)
– c(G): loop number
Application: Map the RC model onto the
8-vertex model.
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Recent Development: Stochastic Lowner Evo-
lution (SLE), by Oded Schramm (1999).
In the scaling limit, critical systems are not
only scale invariant, but also conformally in-
variant.
In the scaling limit, BKW curves with appro-
priate boundary conditions are described by
SELκ in Z+:
∂gt(z)
∂t=
2
gt(z) −√
κBt, g0(z) = z ,
— Bt, t ∈ [0,∞): Brownian motion.
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