Continuum Branching Observable in Higher Genus (Based on Discussion - - PowerPoint PPT Presentation
Continuum Branching Observable in Higher Genus (Based on Discussion with Nicolai Reshetikhin) Matthew Bernard mattb@berkeley.edu Advanced Computational Biology Center , Berkeley Supported by The Lynn Bit Foundation , State of California
Supported by The Lynn Bit Foundation, State of California
Abstract We prove: R-valued partition invariant in supersymmetry for all fjxed suffjcient large genus g⩾0 multiedge embedding; and, the O(n3) bipartite observable in Grassmann kernel transfer matrices on discrete functions for all spanning dual trees. In special hexagonal domain, we prove the free Dirac Fermion convergence Ψ=f ·(1 + O(1))
conjecture: In large deviation, Green’s function Gfor Dirichlet problem of variational principle minimizer is observable in the kernel asymptotics.
Keywords: Continuum-branching, Higher-genus, Observable
2
Bipartite implies no adjacent-black (-white) vertices for all V
X = V
M×N vertices, (M −1)×(N −1) path cartesian latticial edges; 2n=MN. Instance.
1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
. Non-instance.
1 2 3 4 5 6 7 8 9 10 11 12
no bipartite structure for triangular grids
3
∀ g ≫, an embedding X ⊂ Mg | V
X = (iℓ; iℓ=iξ=jℓ, ∀ i=j) is partition
σ ∈ Aut(D) ⇐ ⇒ perfect-matching D ⇐ ⇒ | iℓ ∩ D | = 1, ∀ iℓ ∈ V
X, and
1 2 3 4 5 6 7 8
i.e., 1
2
X
ℓ = ξ σD(iℓjℓ) defjned as:
|Aut(D)| · |σ
exp{n ln 2 + n−1
k=2 ln k}
= 0 ⇐ ⇒
(iℓ, jξ) = ∅
1 if ℓ ∈ D 0 if ℓ ∈ D for all V
X = (iℓ | i=1, . . . , 2n; ∀ n⩽|ℓ|<∞, n ∈ N+), and
X ⊂ Mg is CW cell-complex i.e. face ≈ topological disk i.e. no hole.
4
1 2 3 4 5 6
0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 (σD(iℓjℓ)) := 0 0 ⇐ ⇒ ℓ ∈ D, & 1 1 1 1 1 1 1 1 1,1 1 1 1 1 1 1 1 1,1 1 1 1 1 1 1 1 1 ⇐ ⇒ ℓ ∈ D.
1 2 3 4 5 6 7 8 9 10
0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 .
5
= (Sn×Sn
2 ) (Aut(D)/(Sn×Sn
2 )) ∼
= σ
∼ = { σ}
(ii) |{ σ}|1/|D| ⩽
for min(deg(XK)) ⩾ n! · a(XK) · b(XK) ⌊2n−3⌋!! = ⌊n⌋−2
k=0
2k+1
Im(Aut(D))← −XK∋σ:=(σ(1), . . . , σ(2n)); σ = σ |σ(2ℓ)>σ(2ℓ−1); Sn := {(σ(1), . . . , σ(2n)), . . . , (σ(2n−1), σ(2n), . . . , σ(1), σ(2))} Sn
2 := {(σ(1), . . . , σ(2n)), . . . , (σ(2), σ(1), . . . , σ(2n), σ(2n−1))}
for objects:
1 2 4 5 6 9 10 11 13 14 15 16 19 29 30 66
1 2 3 4 5 6 7 8 9 10 11 12
6
By E[σD(iℓjℓ) σD(iξjξ)] = E[σD(iℓjℓ)] ⇐ ⇒ ℓ = ξ, resp. 0 ⇐ ⇒ σD(iℓjℓ) = 0
correlation (conditional probability) k
σD(iℓjℓ)
= = Prob(i1j1 ∈ D, . . . , ikjk ∈ D) = E
σD(iℓjℓ)
k
σD(iℓjℓ) × Prob(D) =
k
σD(iℓjℓ)
ℓ ∈ D
ωℓ
ωℓ = 0 ⇐ ⇒ 1 Z
ωD = Prob(D) ⇐ ⇒
D ∈ D
(iℓ jℓ)
e−
Ξ(·) K T ,
Ξ(·)=
Ξℓ Z def = =
ωℓ for strict-sense positive partition function Z on Boltzmann weights ω(·) by Ξ : E
X −
→ R+ | ℓ − → Ξℓ .
7
Proposition (combinatorial-equivalence). Given a dimer space (resp. tiling space), there exits one-to-one combinatorial correspondence: family (Dimers) ← → family (Tilings).
(i) 2D cell complex on X (union of all spanning trees T): 0-cells, 1-cells, 2-cells = vertices, edges, faces, resp.
1 2 3 4 5 6 7 8 9 10 11 12 13 14
Disjoint interiors. dim(∂X(k)) = (k−1) mod 2 ∂X(k) = boundary of two k-cells, k = 0, 1, 2.
compact genus g cell-decomposition).
8
(ii) 2D dual cell complex on X∗ (union of all spanning dual trees T∗): 0-cells,1-cells,2-cells = resp.“centers”of 2-cells,1-cells,0-cells of X:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 21 22 23 25 26 27
X∗ = dual cell complex to X. (iii) For a dimer on X: Unique pair of 2-cells on X∗ share:
1 2 3 4 7 10 11 1 2 3 4 7 10 11 15 21 22 23 25
(iv) Therefore, the global bijection: (Dimers on X) ← → Tilings of X∗ by unique pair of 2-cells
□
9
1 2 3 4 7 10 11 15 21 22 23 25 1 2 3 4 7 10 11 15 21 22 23 25
(Below: one-color tiles to the left, and two-color tiles to the right)
43 35 44 31 14 15 31 32 35 34 10 13 35 34 44 45 16 19
.
10
Cubes: 3D boxes out of 2D rhombus-tiling projection
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 39 41 42 43 44 45 46 47 48 51 52 57 59 60 62 64 65 66 68 69 70 71 72 74 76 78 79 80 81 83 84 85 90 91 100 102 103 105 108 113 123 125 126 127 136 137 138 139
.
11
The parameterization space of spanning dual trees T∗ are height functions HX
def
= = {π : F
X −
→ Z} with respect to boundary-normalization π(F
0)=0 | f0=reference face, by
Dimers ← − Discrete surfaces.
π + 1
3
π + 2
3
9 5 13 10 16 14
π π + 1
3
π + 2
3
π + 1
3
9 5 13 10 16 14
π π + 2
3.
Proposition (πD). (i) πD
(ii) πD1D2 = πD1−πD2 .
12
Z
)
F
D
· ·
Prob(π) = 1 Z
qπ(F
)
F
, Z =
· ·
ω
εK
ℓ (F
) ℓ
X −
→Z where qF= invariant “essential” parameter for orientation εK
ℓ fjxed counter
(counterclockwise) ε−
ε∂F= ε− ε∂X boundary orientation ε∂F, ∀ F∈ F X |iℓ=jℓ;
εK
ℓ (F
) = −1 ≡ ℓւ(F ) (or +1 ≡ ℓր(F )) if εK
ℓ ∈ ε− ε∂F (resp. εK ℓ ∈ ε∂F).
Dimers on X
= bijection
= bijection
= bijection
→ s(ℓ+) ωℓ s(ℓ−). Cases. (i) Uniform distribution:
2 4 9 13 10 5
b c
b c
q = a−1b c−1a b−1c = 1.
13
(ii) Generally: qF = qt, Prob(π) ∝
t
q π(t)
t
.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 87 90 91 92 93 94 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 123 124 125 126 127 128 129 130 131 132 136 137 138 139 140 141 142 145 146
x t
14
∀ Prob(π) ∝
t
q π(t)
t
.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 123 124 125 126 127 128 129 130 131 132 136 137 138 139 140 141 142 145 146
x t
15
∃ unbounded stack ∀ Prob(π) ∝
t
q π(t)
t
.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 123 124 125 126 127 128 129 130 131 132 136 137 138 139 140 141 142 145 146
x t
16
Kasteleyn (1963). For g=0, Z = ± Pfaffjan of Kasteleyn matrix. Kasteleyn (1963). For g=1, Z = linear in 4 Pfaffjans; 3“+”, 1“−”. Kasteleyn (1963). For g > 1, Z = conjecture: 22g Pfaffjans, appearing mysteriously i.e. proof was not given, at least not published.
Gallucio & Loebl (1999). Z := ±1; Mg compact orientable. Tesla (2000). Z :=√−1 and ±1; Mg non-orientable. Cimazoni & R. (2004, 2005). Z := ±1 by spin-structure. Cimasoni (2006). Z := √−1 by pin-minus structure for double-cover; Mg non-orientable; a Tesla (2000) topological model∼ = spin structure’s ±1.
17
X −
→ Z, face-weights qF, Z(bipartite) = Const. ×
qπ(F
)
F
2g
Arf(qK
ξ ) · Pf(XK ξ ).
And, as |X|− →∞, qF− →1, in Seiberg-Witten conjecture (Gaussian fjeld theory) entropy, Z equates to path integral of scaling limit: Z =
2
Mg
(∂Φ)2 d2x +
λ(x) Φ(x)
) F
contributes to the R.H.S R.H.S R.H.S linear multiple λ(x) Φ(x) by: qx = ℓ−ε · λ(x)
→ 0. Moreover, in Alvarez-Gaumé, Moore, Nelson & Vafa (1986), studying Fermi and Bose partition correspondence on Riemann surfaces, R.H.S. R.H.S. R.H.S. ∼
Arf(ξ) × |Θ(z | ξ)|2
18
scaling, the asymptotics of observable linearly decaying goes to eVolume × the free energy eVolume × the free energy eVolume × the free energy where next leading term is sum of theta functions; and, every theta function square is next leading asymptotics of a Pfaffjan, resp. The conjecture was confjrmed by: (i) Ferdinand (1967). On square-grid torus. (ii) Costa-Santos & McCoy (2002). Numerically: Arf(ξ) × |Θ(z | ξ)|2 , ∀ g⩾2 . That is, the conjecture works, but still a conjecture i.e. no proof yet.
a sophistication i.e. disorder-type correlation. (ii) Z is glueable (summable) on boundary spins, for surfaces with boundary. (iii) “Higher” spin-structure is unknown, perhaps a para-polynomial theory.
19
(i) Prove Z invariant for all genus g multiedge bipartite embedding T∗ (ii) Prove the O(n3) observable for all fjxed suffjcient-large genus g⩾0
(i) Prove Grassmann kernel convergence for special genus g domain T∗ (ii) Obtain the R logarithmic scaling asymptotics by variational principle (iii) State conjecture for the Green’s function · in large-deviation
20
X ⊂ Mg is Kasteleyn XK if ∀ F ∈ F
X orientation εK ℓ | iℓ = jℓ, fjxed
counter (counterclockwise) ε−
ε∂F= ε− ε∂X boundary orientation ε∂F,
(mod 2) i.e. εK
F =
εK
ℓ (F
) = −1
ℓ (F
) = −1 ≡ ℓւ(F ) if εK
ℓ ∈ ε−
ε∂F
+1 ≡ ℓր(F ) if εK
ℓ ∈ ε∂F.
4 3 2 1 6 1 8 4 5 7 3 2 9 10Given XK for ωℓ trivial otherwise, ∀ ℓ connecting iℓ and jℓ, XK
ij =
εK
iℓjℓ ωℓ = −XK ji
ij = 0
εK
iℓjℓ =
−1 if εK
ℓ is jℓ to iℓ
+1 if εK
ℓ is iℓ to jℓ.
21
iℓjℓ = εK jℓiℓ = 1, then (XK ij ) is called: adjacency matrix
(resp. weighted adjacency matrix) ∀ ωℓ=1 (resp. ∀ ωℓ>1).
XK
ij = −XK ji =
ωℓ if iℓ jℓ or iℓ jℓ −ωℓ if iℓ jℓ or iℓ jℓ if iℓ, jξ such that iℓ=jℓ or ℓ=ξ .
22
The transition subgraph is symmetry D1∆D2=D1∪D2\D1∩D2 of 1-chain complex C1(XK ; Z2); 1-cycle homology H1(XK ; Z2) = H1(Mg; Z2) class; all ordered even-length η=
CασD1∆D2(Cα) simple closed transition paths
Cα = (σ(nα−1+1), . . . , σ(nα)), ∀ α∈N+ | 1⩽α⩽η, n0=0, traversing σ(nα−1+1),
1 2 3 4 5 6 7 8
→ Z2; D1, D2 = 1-chain in cell-complex C1(Mg; Z2); ∂D1, ∂D2 = C0(Mg; Z2).
23
Lemma (sign). For fjxed suffjcient large genus g, the monomial sign εK
D = (−1)t(σ) ℓ ∈ D
εK
σ(2ℓ−1)σ(2ℓ)
→ (1, . . . , 2n) is invariant of Aut(D).
D is Aut(D) invariant by (−1)t(σ) and σ(2ℓ−1)σ(2ℓ) transposition.
Now, let D1, D2 ∈ D orient from σ(2ℓ−1) to σ(2ℓ), resp. τ(2ξ−1) to τ(2ξ), in cyclic order of σ, resp. τ, ∀ Cα (transition even cycles). Then, exactly one εK
ℓ∗∨ ξ∗(Cα) is + (resp. −) in clockwise (resp. counterclockwise) ∀ α. Hence,
for all composition γ = σ ◦ τ | σ(2ν−1)(2ν) = τ(2ν−1)(2ν), +1 = εK
D1 εK D2 =
εK
σ(2ℓ − 1)σ(2ℓ) εK τ(2ξ − 1)τ(2ξ)
εK
γ(2ν − 1)γ(2ν)
=
εK
σ(2(ℓ ∨ ℓ∗) − 1) σ(2(ℓ ∨ ℓ∗)) εK τ(2(ξ ∨ ξ∗) − 1) τ(2(ξ ∨ ξ∗))
= ⇒ εK
D1 = εK D2, for 1 Cα
= 1 (mod 2), ∀ α, by ℓ∗∨ ξ∗ i.e. εK
D1 = εK D2, ∀ ρ− = 1 Cα
≡ 1
Cα
= ρ+ through Aut(D1) invariance, resp. Aut(D2) invariance, ∀ D1, D2 ∈ D. □
24
147
− →
148
Proof. Given two Kasteleyn orientations K
−, K + marked by K − (resp.
K
+) on ith end (resp. jth end) of ℓ, ∀ F
, with respect to ε∂F= ε∂X, εK
−
ℓ
= εK
+
ℓ
· σK
−K +
ℓ
similarly εK
+
ℓ
= εK
−
ℓ
· σK
−K +
ℓ
−K +
ℓ
= εK
−
ℓ
· εK
+
ℓ
i.e. K
−−
→K
+ (resp. K +−
→K
−) by σK
−K +
ℓ
multiplying K
− (resp. K +)
at every vertex; and, K
− ←
→ K
+ ←
→ equivalence class [K] in simple reversal of orientations around vertices by −1 = σK
−K +
ℓ
:= ±1. □
25
Proof. {[K]} is isomorphic to affjne closure of characteristic-2 fjeld κ non-degenerate skew-symmetric quadratic bilinear form Sym2
κ(V ∧):
q(α+β) = q(α)+q(β)+α·β
→κ , ∀ α, β ∈ H1 = V ⊗V for fjrst homology space H1 ∋ α classifjed by: 1
(−1)Arf (q)+q(α) = 1
q(ℓi)q(ℓj) ∈ κ/ f(κ) ⊂ Z2 where {ℓi, ℓj} = symplectic basis-pairs for symplectomorphisms V − → V, Lang’s isogeny f : κ− →κ | x− →x2−x ∈ Gal/F2 (2-element Galois fjeld). Under continuous ψ: XK− →Mg, all Mg\ψ(XK) connected-components (ψ-faces F ) ≈ open disk i.e. χ(XK) = χ(Mg) in Euler-Poincaré bound |V
XK|−|E XK|+|F XK| = χ(XK) ⩾ χ(Mg). Vanishing composition ∂1◦∂2
→C1, ∂1:C1 − →C0 for basis C0, C1, C2 of 2D cell-complex V
XK, E XK, F XK, resp. =
⇒ 1-cycle space Ker(∂1) contains 1-boundary space ∂2(C2). Hence, independent of XK, depending only on genus g: |H1(Mg; Z2)| = |H1(XK; Z2)| = |Ker(∂1)/∂2(C2)| = 22g. □
26
Theorem (existence). Kasteleyn orientation exists ⇐ ⇒ |V
XK| = even.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 29 30 31 32 33 34 35 36 37 39 42 43 44 45 46 47 48 57 59 60 62 64 65 66 68 69 70 71 72 74 76 78 79 80 81 83 84 85 90 91 100 102 103 105 108 113 136 137 138 139
Reduce X to ≪ by n×n − → exp(αn2); and, arbitrarily orient every ℓ not crossing T∗. Then, deleting ℓ∗ from leaves starting at root, make εK
F, ∀ F
.
27
Precisely, XK closed = ⇒ |V
XK| mod 2
= = = = |E
XK−F XK|. That is,
+1 (or −1) = (−1)|E
XK| = (−1)
XK
)
XK
(−1)
)
⇒ |E
XK|, |F XK| = even (resp. odd) ⇐
⇒ |V
XK| = even.
□
1 2
1 2
4
3
5 6 9 10 11 13 14 15 16 19 29 30 66
≡
1
−1
2
−1 −1 −1
5 10 13 14
h0, h1, h2, h3 = K
2 4
3
6 9 10 11 13 14 15 16 19 29 30 66
012
h012 = non-K, h3 = K.
28
h0 = h1 = · · · = h11 = −1.
29
|Pf(XK)| = Z
def
= =
ωℓ where Quot(K[D]) ∋ Pf(XK) = 1 n! 1 2n
sgn(σ) XK
σ(1)σ(2) · · · XK σ(2n−1)σ(2n)
sgn(σ) = (−1)t(σ) | t(σ) := (σ(1), . . . , σ(2n)) − → (1, . . . , 2n).
30
⇒ det XK=det(−(XK)T)=(−1)m det XK=0 ⇐ ⇒ m=odd; but det XK=0 = ⇒ det XK= positive-defjnite, square of rational function of XK
ij | XK= 2n×2n.
In particular, XK
i π(i)= −XK
π(i) i | i⩽π(i) =
⇒ sum of 2-partition monomials:
∈ S2n
2 )
(−1)t(π)
2n
XK
i π(i)
→ i=j ∈{1, . . . , n} = ⇒ XK
i π(i) ≡ XK
π(2ℓ−1)π(2ℓ)
∀ ℓ=1, . . . , n; t(π)= even (odd), for even n (otherwise) t(π) := (π(1), . . . , π(2n)) − → (1, . . . , 2n)
∈ 2 ·
n! 2n
n! 2n − 2
2n
XK
i π(i)
→ i=j ∈{1, . . . , n} = ⇒ XK
i π(i) ≡ XK
π(2ℓ−1)π(2ℓ)
∀ ℓ=1, . . . , n; t(π)= odd (even), for even n (otherwise). by Leibniz’s second-index permutations.
31
And, t(σ):= (σ(1), . . . , σ(2n))− →(1, . . . , 2n) implies the quadratic:
σ
∈ S2n
2 )
(−1)t(π) + n + t(σ)
ℓ ∈ D
XK
σ(2ℓ−1)σ(2ℓ)
for even n (otherwise)
2 ×
σ = σ = τ = τ
∈ S2n
2 )
∼ =
n! 2n
n! 2n − 2
ℓ ∈ D
XK
σ(2ℓ−1)σ(2ℓ)
XK
τ(2ξ−1)τ(2ξ)
=
σ = σ
(−1)t(σ)
ℓ ∈ D
XK
σ(2ℓ−1)σ(2ℓ)
= Pf 2(XK)
− → (1, . . . , 2n) ∀ min(deg( XK))⩾n!a( XK)b( XK)
)⊆S2n.
32
Now, ∀ ξ connecting σ(2ℓ−1) and σ(2ℓ), and by εK
D invariant of Aut(D),
where XK
σ(2ℓ−1)σ(2ℓ) =
εK
σ(2ξ−1)σ(2ξ) ωσ(2ξ−1)σ(2ξ)
then Pf(XK) =
σ = σ
sgn(σ)
εK
σ(2ξ−1)σ(2ξ)
ωσ(2ξ−1)σ(2ξ) =
σ
sgn(σ)
εK
σ(2ℓ−1)σ(2ℓ)
ωℓ = 1 n! 1 2n
∈
Aut(D)
εK
D
ωℓ = sgn(σ)
εK
σ(2ℓ−1)σ(2ℓ) ·
ωℓ = (±)
ωℓ = ± Z
33
i.e., Pf(XK) =
sgn(σ)
XK
σ(2ℓ−1)σ(2ℓ)
therefore, such that all S2n\Aut(D) monomials vanish, Pf(XK) = 1 n! 1 2n
sgn(σ) XK
σ(1)σ(2)· · ·XK σ(2n−1)σ(2n)
difgering only due to orientation, independent of σ ∈ Aut(D). □
Proof. k
σD(iℓjℓ)
ξη
|Pf(XK)| = partition function. □
Proof. Pf( AXK AT) = det( A)Pf( XK) − → O(n3), in diagonalization by skew symmetric Gaussian elimination, for graph spectrum analysis.
34
, ∀ XKbasis (x1, . . . , x2n) is given by 22n=2n
k=0(dim kXK) =2n k=0
2n
k
x0=1; xσ(k)< = xσ(1)⊗· · ·⊗xσ(k) | xσ(ξ)⊗xσ(η) + xσ(η)⊗xσ(ξ) = 0; σ(k)< = (σ(1) · · · σ(k)) | σ(1) < · · · < σ(k), ∀ 1, . . . , 2n
Element is graded by
2n
y(i) xi ⊕
2n
yσ(k)< xσ(2)< =
2n
1 k!
y(σ(1)···σ(k))xσ(1)⊗· · ·⊗xσ(k)
xσ(0)=x0. Multiplication y1(x) y2(x) is given by y(0)
1 y(0) 2
⊕
2n
1 y(i) 2 + y(i) 1 y(0) 2
2
1 y(σ(1) σ(2)) 2
+ + y(σ(1))
1
y(σ(2))
2
− y(σ(2))
1
y(σ(1))
2
+ y(σ(1) σ(2))
1
y(0)
2
35
Derivation.
2XK∋w= ijXK ij xi⊗xj =
⇒
2nXK∋wn=Pf(XK)xσ(2n)<.
kXK− →kXK: w1 ∧ · · · ∧ wk = 1 k!
(−1)t(σ) wσ(1)⊗· · ·⊗wσ(k).
2nXK ∼
= R,
f = fθ
by formal rule
xi ⊗
2n
dxi = (−1)n(2n−1)
(xi ⊗ dxi) = (−1)n(2n−1).
⇒ deg(x) < deg(dx),
xσ(i) ⊗ dx = (−1)t(σ) if k=2n if k<2n
i=1 dxi
t(σ):= (σ(1), . . . , σ(2n)) − → (1, . . . , 2n).
36
XK(x) =
1
2
ij xj
XKuniquely maximizes −
such that: (i) Pf(XK) =
exp 1 2
xiXK
ij xj
(ii) Pf
−(XK)
T
(iii) (Pf(XK))2 = det(XK) (iv) ∂ ∂XK
i1j1
· · · ∂ ∂XK
ikjk
Pf(XK) = Pf(XK) · Pf((XK−1)xy)
y=j1, . . . , jk.
37
Proof. (i). Write:
exp 1 2
= 1 n! 1 2n
n dx such that x, XKx n dx =
σ(1) τ(1) · · · XK σ(n) τ(n) dx =
= (−1)t(σ)XK
σ(1) τ(1) · · · XK σ(n) τ(n)
− → (1, . . . , 2n). This implies
exp 1 2
= 1 n! 1 2n Pf(XK).
38
(ii). Choosing splitting XK= W K⊕W K for block structure, where XK is isomorphic to algebra (tensor product) generated by ui, vi|i=1, . . . , n with relations uiuj =−ujui, uivj =−vjui, and vivj =−vjvi: (x1, . . . , x2n) = =
, v1, . . . , vn
As a result,
−(XK)
T
2
exp (
= det(XK). (iii). Similar.
39
(iv).
1 2
= =
1 2
2
= Pf(A) exp
2
∂ ∂XKi1j1 · · · ∂ ∂XK
ikjk
Pf(XK) = =
1 2
= ∂ ∂η
exp 1 2
40
Then, by Kullback-Leibler distance D(··) and Jensen’s inequality for any U, − D
=
U log
U ⩽ log
U
U = log
i.e. −
U log U = −
U log U
−
U log XK ⩽ −
U log( XK) = −
U 1 2
xiXK
ij xj
XK) where the equality holds ⇐ ⇒ U(x) = XK(x) almost everywhere. □ Lemma.
subfjeld, is isomorphic to kernel of either Q or prime-ordered fjeld Fq = pm.
□
41
then (i) Z = |det(CXK)|
XK←
֓ RV (XK) = RV•
XK ⊕ RV◦ XK←
֓ where ← ֓ = ⇒ nested (ii)
b1, . . . , bk w = w1, . . . , wk where b = white-vertex identifjed with b.
42
Proof. (i). XK⊂ Mg | g=0 implies Z = εK
X
1 2
xi (XK
ij ) xj
X = (−1)σ εK
σ1σ2 · · · εK σ2n−1σ2n
2n = |V (XK)|. XK⊂ Mg | g=0 bipartite V
XK= V
XK =
−(BXK)T
XK −
→ RV•
XK
RV (XK) = RV•
XK ⊕ RV◦ XK
dim(RV•
XK) = dim(RV◦ XK) = n
|V (XK)| = 2n. Identifying V•(XK), V◦(XK) via a diagram {b} ∼ {w} with “hole” XK =
−(CXK)T
XK ⊕ RV◦ XK←
֓ CXK = RV◦
XK←
֓ where ← ֓ = ⇒ nested i.e. Z = |det(CXK)|. □
43
(ii). Write
∂ ∂ w(b1w1) · · · ∂ ∂ w(bkwk) ln Z = det
b1, . . . , bk w = w1, . . . , wk where b = white-vertex identifjed with b. □
=
b1ψw1· · · ψ∗ bkψwk exp
which corresponds to the free Fermionic observable.
44
The order |σ
σ}| = |D| of an equivalence class D in fjxed g is given by generating function in two variables i.e. k=2 | ω1=1=ω2 for
σ
1 =
ν=1 Nν) = n
(±)
k
ωNν
ν
∀ ξ connecting σ(2ℓ−1) and σ(2ℓ); Nν = |ν-class dimers|. Derivation I. Let X⊂Mg =planar M×N square grid, where ∂X =open. |{ σ(X; M, N)}| = = 2(MN
2 )
M
N 2
M +1
N +1
= |{ σ(X; N, M)}|
= 0
45
Derivation II. Let X⊂Mg =cylindrical M ×N square grid. |{ σ(X; M, N)}| = = 2(MN
2 )
M
N 2
M
N +1
= 2(MN
2
− M
2 + 1)
M
N 2
M
N +1
= 0
46
Derivation III. Let X⊂Mg =toroidal M ×N square grid. |{ σ(X; M, N)}| = = 2(MN
2
− 1)
M
N 2
M
N
M
N 2
M
N
M
N 2
M
N
= |{ σ(X; N, M)}|
= 0
47
Derivation IV. Let X⊂Mg =planar 6×8 square grid, where ∂X =open. |{ σ(X; M, N)}| = = 16777216 1 4 + cos2(π 7)
9) + cos2(π 7)
7) + cos2(2π 9 )
×
7) + sin2( π 18) 1 4 + sin2( π 14)
9) + sin2( π 14)
×
9 ) + sin2( π 14)
18) + sin2( π 14) 1 4 + sin2(3π 14)
×
9) + sin2(3π 14)
9 ) + sin2(3π 14)
18) + sin2(3π 14)
48
Derivation V. Let X⊂Mg =cylindrical 6×8 square grid. |{ σ(X; M, N)}| = = 5242880 1 4 + cos2(π 9) 2 1 + cos2(π 9) 1 4 + cos2(2π 9 ) 2 × ×
9 ) 1 4 + sin2( π 18) 2 1 + sin2( π 18)
Derivation VI. Let X⊂Mg =toroidal 6×8 square grid. |{ σ(X; M, N)}| = = 8388608 18225 131072 + cos4(π 8) 3 4 + cos2(π 8) 4 sin4(π 8) 3 4 + sin2(π 8) 4 + + 1 4 + cos2(π 8) 4 1 + cos2(π 8) 21 4 + sin2(π 8) 4 1 + sin2(π 8) 2 .
49
Corollary (dimer-monomer problem).
remove vertices and adjacent edges
Monomers ← → Dimers.
50
Taking monomer cover
the monomer-monomer observable Mb1···bnw1···wn is given by Z(XK
b1···bnw1···wn)
Z(XK) .
51
In particular, adjacent monomers (bℓ, wℓ) = ⇒ dimer (ibℓ jwℓ), ∀ i, j |ℓ⊆D: X ⊂ Mg .
for nontrivial fundamental-group surfaces:
52
Lemma. Z = 1 2g
Arf(qK
D0) · εK
(D0) · Pf(XK)
2g
(−1)q(α) 2g = |H1(XK; Z2)| where [K] = all equivalence classes of Kasteleyn orientations, 22g in total qK
D0 = quadratic form on H1(Mg; Z2), corresponding to Kasteleyn
εK (D0) = (−1)σ εK
σ1σ2· · · εK σ2n−1σ2n
σ
= Aut(D)
2 ).
53
Theorem. Z = 1 2g
Arf(qK
ξ ) · Pf(XK ξ )
2g
(−1)q(α) 2g = |H1(XK; Z2)| where Arf(qK
ξ ) := quadratic form qK ξ on H1(Mg; Z2) for spin structure ξ
XK
ξ
= Kasteleyn matrix corresponding to spin structure ξ S(Mg) = set of all spin structures on Mg.
54
height function = = section of the non-trivial Z-bundle. where ∆Ch = change in height function along Mg noncontractible cycle C, then Z
=
ω(ℓ)
g
exp
i
Hxi∆xih + +
Hyi∆yih
55
N N N uniform measure Prob(h) = 1 |HX| N − →∞ .
56
Theorem (Schur process; Okounkov & R). Let ϕε:Z2֒ →R2|D⊂R2; ε ε
D
such that: ε− →0, |Dε|− →∞ Dε = ϕε
∩ D . Then, for cube-stack with measure Prob(π) =
qπ(t)
t
qπ(t)
t
π ∼ = D, there is existence of: Thermodynamic limit (|Dε|− →∞) + + Scaling limit (q=e−ε, ε− →+0).
57
x1N y1N x2N y2N
uN | vN
where u + v = = x1 + x2 + y1 + y2; N = ε−1, q = e−ε.
58
Points: (i) Prove Grassmann kernel convergence for special genus g domain T∗ (ii) Obtain the R logarithmic scaling asymptotics by variational principle (iii) State conjecture for the Green’s function · in large-deviation
59
Pairing
→R: σ(k)> =(σ(1), . . . , σ(k))
def = = ϕ0 ψ0 +
2n
ϕkψk +
2n
ϕσ(k)···σ(1) ψσ(1)···σ(k) = = |ψ0|2 +
2n
|ψσ(1)···σ(k)|2 d 2nx, ∀ |ψ|2 ∝ |ϕ|2 ∈ R such that for the dual space, graded basis x∗
σ(k)>,
2n
ψσ(k)< xσ(k)<
2n
ϕσ(k)> x∗
σ(k)>
σ(k)>
where
x0=1; xσ(k)< = xσ(1)⊗· · ·⊗xσ(k) | xσ(ξ)⊗xσ(η) + xσ(η)⊗xσ(ξ) = 0; σ(k)< = (σ(1) · · · σ(k)) | σ(1) < · · · < σ(k), ∀ 1, . . . , 2n
60
Fixing integrals on
x1, . . . , x2n ∈
2nXK, x∗ 2n, . . . , x∗ 1 ∈ 2nXK∗
and x∗
2n, . . . , x∗ 1 , x1, . . . , x2n ∈ 2nXK∗⊗ 2nXK
then
x∗
σ(i) η
xτ(i) dx∗ dx = , η = 2n (−1)(σ + τ + n(2n−1)) , η = 2n σ : (σ(1), . . . , σ(2n)) − → (1, . . . , 2n) τ : (τ(1), . . . , τ(2n)) − → (1, . . . , 2n) . Lemma.
x∗
i xi
61
→XK by ψY K(x) =
x{i}< Y{i}<{j}< ψ{j}< = ψ0 ⊕ Y ψ1 ⊕ Y ⊗2 ψ2 ⊕ · · · then ψY K(w) = =
−x∗x) ψ(x)dx∗ dx.
Lemma.
−x∗x) exp
= exp
−w∗Y Kw) is Y K “integral kernel” acting on
2nXK.
62
(i). The Fermionic Fock space F i.e. XK
m
2
is given by F =
m1 ∧ XK m2 ∧ · · ·
2
mi+1 = mi−1 i ≫ 1
(ii). The Clifgord algebra is given by ClZ =
m
2
ψm ψm′ + ψm′ ψm = ψ∗
m ψ∗ m′ + ψ∗ m′ ψ∗ m = 0
ψm ψ∗
m′ + ψ∗ m′ ψm = δm m′ .
(iii). The Clifgord algebra acting on the Fock space F : ψm xm1 ∧ xm2 ∧ · · · = xm ∧ xm1 ∧ xm2 ∧ · · · ψ∗
m xm1 ∧ xm2 ∧ · · ·
=
∞
(−1)i δmi, m xm1 ∧ · · · ∧ xm1 ∧ · · ·
63
(iv). The Heisenberg algebra is given by
[αn, αn′] = −n δn, −n′ . (v). The Heisenberg algebra acting on the Fock space F :
αn − →
2
ψm+n ψ∗
m .
ξ
ξ−n .
(vi). The vertex operators in F are given by XK
± (x) = exp
∞
xn n α±n
− (x)v, w
=
+ (x)w
=
+ (x)w, v
64
(vii). The commutation relations are given by XK
+ (x) XK − (y)
= (1−x) · XK
− (y) XK + (x)
XK
+ (x) ψ(z)
= (1−z−1 x)−1 · ψ(z) XK
+ (x)
XK
− (x) ψ(z)
= (1−x z)−1 · ψ(z) XK
− (x)
XK
+ (x) ψ∗(z)
= (1−z−1 x) · ψ∗(z) XK
+ (x)
XK
− (x) ψ∗(z)
= (1−z x) · ψ∗(z) XK
− (x).
(viii). The eigenvectors are given by XK
− (x)
ψ∗(wi)
ψ∗(zj) v(n) = =
(1−x zi)−1
j
(1−x wj)
ψ∗(wi)
ψ∗(zj) v(n) where v(n) = vn−1
2 ∧ vn−3 2 ∧ · · ·
65
For the one cube X∗ of two-color tiles on bipartite hexagonal lattice X
2 3 4 5 7 8 9 10 12 13 14 15 16 17 18 19 22 23 25 26 62 68 70 71
x t let the general parameterization for bipartite hexagonal lattice be given by: b(h, t) = (h, t− 1
2)
w(h, t) = (h, t+ 1
2).
66
Kasteleyn matrix by the above-given b ∼ w diagram is then given by K(h, t) = (h, t) −
2, t+1
2, t+1
Placing Fermions x∗
h, t , xh, t respectively at b(h, t) and w(h, t):
4 8 9 10 12 13 16 17
(h, t) (h− 1
2, t+1)
(h+ 1
2, t+1)
x∗Kx =
x∗
h,t xh,t −
x∗
h+1
2, t+1 xh,t +
x∗
h−1
2, t+1 xh,t yh,t =
=
t xt + xt V x∗ t+1 + xt V −1 xt x∗ t+1
67
[Diagram]
∝
t
q |π(t)|
t
the boundary conditions imply Z =
=
−
2
−
2
+
2
+
2
0 , v(0)
68
Proof (outline).
t−1 xt−1
t
t
· exp
t xt
t
t+1
= = · · ·
t−1
+ (xt)
·
t
− (xt)
· · · where XK
+ (xt) and XK − (xt) each depends on t such that
֓ is lifted to
∞ 2 V
2
C vh under boundary conditions, etc. □
Corollary. Z =
u1 − 1
2
2
−1
2
2
m′ x+ m
−1 .
69
K((ti, hi), (tj, hj)) = = 1 (2πi)2
R(t2) Φ−(z, t1) Φ+(w, t2) Φ+(z, t1) Φ−(w, t2) · · 1 z − w · z
2
2
where |w|<|z|, t1⩾t2 |w|>|z|, t1<t2
m > t((x+ m)−1),
R(t)=max
m < t(x− m), B(t)= |t| 2 − |t−u0| 2
Φ+(z, t)=
m > max(t, 1
2)
(1 − z x+
m), Φ−(z, t)= m < max(t, −1
2)
(1 − z−1 x−
m).
70
[Diagram] x+
m = aqm
x−
m = a−1qm
corresponding to Prob(π) ∝ q |π| .
71
Considering limit ε− →0, q=e−ε, u1=ε−1v1, u0=ε−1v0 for fjxed v1, v0: Z =
0 < m < u1
(1 − x−
m x+ n )−1
=
0 < m < u1
(1 − qm−n)−1 |π| = q ∂ ∂q ln Z = ε−3 u1
s−t 1−e t−s 3D volume function ds dt + · · · where ln Z = ε−2 u1
ln ( 1−e−s+t 2D partition function ) ds dt + · · ·
72
Consider limit ε− →0 where ti = ε−1τi, h1 = ε−1χi, for fjxed τi, χi: [Diagram] (τi, χi) in the bulk K((t1, h1), (t2, h2)) − → − → 1 (2πi)2
exp
· (zw)1/2 (z−w)−1 dz dw where S(z, t, χ) = = − (χ + τ 2 − u0) ln Z + Li2(ze−v0) + Li2(ze−v1) − Li2(z) − Li2(ze−τ) and Li2(z) = z
73
exp
2
(1−z)(1−ze−τ) gives quadratic equation, implying a discriminant for two real solutions or two complex-conjugate solutions, or a zero-discriminant. [Diagram] ∂χh0 (τ, χ) = 1 π arg(z0)
→ ε ∂χh0(τ, χ)
74
K((t1, h1), (t2, h2)) = − ε 2π ·
(z1 − w2)
2(w2)
1(z1)
− − exp{ε−1(S1(z1) − S2(w2))} (z1 − w2)
2(w2)
1(z1)
+ c. c.
That is, for H+ = {z ∈C, Im z >0} | z0(χ, τ) = inner process, such that z1 = z0(χ1, τ1) w2 = z0(χ, τ), K((t1, h1), (t2, h2)) = = ε 2π exp{ε−1(Re(S(z0(χ1, τ1))) − Re(S(z0(χ2, τ2))))} · · exp{i ε−1(Im(S′(z1)) − Im(S(w2)))} (z1−w2) + + exp{i ε−1(Im(S′(z1)) − Im(S(w2)))} (z1−w2) + c. c.
(∗).
75
Hence, solution of Kasteleyn-Fermions convergence to free Dirac-Fermions: 1 √ε ψ x = exp(ε−1Re(S(z0))) ·
+ ψ−(z0) exp(iε−1Im(S(z0)))
1 √ε ψ∗
+(z0) exp(iε−1Im(S(z0))) + + ψ∗ −(z0) exp(iε−1Im(S(z0)))
such that E(ψ∗ ±(z) ψ±(w)) = 1 z − w E(ψ∗ ±(z) ψ∓(w)) = E(ψ∗ ψ∗) = E(ψ ψ) = 0 where ψ∗ ±(z), ψ±(w) are spinors: ψ∗ ±(z) = ψ∗ ±(w)
∂z , ψ±(z) = ψ±(w)
∂z .
76
9 5 13 10 16 14
x1−σ x1) (σ x2−σ x2)
= ε2 (2π)2 ∂z1 ∂x1 ∂w2 ∂x2 (z1−w2)2 − ∂z1 ∂x1 ∂ w2 ∂x2 (z1−w2)2 + c. c.
× (1 + O(1)). In particular, σ x1−σ x1 = ε ∂x ϕ(z0(τ, x)) + · · ·
such that the Green’s function of Dirichlet problem on H+ is given by ϕ(z) ϕ(w) = 1 2π ln
z−w
∂xϕ = : ψ(z, z) ψ(z, z) : · · · .
77
Let X =Dε=ϕε(L) ∩ D, for arbitrary lattice L |AK
G = difgerence operator,
.
where ε − → 0 in the asymptotics of the equation for G
x,y given by
(AK
X)x · G x,y = δx,y
Cases. (i) Hexagonal lattice: Utilizes the weighted as above, for qt = e−ε f(t), t = τ ε , ε− →0.
x,y = same as (∗), with difgerent z0(τ, x).
(ii) Periodic lattice: Utilizes variational principle.
78
(i). For the N ×M torus [Diagram] Z(H, V ) =
ω(ℓ) exp(H∆ahD + V ∆bhD) = 1 2
+ Pf
+ Pf
− Pf
where N, M − →∞, for fjxed N
M.
And, ω(ℓ)=1 = ⇒ eigenvalues of Kasteleyn matrices by Fourier transform.
lim
N,M− →∞
1 NM ln ZNM = ln |1+zw| dz z dw w = f(H, V ) = |z| = eH |w| = eV .
79
(ii). Taking Legendre transform σ(s, t) = max
H,V (Hs + Vt − f(H, V ))
then
1 =
w (e) = exp(MN σ(s, t) · (1 + O(1))) where ∆ahD M = s, ∆bhD N = t, M, N → ∞, N M fjxed. (iii). For domain [Diagram] ∆ah = sM, ∆bh = tN.
1 = exp(MN σ(s, t) · (1 + O(1))) with the boundary conditions of height function hD.
80
(iv). For domain [Diagram] Mi × Nj ZDε =
values of height functions
boundaries between rectangles
Z Mi Nj (h bound) =
exp Mi Nj Mi Mj σ ∆xh Mi , ∆yh Nj
σ(∂xh0, ∂yh0) dx dy (1 + O(1))
S[h] =
σ(∂xh0, ∂yh0) dx dy.
81
lim ε→0 ε2 ln ZDε =
σ − → ∇h0
for 0 < ∂xh, ∂yh < 1 | h0 = minimizer h0
→ 0 [Diagram] for height function h = ε−1h0 + ϕ = ε−1 (h0 + εϕ) with respect to h0 = limit shape, and ϕ = distribution (factor).
82
S[h0 + εϕ] = S[h0] + ε2 2
aij(x)∂iϕ ∂jϕ d2x aij(x) = ∂i∂j ϕ(s, t)
s = ∂1h0 t = ∂2h0 such that:
Z = exp(ε−2S(h0))
1 2
aij(x)∂iϕ ∂jϕ d2x
where D = scalar fjeld with Riemannian metric induced by h0;
ϕ(x) ϕ(y) = G (x, y) where G= Green’s function for ∆ = ∂i(aij∂j).
83
(Chebotarev, Guskov, Ogarkov & Bernard, 2019). For free-action
S[g, ¯ ϕ] ≡ Z[g, j = ˆ G−1 ¯ ϕ] Z[j = ˆ G−1 ¯ ϕ] = =
∞
(−1)n n! n
ϕ(xa)
−1
2 n
λaλbG(xa−xb)
. (Guskov, Kalugin, Ivanov, Ogarkov & Bernard, ’19-submitted). For nonpolynomial theory, Z [g; dµ] =
dµ(x)
= C1 [g; dµ]
Γ 2 1
4
+ O 1 g
84
(i). Monte Carlo for exp
(ii). Sampling around most probable region by MCMC
(i). Equipartition Pfaffjan asymptotics with boundary conditions (ii). Variational principle: Minimizer functional in large deviation
85
[BGI+19]
integration, basis functions representation and strong coupling
[CGOB19]
S-Matrix of Nonlocal Scalar Quantum Field Theory in Basis Functions Representation. Particles, 2(1):103–139, 2019. [Kas63]
[OR07]
and the Pearcey process. Comm. Math. Phys., 269:571–609, 2007.
86
87