Limit Topology for Dimer Algebra Random...
Transcript of Limit Topology for Dimer Algebra Random...
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Limit Topology for Dimer AlgebraRandom R-Trees
(Based on Joint Discussion with Nicolai Reshetikhin)Matthew Bernard
May 2018
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Abstract
We prove inverse partition correlation of Z projective limit topology by KasteleynPfaffian Pf(ΣK)∈Quot(K[D]) Grassmann field framework of bipartite (Z+)d dimer‘‘tree-like object” equipped with dual lattice height. We obtain the asymptoticscritical points of real and complex discriminants for the Grassmann-integral FermionKernel of special operators, and formulate conjecture for correlation function on largedeviation functional of height function fluctuation φ, in asymptotic behavior of largerandom spanning tree and tree-valued measure.
Keywords: Random-Dimer, Kasteleyn, Grassmann, Fermionic-Correlation
Grateful to all the conference organizers/participants
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1 Characterizations
1.1 Basic definitions and observations
Graph Γ:={i∈N |1≤ i≤n}⊂ (Z+)d |d=2,3,. . . in topological surfaceMg
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is partition [σ] if, ∀ ℓ :={i= i1· · · id, j=j1· · · jd}∈D, ∃ perfect matching:• Dimers (ℓ)ℓ∈D do not overlap, ∀D:= set of all configurations D of Γ• All vertices (i)i∈Γ :=∂D are covered: |∂D|=2
∑ℓ σD(ℓ), σD :={0, 1}
∀ |[σ]|≤|D| partitions [σ].Remark. Γ:= closed, connected; n := even; dimer is undefined ∀ i=j loop.
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Symmetric difference D∆D′ :=D∪D′\D∩D′ ⊇ (Cα | α∈N, 1≤α≤m).
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One or more circuit-coverings ⇐= transition cycles := superposition cycles:
• Disjoint even-length (simple closed curves) cycles Cα | α∈N, 1≤α≤m.• Alternating sequence of configuration-pair by ordered cyclic sequence:
elementary circuit := (ℓ1, ℓ2, . . . , ℓr) of (ℓ1, ℓ3, . . . , ℓr−3, ℓr−1) ∈D, resp.(ℓ2, ℓ4, . . . , ℓr−2, ℓr) ∈D′, for sequence i1, ℓ1, i2, ℓ2, . . . , ℓr joining vertexi1 after finite edges, starting at i1. Another elementary circuit starting atvertex ir+1 at end of edge ℓr+1∈D for Γ of vertices > r, etc.
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Proposition. Aut(D) :=subset of automorphism group Aut(D) is given by|Aut(D)|=
(n2
)! 2
n2 , where |Aut(D)|≤|Aut(Γ)|=n! .
Proof. Preserving contiguity (adjacency+distribution), ∀ [σ]∼σ∈Aut(D) :
[σ] :Γ→Γ{ σ(1, . . . , n) = (σ(1),σ(2), . . . ,σ(n−1),σ(n))
s.t. σ(2ℓ−1)=ℓ, ℓ
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The local observables :=D-D correlation functions (conditional probabilities)⟨ ∏ℓ∈D
σD(ℓ)⟩
def== Prob(ℓ1 ∈ D, . . . , ℓn ∈ D) := E
[ n∏i=1
σD(ℓi)]
=∑D∈D
n∏i=1
σD(ℓi)× Prob(D) =
∑D∈D
∏ℓ∈D
εℓ ωℓn∏
i=1
σD(ℓi)∑D∈D
∏ℓ∈D
εℓ ωℓ
where εℓ∈{−1,+1}, and σD : Γ→ {0, 1} by
σD(ℓ) :={1 if ℓ ∈ D0 if ℓ ̸∈ D
∀D ∈ D
is the indicator function of ℓ on the support D for all dimer weights ω(·)and,
the numerator =∑
D∋ ℓ1, ..., ℓn
εD ωD∣∣∣ εD∈{−1,+1} ∼ ∏
ℓ∈D
εℓ .
Remark. If any ℓξ ̸= ℓη | ξ,η ∈ {1, . . . , n} have a common vertex, then⟨σD(ℓ1) · · ·σD(ℓn)⟩=0; more so, σD(ℓ)σD(ℓ)=σD(ℓ); so, ℓξ ̸=ℓη, ∀ ξ ̸=η.
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That is, in the σ-finite energy Ξ(·) measure δ(·) ∈ Ξ :E(Γ)→R, ℓ 7→ Ξℓ,probability exists by⟨∏
ℓ∈D
σD(ℓ)⟩:=
∏ℓ∈D
εℓ ωℓ∑D∈D
∏ℓ∈D
εℓ ωℓ=
εD ωDZ(Γ;ω)
=:Prob(D)∣∣∣∣ωℓ=e− Ξℓk T , εD∼∏
ℓ∈D
εℓ
=1
Zexp(−ΞDk T
) ∣∣∣ ΞD=∑ℓ∈D
Ξℓ , Z=∑D∈D
exp(−ΞDk T
)where Z := strict-sense positive continuous partition function on the objects:
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• Domains in hexagonal lattice.
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• Domains in square lattice.
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The bipartite graph (of no black-black nor white-white edge) is disjoint unionV (Γ) = V•(Γ) ⊔ V◦(Γ).
Instance.
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Non-instance. 1 2 3
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(no bipartite structureon triangular lattices
).
Remark.D := 1-chain in cell complex of GF2 field: (ℓ)ℓ∈D∈H1
(Mg;GF2
)∼=(Z2)2g.∂D := 0-chain in cell complex of GF2 field: (i)i∈Γ∈H0
(Mg;GF2
)∼=Z2.8
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1.2 Combinatorial equivalence
Proposition. Two families, respectively, on space of dimers and space oftilings, are combinatorially equivalent; that is, ∃ bijective correspondence
Dimers ←→ T ilings.Proof. Let all (v)v ∈Γ⊂Mg := closed traversal (walk); Γ finite, connected,and planar (non-intersecting edges). The set of all Γ spanning trees defines:I. The 2D cell complex Γ:
vertices, edges, faces := 0-cells, 1-cells, 2-cells, resp.1
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Disjoint interiors.
dim(∂Γn) = (n−1) /GF (2)∂Γn := boundary between
two n-cells.
Remark. Γ⊂Mg := embedding by 1-skeleton CW-complex (resp. cellulardecomposition of an oriented closed connected surface).
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II. The dual cell complex Γ∗:0-cells, 1-cells, 2-cells := ‘‘centers” of 2-cells, 1-cells, 0-cells of Γ, resp.
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Γ∗ := dualcell complexto Γ.
III. Dimer Unique pair of 2-cells on Γ∗on Γ: share a dimer:
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IV. Therefore, the global bijection:
(Dimers on Γ) ←→(
Tilings of Γ∗ byunique pair of 2-cells
).
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In particular, on the dual cell complex of bipartite graph:1
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(two ‘‘colorings” of the same tile).
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Limit topology := stacked3D boxes; projection of2D rhombus tiling:
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1.3 Space of height functions
Definition. Let Γ⊂Mg := planar, bipartite, hexagonal, such thatDimers ←−[ Discrete Surfaces.
The space of height functions is the partially ordered by overlap, given by:
HΓdef== {π : faces (Γ)→ Z}
π+ 13
π+ 239 5
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16 14π
π+ 13 π+23
π+ 139 5
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16 14π
π + 23
with respect to normalization π(f0)=0 of the reference face f0.
Proposition.(i). Let ∂Γ:= boundary faces. Then, πD
∣∣∂Γ
does not depend on D.(ii). πD1D2 = πD1−πD2.
Proof. ♡.
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Theorem. The dimer configuration probability is given by
Prob(D)= 1Z
∏f
qπD(f)f
∣∣∣∣Z=∑π ∈HΓ
∏f
qπ(f)f , qf =
∏ℓ∈ ∂f
ωεℓℓ , π∈HΓ, D∼=π.
Proof. By the combinatorial equivalence,{Dimers on Γ
}∼=
bijection
{height functions
}.
The bijection then gives the measure. □
Remark. Prob(D) :=“gauge” invariant: ω(ℓ) 7−→s(ℓ+)ω(ℓ) s(ℓ−). More so,qf := invariant (‘‘essential” parameters).
Particular cases.
(i).4
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a
bc
acb
=⇒ the uniform distribution:q = a−1b c−1a b−1c = 1.
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(ii). qf :=qt,Prob(π) ∝
∏tqπ(t)t .1
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Remark.0
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1.4 Kasteleyn orientation and matrix
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+ −
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+−
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−−
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++
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−−
−+ .
Definition. Let Γ⊂ (Z+)d :=CW-complex 1-skeleton (resp. genus g compactorientable surface cell-decomposition) closed connected embedding, endowedwith induced (counterclockwise) orientation ε(∂F) on every face boundary;Γ is Kasteleyn if ∀{i, j} edges ℓ, i ̸=j, arbitrary i-to-j orientations εK(iℓjℓ),∏
ℓ∈ ∂F
εKiℓjℓ = −1, ∀F ∈Γ∣∣ εKiℓjℓ :={−1 if ε(∂F) ∈ counter-εK(iℓjℓ)+1 if ε(∂F) ∈ εK(iℓjℓ).
Let Γ:=Kasteleyn, weighted ω>0, ∀{i, j} edges ℓ, i ̸=j,ΣKij =
∑ℓ
εKiℓjℓ ωiℓjℓ∣∣ ΣKij :=0 for i, j not adjoined and i=j loop.
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Proposition (existence). Kasteleyn orientation exists.Proof. ♡.Construction (bipartite case). Following Γ∗ spanning tree:
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−−− +++
−−−
+++
−−− +++
+++ −−−−−−+++
+++−−−
−−− +++
−−−+++
−−−+++ −−−
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−−−
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−−−+++
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−−− +++
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−−− +++
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−−−+++
−−− +++
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−−−+++
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+++ −−−−−−
−−− +++
+++−−−−−−+++
−−− +++
−−−
+++
−−−+++ −−−
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+++−−−
+++−−−
−−−+++
+++ −−−
−−− +++
−−−+++
+++−−−
−−− +++
−−−+++
−−−+++
+++
−−−
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−−−
−−−
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+++−−−
+++
−−−
Reduce Γ to small N×N→ exp(αN 2); choose ε(∂F); follow Γ∗ spanningtree, rooted outside Γ, from root; make εK(F) with each Γ∗ edge deletion.
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Definition. Two orientations are equivalent, if each is obtainable from theother through a sequence of orientation-reversing maps:
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−→148
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−−−
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Theorem. All Kasteleyn orientations of Γ planar are equivalent.Proof. Induction, ∀n
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Remark. Deleting a vertex changes orientation to non-Kasteleyn at ‘‘holes”:
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A
B
C
≡
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−1
−1
hAhBhC=(−1)3=−1
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That is, hAhBhC=−hABC.
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Remark. To convert the non-Kasteleyn orientation back to Kasteleyn:
hAhBhC=1.
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Derivation. Given Γ⊂ (Z+)d := cycle-graph in plane, ∀ i, j=1, . . . , n
ΣKij =
{−ωij = ωji for ε(∂F) ∈ counter-εKij , ∀ i i j j, i i j jωij = −ωji for ε(∂F) ∈ counter-εKij , ∀ i i j j, i i j j0 for i, j not adjoined and loop i=j.
Theorem (Kasteleyn). Every ±|detΣK|12 =: Pf(ΣK) ∈ Quot(K[D]),Γ ⊂ (Z+)d, non-vanishing monomial corresponds 1-1 to [σ], ∀D on Nνedges of class Cν | ν = 1, . . . , d, with respect to sgn(σ) := (−1)t(σ) wheret(σ) :=(σ(i∈N) |1≤ i≤n)−→ (i∈N |1≤ i≤n), ∀σ∈Aut(D), such that:
(i) |Pf(ΣK)|=∑D∈D
∏ℓ∈D
εℓ ωℓ∣∣∣Pf(ΣK)=∑
σ= [σ]
(−1)t(σ)n2∏
ℓ=1
ΣKσ(2ℓ−1)σ(2ℓ).
(ii) Pf(ΣK)= 12n2
1(n2
)!
∑σ
(−1)t(σ)n2∏
ℓ=1
ΣKσ(2ℓ−1)σ(2ℓ) =:∑
N1, ..., Nd
∑D
d∏ν=1
(±)ωNνν .
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Proof. (ii) follows from (i) by Aut(D).
To see (i): detΣK = det(−(ΣK)T) = det(−ΣK) = (−1)n detΣK implies
detΣK:= strictly positive-definite square of a rational function of array ΣKij ,∀ i, j=1, . . . , n, n := even, on the Leibniz’s second-index permutations:
∑σ∈Sn
n∏i=1
(−1)t(σ)ΣKiσ(i)=n!/2∑σ=1
n∏
i=1
σ(i) := given byeven permutations
σ∈Sn
(−1)t(σ)ΣKiσ(i) +n∏
i=1
σ(i) := given byodd permutations
σ∈Sn
(−1)t(σ)ΣKiσ(i)
where
σ : {(i∈N | 1≤ i≤n)} −→ {(σ(i∈N) | 1≤ i≤n)}
and,t(σ) := (σ(i∈N) | 1≤ i≤n) −→ (i∈N | 1≤ i≤n).
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In particular, skew-symmetry ΣKiσ(i)=−ΣKσ(i) i | i≥σ(i) implies non-vanishingmonomials given by:
n!/(n
2
)! 2
n2∑
σ=1
(−1)t(σ)n2∏
i=1
ΣKτ1(2i−1)τ1(2i)
n2∏
i=1
ΣKτ2(2i)τ2(2i−1)
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
∃ΣKτ1(2i−1)τ1(2i)=−ΣKτ2(2i)τ2(2i−1),∀ΣKτ2(2i)τ2(2i−1);
t(σ)=even for n2 :=even,t(σ)=odd for n2 :=odd
+
2
(n!/(n2
)! 2
n2
2
)∑σ=1
(−1)t(σ)n∏
i=1
ΣKiσ(i)
∣∣∣∣∣∣t(σ)=odd forn2 :=even,
t(σ)=even for n2 :=odd.
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That is,
n!/(n
2
)! 2
n2∑
σ=1
(−1)t(σ)+n2
( n2∏i=1
ΣKτ1(2i−1)τ1(2i)
)2 ∣∣∣∣∣∣t(σ)=even forn2 :=even,
t(σ)=odd for n2 :=odd
+
2
(n!/(n2
)! 2
n2
2
)∑σ=1
(−1)t(σ)n∏
i=1
ΣKiσ(i)
∣∣∣∣∣∣t(σ)=odd forn2 :=even,
t(σ)=even for n2 :=odd.
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Furthermore, the non-vanishing monomials are given by
n!/(n
2
)! 2
n2∑
σ̃=[σ̃]=1
(−1)t(σ)+n2+2×t(σ̃)
( n2∏ℓ=1
ΣKσ̃(2ℓ−1)σ̃(2ℓ)
)2 ∣∣∣∣∣∣t(σ)=even forn2 :=even,
t(σ)=odd for n2 :=odd
+
2 ×
(n!/(n2
)! 2
n2
2
)∑
{σ̃= [σ̃] ̸= τ̃= [̃τ]}=1
(−1)t(σ̃)n2∏
ℓ=1
ΣKσ̃(2ℓ−1)σ̃(2ℓ) × (−1)t(̃τ)
n2∏
ℓ=1
ΣKτ̃(2ℓ−1)̃τ(2ℓ)
∀ [σ] :=Sn/(Sn
2×S
n22
).
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That is,
n!/(n
2
)! 2
n2∑
σ̃=[σ̃]=1
(−1)2×t(σ̃)( n2∏
ℓ=1
ΣKσ̃(2ℓ−1)σ̃(2ℓ)
)2 ∣∣∣∣∣∣t(σ̃) :=(σ̃(i∈N) |1≤ i≤n)−→ (i∈N |1≤ i≤n)+
2 ×
(n!/(n2
)! 2
n2
2
)∑
{σ̃= [σ̃] ̸= τ̃= [̃τ]}=1
(−1)t(σ̃)+t(̃τ)n2∏
ℓ=1
ΣKσ̃(2ℓ−1)σ̃(2ℓ)
n2∏
ℓ=1
ΣKτ̃(2ℓ−1)̃τ(2ℓ)
∀ [σ] :=Sn/(Sn
2×S
n22
)=
( ∑σ= [σ]
(−1)t(σ)n2∏
ℓ=1
ΣKσ(2ℓ−1)σ(2ℓ)
)2 ∣∣∣∣∣∣t(σ) :=(σ(i∈N) |1≤ i≤n)−→ (i∈N |1≤ i≤n)∀ [σ] :=Sn
/(Sn
2×S
n22
). □
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0 := non-adjoined i, j1,1,1 := adjoined i, j.
To see |Pf(ΣK)|=Z, write on the non-vanishings, by partitions σ=[σ], ∀D :
Z =∑
σ= [σ]
∑D(σ)
εKσ(1)σ(2) ωσ(1)σ(2) · · · εKσ(n−1)σ(n) ωσ(n−1)σ(n) =
=∑
σ= [σ]
∑D(σ)
n2∏
ℓ=1
εKσ(2ℓ−1)σ(2ℓ) ωσ(2ℓ−1)σ(2ℓ).
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0 := non-adjoined i, j1,1,1 := adjoined i, j
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.
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But the derived Pfaffian is given by
Pf(ΣK)=
∑
σ ∈ Sn/(Sn
2×S
n22
)
=
[σ]
(−1)t(σ)n2∏
ℓ=1
ΣKσ(2ℓ−1)σ(2ℓ)︸ ︷︷ ︸depends only on σ
=∑
σ= [σ]
depends only on σ︷ ︸︸ ︷∑D(σ)
(−1)t(σ)n2∏
ℓ=1
εKσ(2ℓ−1)σ(2ℓ)
n2∏
ℓ=1
ωσ(2ℓ−1)σ(2ℓ).
Therefore, write
Pf(ΣK) =∑D(σ)
(−1)t(σ)n2∏
ℓ=1
εKσ(2ℓ−1)σ(2ℓ)
n2∏
ℓ=1
ωσ(2ℓ−1)σ(2ℓ) = (sgn) · Z=
(−1)t(σ) .
where the first equality is well-defined by all partitions [σ], ∀ ℓ∈D. □
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Corollary. ⟨ k∏i, j=1
σ(i)σ(j)⟩
= Pf((ΣK)−1ξη
) ∣∣∣ ξ=1, . . . , kη=1, . . . , k.
Proof. ♡.
Remark. Pf(ΣK) := partition function, (ΣK)−1 := correlation functions.
Remark. Kasteleyn theorem allows polynomial-time; skew-symmetric Gausselimination algorithm, by Pf(BABT )=det(B)Pf(A), does O(m3) time forPfaffian of size 2m matrix.
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1.5 Grassmann integral
Definition. Let (v1, . . . , vn) ∈ V := vector space, where∧nV := exterior
algebra of V is defined by:(i) Elements:
n∧i=1
vi := v1∧· · ·∧vn =
=1
n!
∑σ∈Sn
(−1)t(σ)n⊗
i=1
vσ(i) :=1
n!
∑σ∈Sn
(−1)t(σ)vσ(1)⊗ · · · ⊗ vσ(n);
(ii) Multiplication:( n∧i=1
vi
)∧( n∧i=1
wi
):= (v1∧· · ·∧vn) ∧ (w1∧· · ·∧wn) == v1∧· · ·∧vn ∧ w1∧· · ·∧wn.
Then, with chosen basis a=(a1, . . . , an)∈V, the exterior algebra∧nV := {ai | aiaj+ajai = 0}is a Grassmann algebra.Proposition. The Grassmann algebra
∧nV is generated by V.Proof. ♡.
32
-
Definition. The Grassmann integral on∧nV with respect to x is given by∫
∧nV f = fx , f = fx x + · · ·︸︷︷︸lowerorder termswhere x∈
∧nV ⊂R is an orientation.Lemma. x=
∏ni=1 ai if (a1, . . . , an) is a basis in V.
Proposition. The formal Grassmann constraints are given by the integral:∫n∏
i=1
ai
n∏i=1
dai = (−1)n(n−1)
2
∫n∏
i=1
ai dai = (−1)n(n−1)
2 .
where:(i).
∫n∏
ν=1
aiν da ={0 , k
-
Theorem. Let a=(ai)∈V be basis, where A∗=Aa=exp(12
∑ij aiAijaj
)satisfies the Grassmann constraints, uniquely generalizing A∗ for allKasteleyn matrices A satisfying the Grassmann constraints. Then:
(i) Pf(A) =∫∧nV
exp(1
2
∑ij
aiAijaj
)da.
(ii) Pf(
0 A−At 0
)= det(A).
(iii) (Pf(A))2 = det(A).
(iv) ∂∂Ai1j1
· · · ∂∂Aikjk
Pf(A) =
= Pf(A) · Pf((ΣK)−1ab )∣∣∣ a = i1, . . . , ikb = j1, . . . , jk .
34
-
Proof.(i). Write:∫∧nV exp
(1
2⟨a,Aa⟩
)da =
1(n2
)!
1
2n2
∫∧nV ⟨a,Aa⟩
n2 da
such that∫⟨a,Aa⟩
n2 da =
∫ai1aj1 · · · ain
2ajn
2Ai1 j1 · · ·Ain
2jn2
= (−1)σ Ai1 j1 · · ·Ain2jn2
σ :((i1, j1) , . . . ,
(in2, jn
2
))→ (1, . . . , n) .
This implies ∫∧nV exp
(1
2⟨a,Aa⟩
)da =
1(n2
)!
1
2n2
Pf(A) .
Use the integral formula to prove II, III, IV. ♡.
35
-
Proof (hints).
(ii). Choosing splitting V = W ⊕W∗ by matrix block structure, suchthat the Grassmann algebra of V ∼= algebra (tensor product) generated byci, bi | i=1, . . . , n2 , with relations cicj=−cjci, cibj=−bjci, and b1bj=−bjbi:
(a1, . . . , an) =
=(c1, . . . , cn2︸ ︷︷ ︸
basis inW
, b1, . . . , bn2︸ ︷︷ ︸basis inW ∗
)then ⟨
a ,
(0 A−At 0
)a
⟩= 2 ⟨c, Ab⟩ .
Hence, prove ∫∧n(W⊕W∗)
exp (⟨c, Ab⟩) dc db = det (A).
(iii). Similar.
36
-
(iv).∫
exp(1
2⟨a,Aa⟩ + ⟨a,η⟩
)da =
=
∫exp(1
2
⟨a + A−1η, A
(a + A−1η
)⟩− 1
2
⟨η, A−1η
⟩)da
= Pf(A) exp(−12
⟨η, A−1η
⟩).
∂
∂Ai1j1· · · ∂
∂AikjkPf(A) =
=
∫ai1aj1 · · · aikajk exp
(1
2⟨a,Aa⟩
)da
=
(∂
∂η
)2k ∫exp(1
2⟨a,Aa⟩ + ⟨η, a⟩
)da.
Prove the Pfaffian formula for the correlation functions. ♡.
37
-
1.6 Kasteleyn solution for bipartite graph
Given Γ⊂R2 endowed with configuration and Kasteleyn orientation
ZΓ= εKΓ
∫exp(12
∑ij
ai (ΣKΓ )ij aj
)da
∣∣∣∣ εKΓ :=(−1)σ εKσ1σ2· · · εKσn−1σn∈{±1}n= |V (Γ)|.In addition, Γ:= bipartite implies
ΣKΓ =
(0 BKΓ
−(BK)tΓ
0
) ∣∣∣∣∣∣∣∣∣∣BK : RV◦(Γ)→RV•(Γ)
RV (Γ)=RV•(Γ)⊕RV◦(Γ)
dim(RV•(Γ))=dim(RV◦(Γ))= n2V (Γ)=V•(Γ) ⊔ V◦(Γ), |V (Γ)|=n.
Identifying V•(Γ), V◦(Γ) via a diagram {b}∼{ω} with ‘‘hole”
ΣKΓ =
(0 CKΓ
−(CKΓ )t 0
)for RV•(Γ)⊕ RV◦(Γ)←↩
∣∣∣CKΓ :=RV◦(Γ)←↩where ←↩ denotes recursive invocations.
38
-
Then:
(i). ZΓ = |det(CKΓ )|
(ii).⟨
σb1w1 · · ·σbkwk⟩=
∂
∂ ω(b1w1)· · · ∂
∂ ω(bkwk)ln ZΓ =
= det(((CKΓ )
−1)b̂ w
) ∣∣∣∣ b̂ = b̂1, . . . , b̂kw = w1, . . . , wkwhere the inverse-matrix element b̂ is the white vertex identified with b.
Remark. The ‘‘Physical” meaning of (ii) is given by⟨σb1w1 · · ·σbkwk
⟩=
=
∫ψ∗b1w1· · ·ψ
∗bkwk× exp
(ψ∗CKΓ ψ
)×(∫
exp(ψ∗CKΓ ψ
)dψ∗ dψ
).
That is, it corresponds to correlation functions in free fermionic model.
39
-
1.7 Dimer monomer relation
Γ
removevertices andadjacent edges
2
4
5
29 32
36
37
229
36
3765
66
145
3
7 8
12
21 22
33
39
41
42
17
22
2342
46
51
52
62
68
126
23
46
51
68
70
74
76 81
87127
Monomers Dimers.
40
-
For monomer coverΓw1w2b1b2
b1
b2
w1
w2
the monomer-monomer correlation functions are given by
Mb1···bnw1···wn =ZΓw1···wnb1···bn
ZΓ.
41
-
Remark. If w1 and b1 are adjacent, then we have a dimerΓ
.
Remark. The monomer-monomer correlation functions are a special case ofdimer models on surfaces with nontrivial fundamental group.
.In this case, #([K])=22g+n−1.
42
-
1.8 Partition function as sum of Pfaffians
Theorem.Z
dimers
Γ⊂Mg =1
2g
∑[K]
Arf(qKD0)εK(D0) · Pf(ΣK)
∣∣Arf(·) := sgn.where:
[K] := equivalence classes of Kasteleyn orientations, 22g in total
[qKD0] := quadratic form on H1(Σ,Z2) associated with Kasteleyn
orientation on reference configuration D0εK(D0) := (−1)σεKσ1σ2· · · ε
Kσn−1σn
σ∈Sn, [σ] := Sn/(S
n22 ×Sn2
)n := number of vertices.
Proof. ♡.
43
-
Remark. In particular:
I. For bipartite graphs on Mg :
height function :== section of the non-trivialZ-bundle.
II. For fundamental cycles (a1, . . . , ag, b1, . . . , bg) :
Z(Ha1, . . . ,Hag,Hb1, . . . ,Hbg
)=
=∑D
∏ℓ∈D
ω(ℓ)g∏
i=1
exp(∑
i
Hai∆aih +
+∑i
Hbi∆bih)
where ∆Ch := change in height function along every noncontractible cycleC onMg.
44
-
1.9 Thermodynamic limit of bulk interactions
N
N
N
uniform measureProb(h)= 1|HΓ|N→∞ .
45
-
Theorem (Schur process; Okounkov & R). Let φε :Z2 ↪→R2 |D⊂R2.
ε
ε
D such that :{ ε→0, |Dε|→∞Dε := φε
(Z2)∩D
.
Then, for stacks of cubes with measure
Prob(π) =
∏tqπ(t)t∑
π
∏tqπ(t)t
∣∣∣∣ π∈HΓ , π∼=D,there is an existence of
Thermodynamic limit (|Dε|→∞) ++ Scaling limit (q=e−ε, ε→+0).
Proof. ♡.
46
-
a1N b1N a2N b2N
| uN | vN |where u+v = a1+a2+b1+b2, N =ε−1, q=e−ε.
47
-
1.10 Enumeration of perfect matchings in graphs
Involves computation of matching polynomials using the derived Kasteleyndeterminantal method:
• Γ⊂Mg :=m×n planar square lattice without closed boundary:µ(Γ;m,n) :=
2mn2
m∏i=1
n2∏
j=1
√cos2
(πi
m + 1
)+ cos2
(πj
n + 1
)n := even.
Or,= 0, for n := odd.
Proof. ♡.
48
-
• Γ⊂Mg :=m×n cylindrical square lattice:µ(Γ;m,n) :=
2mn2
m∏i=1
n2∏
j=1
√sin2
(π(2i−1)
m
)+ cos2
(πj
n + 1
)n := even.
Or,µ(Γ;m,n) :=
2mn2 −
m2 +1
m∏i=1
n2∏
j=1
√sin2
(π(2i−1)
m
)+ cos2
(πj
n + 1
)n := odd.
Proof. ♡.
49
-
• Γ⊂Mg :=m×n toroidal square lattice:µ(Γ;m,n) :=
2mn2 −1
m∏i=1
n2∏
j=1
√sin2
(π(2i−1)
m
)+ sin2
(2πj
n
)+
m∏i=1
n2∏
j=1
√sin2
(2πi
m
)+ sin2
(π(2j−1)
n
)+
m∏i=1
n2∏
j=1
√sin2
(π(2i−1)
m
)+ sin2
(π(2j−1)
n
)
n := even.
Or,= 0, for n := odd.
Proof. ♡.
50
-
• Γ⊂Mg := 6×8 planar square lattice without closed boundary:µ(Γ;m,n) :=
16777216
(1
4+ sin2
( π14
))(sin2
( π18
)+ sin2
( π14
))(14+ sin2
(3π
14
))×(
sin2( π18
)+ sin2
(3π
14
))(1
4+ cos2
(π7
))(cos2
(π9
)+ cos2
(π7
))×(
cos2(π7
)+ cos2
(2π
9
))(sin2
( π18
)+ cos2
(π7
))×(
sin2( π14
)+ cos2
(π9
))(sin2
( π14
)+ cos2
(2π
9
))×(
sin2(3π
14
)+ cos2
(π9
))(sin2
(3π
14
)+ cos2
(2π
9
))
51
-
• Γ⊂Mg := 6×8 cylindrical square lattice:µ(Γ;m,n) :=
5242880
(1
4+ sin2
( π18
))2 (1 + sin2
( π18
))(14+ cos2
(π9
))2(1 + cos2
(π9
))(14+ cos2
(2π
9
))2(1 + cos2
(2π
9
))• Γ⊂Mg := 6×8 toroidal square lattice:µ(Γ;m,n) :=
8388608
(18225
131072+
(1
4+ sin2
(π8
))4(1 + sin2
(π8
))2(14+ cos2
(π8
))4(1 + cos2
(π8
))2+ sin4
(π8
)(34+ sin2
(π8
))4cos4
(π8
)(34+ cos2
(π8
))4)
52
-
2 Special cases
Points:(i). Reformulate Kasteleyn Grassmann integral by transfer-matrices (for
special domains)
(ii). Compute inverse of Kasteleyn operator
(iii). Find scaling limit
(iv). Derive variational principle for limit topologies.
53
-
2.1 Grassmann integral kernel
Let∧nV :={ai | aiaj+ajai = 0} be Grassmann algebra generated by V, for
a chosen basis a=(a1, . . . , an)∈V.For vectors ψ∈
∧nV , writeψ(a)=
n∑k=0
n∑i1 .
Pair∧nV ∗⊗∧nV →R by⟨
φ(a∗), ψ(a)
⟩ def==
n∑{i}<
φik···i1 ψi1···ik .
54
-
Choosinga1, . . . , an ∈
∧nV , a∗n , . . . , a∗1 ∈ ∧nV ∗and
a∗n , . . . , a∗1 , a1, . . . , an ∈∧nV ∗⊗∧nV
then ∫n∏
ν=1
a∗iνn∏
ν=1
ajν da∗ da =
{0 , k ̸= n
(−1)(
σ + τ + n(n−1)2)
, k = n
σ : (i1, . . . , in)→ (1, . . . , n)τ : (j1, . . . , jn)→ (1, . . . , n).
Proposition.⟨φ(a∗), ψ(a)
⟩=
∫exp(∑
i
a∗i ai)
φ(a∗) ψ(a) da∗ da .
Proof. Exercise.
55
-
Proposition. Let A :V →V such thatψA(a) =
∑{i}< , {j}<
a{i}< A{i}
-
2.2 Vertex operators
(i). Fermionic Fock space: For ⟨Vm⟩∈CZ+12
F :=
{Vm1 ∧ Vm2 ∧ · · ·
∣∣ mi ∈ Z+ 12mi+1 = mi−1, i≫ 1
}.
(ii). Clifford algebra:
ClZ :=
⟨ψm, ψ∗m
∣∣∣ m ∈ Z+12
⟩ψm ψm′ + ψm′ ψm = ψ∗m ψ∗m′ + ψ∗m′ ψ∗m = 0ψm ψ∗m′ + ψ∗m′ ψm = δmm′ .
(iii). Clifford algebra acts on Fock space F :ψm vm1 ∧ vm2 ∧ · · · = vm ∧ vm1 ∧ vm2 · · ·
ψ∗m vm1 ∧ vm2 ∧ · · · =∞∑i=1
(−1)i δmim ×
× vm1 ∧ · · · ∧ v̂m1 ∧ · · ·
57
-
(iv). Heisenberg algebra:⟨αn∣∣ n∈Z\{0}⟩ : [αn,αn′] = −n δn,−n′ .
(v). Heisenberg algebra acts on F :• As part of Bose-Fermi correspondence in 1D:
αn 7−→∑
m ∈ Z+12
ψm+n ψ∗m .
• As operator in F :[αn,ψk] = ψk+n[αn,ψ∗k
]= −ψ∗k−n .
(vi). Vertex operators (in F ) :
Γ±(x) = exp( ∞∑
n=1
xn
nα±n
)(Γ−(x) v, w) = (v, Γ+(x)w) = (Γ+(x)w, v).
58
-
(vii). Commutation relations:
Γ+(x) Γ−(y) = (1−x) · Γ−(y) Γ+(x)
Γ+(x)ψ(z) = (1−z−1 x)−1 · ψ(z) Γ+(x)
Γ−(x)ψ(z) = (1−x z)−1 · ψ(z) Γ−(x)
Γ+(x)ψ∗(z) = (1−z−1 x) · ψ∗(z) Γ+(x)
Γ−(x)ψ∗(z) = (1−z x) · ψ∗(z) Γ−(x).
(viii). Eigenvectors:
Γ−(x)∏i
ψ∗(wi)∏j
ψ∗(zj)V (n)0 =
=∏i
(1−x zi)−1∏j
(1−xwj)∏i
ψ∗(wi)∏j
ψ∗(zj)V (n)0
where V (n)0 = vn−12 ∧ vn−32 ∧ · · ·
59
-
2.3 Fermionic Kasteleyn operator
[Diagram] [Diagram]
where b(h, t)=(h, t− 12
), and w(h, t)=
(h, t+ 12
).
The K-matrix over (b, w), using above-chosen diagram b ∼ w is:
K(h, t) = (h, t)−(h+
1
2, t+1
)+ xh,t
(h−1
2, t+1
).
[Diagram]
60
-
Placing fermions a∗h, t and ah, t , respectively, at b(h, t) and w(h, t), then
a∗Ka =∑h,t
a∗h,t ah,t −∑h,t
a∗h+12 , t+1
ah,t +∑h,t
a∗h−12 , t+1
ah,t xh,t =
=∑t
(a∗t at + atV a∗t+1 + atV −1xt a∗t+1
)in addition to considering boundary conditions
[Diagram]
Prob (π) ∝∝∏tq|π(t)|t in previousnotations
qh,t=qt
where the assumption is that xh,t=xt .
61
-
Theorem. WriteZ =
∫exp(a∗Aa
)da∗ da =
=⟨Γ−(x−12
)· · ·Γ−
(xu0+
12
)Γ+(x1
2
)· · ·Γ+
(xu1+
12
)V
(0)0 , V
(0)0
⟩.
Proof. Outline:∫· · · exp
(a∗t−1 at−1
)· exp
(at−1
(V −V −1Xt
)a∗t)·
· exp(a∗t at
)· exp
(at(V −V −1Xt
)a∗t+1
)· · · =
= · · ·(V −V −1Xt−1
)∼−1︸ ︷︷ ︸
νt−1
·(V −V −1Xt
)∼−1︸ ︷︷ ︸
νt
· · ·
where Γ+(xt)=νt−1 or Γ−(xt)=νt depending on t, Ã :=A such that V ←↩is lifted to
∧∞2 V , given by V :=
⊕m∈Z+12
C vh , with boundary conditions, etc.
Remark. Direct proof exists combinatorially, without K-orientation formula.
62
-
Remark. Write
Z =
u1−12∏m=12
−12∏m′=u0+
12
(1−x−m′ x
+m
)−1.
Theorem. (Okounkov & R., 2005).⟨σ(h1t1) · · ·σ(hktk)
⟩= det(K((ti, hi), (tj, hj)))1⩽i, j⩽k
K((ti, hi), (tj, hj)) =
=1
(2πi)2
∫|z|
-
2.4 Thermodynamic limit with scaling
[Diagram]
x+m = aqm
x−m = a−1qm
}assumed
corresponds to Prob(π) ∝ q |π| .
64
-
2.5 Asymptotics of partition functions
Consider limit ε→0, q=e−ε, u1=ε−1v1, u0=ε−1v0, where v1, v0 are fixed:
Z =∏
u0
-
Consider limit ε→0 for ti=ε−1τi, h1=ε−1χi, where τi, χi are fixed:
[Diagram](τi, χi)
in the bulkK((t1, h1), (t2, h2)) =
=1
(2πi)2
∫Cz
∫Cw
exp(ε−1 (S(z, t1, χ1)− S(z, t2, χ2))
)·
· (zw)1/2 (z−w)−1 dz dw .S(z, t, χ) =
= −(
χ+τ2−u0
)lnZ +
+ Li2(ze−v0
)+ Li2
(ze−v1
)− Li2(z)− Li2
(ze−τ
)where
Li2(z) =z∫0
t−1 ln(1−t) dt .
66
-
2.6 Critical points of S (z)
exp(
χ+τ2
)=
(1−ze−v0) (1−ze−v1)(1−z)
(1−ze−τ
)gives quadratic equation. That is, 2 real solutions, or 2 complex conjugate,or zero discriminant.
[Diagram]
∂χh0 (τ, χ) =1
πarg(z0)
⟨σ(h, t)
⟩= K((t, h), (t, h))→ ε ∂χh0(τ, χ)
67
-
2.7 Results of steepest descent
K((t1, h1), (t2, h2)) = −ε2π·( exp(ε−1(S1(z1)−S2(w2)))(z1−w2)
√−w2S ′′2 (w2)
√z1S ′′1 (z1)
−
−exp(ε−1(S1(z1)−S2(w2))
)(z1−w2)
√−w2S ′′2 (w2)
√z1S ′′1 (z1)
+ c. c.
)· (1 +O(1))
where z0(χ, τ) := inner of limit shape H+ :={z∈C, Im z>0} such thatz1 :=z0(χ1, τ1), w2 :=z0(χ, τ).
Therefore,
K((t1, h1), (t2, h2)) =ε2π
exp(ε−1(Re(S(z0(χ1, τ1)))−Re(S(z0(χ2, τ2))))
)·
·(exp(iε−1(Im(S ′(z1))−Im(S(w2))))
(z1−w2)+
exp(iε−1(Im(S ′(z1))−Im(S(w2)))
)(z1−w2)
+ c. c.
)· (1 +O(1)) (∗) .
68
-
This suggests convergence of K-orientation fermions to free Dirac fermions:1√ε
ψx⃗ = exp(ε−1Re(S(z0))
)·(
ψ+(z0) exp(iε−1Im(S(z0))
)+
+ ψ−(z0) exp(iε−1Im(S(z0))
))· (1+O(1))
1√ε
ψ∗x⃗ = exp(ε−1Re(S(z0))
)·(
ψ∗+(z0) exp(iε−1Im(S(z0))
)+
+ ψ∗−(z0) exp(iε−1Im(S(z0))
))· (1+O(1))
whereE(ψ∗±(z) ψ±(w)
)=
1
z − wE(ψ∗±(z) ψ∓ (w)
)= E
(ψ∗ψ∗
)= E(ψ ψ) = 0
ψ∗±(z), ψ±(w) are spinors:
ψ∗±(z) = ψ∗±(w)
öw
∂z
ψ±(z) = ψ±(w)√
∂w
∂z.
69
-
The correlation functions are derived as follows:
9 5
13 10
16 14x⃗
∣∣∣∣∣∣∣∣∣∣∣∣∣∣
⟨(σx⃗1−⟨σx⃗1⟩) (σx⃗2−⟨σx⃗2⟩)
⟩= K12K21 =
= ε2
(2π)2
( ∂z1∂x1
∂w2∂x2
(z1−w2)2−
∂z1∂x1
∂ w2∂x2
(z1−w2)2+ c. c.
)×
× (1+O(1)).
That is, correlation functions are given byσx⃗1−⟨σx⃗1⟩ = ε ∂x φ(z0(τ, x)) + · · ·
where φ(z) := Gaussian free field on H+.
The Green’s function of the Dirichlet problem on H+ is given by
⟨φ(z)φ(w)⟩ = 12π
ln∣∣∣z−wz−w
∣∣∣.And, the Bose-Fermi correspondence is given by
∂xφ = : ψ̃(z, z) ψ̃(z, z) : · · · .
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2.8 Scaling limit in the Kasteleyn operator
For lattice L such that Γ=Dε := φε(L)∩D
Let ΣKΓ := difference operator. Then(ΣKΓ)x·Gx,y = δx,y
where ε→0 in asymptotic for Gx,y.Particular cases.(i). Hexagonal lattices, with weights as above, such that
qt = e−ε f (t), t = τ
ε, ε→0.
Theorem. Gx,y := same as (⋆) , with different z0(τ, x).
(ii). Periodic lattices.
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2.9 Limit shapes and variational principle
(i). For the N×M torus[Diagram]
Z(H, V ) =∑D
∏ℓ
ω(ℓ) exp(H∆ahD + V∆bhD)
=1
2
(Pf(AK1
)+ Pf
(AK2
)+ Pf
(AK3
)− Pf
(AK4
))where N,M →∞, NM :=fixed.And, ω (ℓ)=1 gives Kasteleyn matrices’ eigenvalues by Fourier transform.
Theorem. (McCoy & Wu, 1969; Kenyon & Okounkov, 2005).
limN,M→∞
1
NMlnZNM =
∮ ∮ln |1+zw| dz
z
dw
w
= f (H, V ) :=
{|z| = eH|w| = eV .
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(ii). Taking Legendre transformσ(s, t) = max
H,V(Hs + Vt − f (H, V ))
then ∑D
1 =∑D
∏D
w (e) = exp(NM σ(s, t) · (1 +O(1)))
where∆ahDN
= s,∆bhDM
= t, N,M →∞, NM
fixed.
(iii). For the domain[Diagram]
∆ah = sN, ∆bh = tM .
Theorem. (Cohn, Kenyon, & Propp, 2000). As a result,∑D
1 = exp(NM σ(s, t) · (1 +O(1)))
with those boundary conditions of height function hD.
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(iv). For the domain[Diagram] Ni ×Mj
ZDε =∑
values of
height functionson
boundariesbetween rectangles
Z
Mj
Ni(h bound)
=∑
{∆xh, ∆yh}ij
exp( ∑
Mj
Ni
NiMj σ(∆xhMj
,∆yhNi
))
= exp(
ε−2∫D
σ (∂xh0, ∂yh0) dx dy (1 +O (1)))
where h0 :=minimizer forS [h] =
∫D
σ (∂xh0, ∂yh0) dx dy .
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Theorem. (Cohn, Kenyon, & Propp, 2000). In addition,
limε→0
ε2 lnZDε =∫D
σ(−→∇h0
)dx dy
such that• h0 is a minimizer• 0 < ∂xh, ∂yh < 1• h0
∣∣∂D = b, the boundary condition appearing in the limit ε→0.
[Diagram]
for height functionh = ε−1h0 + φ = ε−1 (h0 + εφ)
where h0 := the limit shape, and φ := fluctuations.
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2.10 Physics way of describing fluctuations
S[h0 + εφ] = S[h0] +ε2
2
∫∫D
aij(x)∂iφ ∂jφ d2x
aij(x) = ∂i∂j φ(s, t)s = ∂1h0t = ∂2h0Partition function is given by
Z = exp(ε−2S(h0))∫
exp(12
∫∫D
aij(x)∂iφ ∂jφ d2x)Dφ
where D := scalar field with Riemannian metric induced by h0.Correlation functions are given by
⟨φ(x)φ(y)⟩ = G(x, y)where G :=Green’s function for ∆=∂i(aij∂j).
Conjecture. G := same as from asymptotics of Kasteleyn operators.
Remark. The conjecture := theorem always in certain cases (R., et. al.).
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2.11 Conclusion - limit topology phenomena, ongoing
1. How to make such pictures of (i.e. simulate) random configuration:(i). Monte Carlo for exp
(∝10002
)(ii). Sampling around most probable region(iii). Markov Chain Monte Carlo method.
2. How to describe limit topologies and fluctuations analytically:(i). Kasteleyn matrix solution and correlation function(ii). Variational principle: Minimizing large deviation functionals(iii). Boundary conditions.
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Thank You!
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