text: R DYNAMICS FOR A CLASS OF POLYNOMIAL MAPS IN 2 25 − × − ⊂ − × − f 1([c, 0] [ 1, c[) ([ 1, c[ [ 1, 0]) and − − × − ⊂ − × ∞ f 1([ 1, c[ [ 1, 0]) ([ 1, 0] ]0, + ]). × − b) Since R = [0, 1 + c] [ 1, 0], it follows that 3 − ∩ ⊂ − × − ∞ ∩ ⊂ − × − f 1(R ) R ([ 1, 0] ] , c]) R ([ 1, c] [ 1, c]). 3 2 2 − − × − ⊂ − × ⊂ (cid:3) c) f 1([ 1, c] [ 1, c]) [ 1, c] [0, 1 + c] R 1 R− { ∈ ∪ ∪ − ∈ ∪ ∪ ∈ N} Let := z (R R R ) : f n(z) (R R R ), for all n . 1 2 3 1 2 3 − R− ⊂ ∪ ∪ Proposition 12. If 1 < c < 0, then W u(p) W u(f(p)) W u(f2(p)). ∈ R− ∈ R− Proof. Let z = (x, y) . Since z , it follows from Lemmas 3.4, 3.2 ∈ N − ∈ − × − and 3.5 that there exists j , such that f j(z) [ 1, c] [ 1, c]. Without ∈ − × − loss of generality, supposes that z [ 1, c] [ 1, c]. Thus, from Lemmas − ∈ − × − 3.2 and 3.5, we have to study the case where f 3n(z) [ 1, c] [ 1, c], f−3n−1(z) ∈ [−1, c] ×√[0, 1 + c] and√f−3n−2(z) ∈ R , for all n √∈ N. − − × − − ⊂ 3 × − − ⊂ Observe that f([ c, c] [ c, c]) [c2 + c, c] [ c, c] R . ∈ ∈ 2 √ √ However, we are considering the points w R such that f(w) R . ∈ − × − \ − −2 × − − 3 √ Therefore, let z ([ 1, c] [ 1, c]) ([ c, c] [ c, c]). − − ∈ − ≥ Let z = c and z [ 1, z ] such that z = g(z ), for all n 0, where g0: R2 −→ R2 is thne+1m√ap deﬁnned by g(x) =nx(x2 +nc+)1+ c. ∈ − − − ≥ √ √ Observe that z [ 1, c] exist and is unique, for each n 0, since g n − − − − − − − is a nondecreasing map at interval [ 1, c] and g([ 1, c]) = [ 1, c]. − ≥ Let f i(x, y) = (x , y ), for all i 0. Thus, from proof of Proposition i i ∈ − × 5, we have that (z ) ≥ is a decreasing sequence and (x , y ) ([ 1, 0] − \ ×n n 0 ≥ 3n 3n √ [ 1, 0]) ([z , 0] [z , 0]), for all n 0. n n − − − Since (z ) ≥ is decreasing and z = c < c, it follows that z + c > n n 0 ≥ 0 n 0. Hence, for all n 0, we have: − − ∈ − × i) If f 3n 3(z) = (x , y ) [ 1, z ] [z , 0], then 3n+3 3n+3 n+1 n − − ∈ − × − ∩ − × − f 3n 2(z) [c, z + c] [ 1, z ] R = [0, z + c] [ 1, z ], − − ∈ n× − n∩+1 3 × n − n+1 f 3n 1(z) [z , c] [0, z + c] R = [z , c] [0, z + c] e − n ∈ −n 1 ×n ∩ n f 3n(z) = (x , y ) [z ( z + c) + c, c] [z , c] R = 3n 3n n −n × n 2 [z ( z + c) + c, c] [z , c]. n n n − ≥ − On the other hand, since z ( z + c) + c z , it follows that [z ( z + × ⊂ × n n n n n ∈ c) + c, c] [z , c] [z , 0] [z , 0], which is an absurd, because (x , y ) − × − n \ n × n ≥ 3n 3n ([ 1, 0] [ 1, 0]) ([z , 0] [z , 0]), for all n 0. − − n n ∈ × − ii) If f 3n 3(z) = (x , y ) [z , 0] [ 1, z ], then 3n+3 3n+3 n n+1 − − ∈ − × ∩ − × − − ∈ f 3n 2(z) [c, z + c] [z , 0] R = [0, z + c] [z , 0], f 3n 1(z) − n× − n ∩ 3 −n n × − [z ( z + c) + c, c] [0, z + c] R = [z ( z + c) + c, c] [0, z + c] n n n 1 n n n and ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: R

text: DYNAMICS

text: FOR

text: A

text: CLASS

text: OF

text: POLYNOMIAL

text: MAPS

text: IN

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text: c[

text: and

text: f

text: c[

text: b)

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text: R

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text: Proposition

text: If

text: c

text: then

text: W

text: u(p)

text: W

text: u(f(p))

text: W

text: R−

text: R−

text: Proof.

text: Let

text: z

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text: y)

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text: z

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text: follows

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text: Lemmas

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text: 26 DANILO ANTONIO CAPRIO − − Figure 10. Region that contains f 3n 3(z) = (x , y ). 3n+3 3n+3 − ∈ f 3n(z) = (x , y ) − − 3n× 3n − ∩ [( z + c)(z ( z + c) + c) + c, c] [z ( z + c) + c, c] R = n− n n− ×n n− 2 [( z + c)(z ( z + c) + c) + c, c] [z ( z + c) + c, c]. n n n n n On the other hand, − − × − ⊂ × [( z + c)(z ( z + c) + c) + c, c] [z ( z + c) + c, c] [z , 0] [z , 0], n n n n n n n ∈ − × − \ × which is an absurd, since (x , y ) [ 1, 0] [ 1, 0] [z , 0] [z , 0], for ≥ 3n 3n n n all n 0. − − ∈ − × Therefore, by the items i) and ii), we have that f 3n 3(z) [ 1, z ] − \ × ≥n [ 1, z ] [z , z ] [z , z ] (hatching region on Figure 10), for all n 0,. n n+1 n− − n+1 ∈n − × − ≥ Hence, since f 3n 3(z) [ 1, z ] [ 1, z ], for all n 0, and z con- − ∞n n − n verges to 1 when n goes to + , it follows that lim f 3n(z) = p, → ∞ − − − − n + (cid:3) lim f 3n 1(z) = f2(p) and lim f 3n 2(z) = f(p). → ∞ → ∞ n + n + − ∩ K− ⊂ Proposition 13. If 1 < c < 0, then L W u(θ). ∈ ∩K− Proof. Let z = (x, y) L . From Lemma 3.2 and by case(cid:88)3) of the proof − ∈ ≥ of Proposition 11, we have to suppose that f n(z) L, for all n 0. − n 1 (cid:107) (cid:107) ≥ w − c Let w = z and for each n 1, let b := ai , A = n an an 2 2n × × 2 2 i=0 [a , b ] [a , b ] and A = [a , b ] [a , b ]. 2 n −2 n+∈1 2n+1 ≥ 2 n+1 2 n+1 Claim: f j(z) A , for all j 2. j ∈ × In fact, we have that z [a , w] [a , w]. Thus, 2 2 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: DANILO

text: ANTONIO

text: CAPRIO

label: caption

text: Figure

text: Region

text: that

text: contains

text: f

text: (x

text: y

text: f

text: (x

text: y

text: z

text: c)(z

text: z

text: c)

text: c)

text: c,

text: c]

text: [z

text: z

text: c)

text: c,

text: c]

text: R

text: n−

text: n

text: n−

text: ×n

text: n−

text: z

text: c)(z

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text: c)

text: c)

text: c,

text: c]

text: [z

text: z

text: c)

text: c,

text: c].

text: n

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text: the

text: other

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text: z

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text: c)

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text: y

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text: for

text: n

text: n

text: all

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text: by

text: the

text: items

text: i)

text: and

text: ii),

text: we

text: have

text: that

text: f

text: z

text: ≥n

text: z

text: [z

text: z

text: [z

text: z

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text: region

text: on

text: Figure

text: for

text: all

text: n

text: n

text: n−

text: ∈n

text: Hence,

text: since

text: f

text: z

text: z

text: for

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text: z

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text: ∞n

text: n

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text: verges

text: to

text: when

text: n

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text: to

text: it

text: follows

text: that

text: lim

text: f

text: p,

text: n

text: lim

text: f

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text: lim

text: f

text: f(p).

text: n

text: n

text: K−

text: Proposition

text: If

text: c

text: then

text: L

text: W

text: u(θ).

text: ∩K−

text: Proof.

text: Let

text: z

text: (x,

text: y)

text: L

text: From

text: Lemma

text: and

text: by

text: of

text: the

text: proof

text: of

text: Proposition

text: we

text: have

text: to

text: suppose

text: that

text: f

text: n(z)

text: L,

text: for

text: all

text: n

text: n

text: w

text: c

text: Let

text: w

text: z

text: and

text: for

text: each

text: n

text: let

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text: ai

text: A

text: n

text: an

text: an

text: [a

text: b

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text: b

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text: A

text: [a

text: b

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text: n

text: −2

text: Claim:

text: f

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text: A

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text: we

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text: w]

text: [a

text: w].

text: Thus,

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text: R DYNAMICS FOR A CLASS OF POLYNOMIAL MAPS IN 2 7 − × ⊂ − × − ⊂ fn([ 1, c] [c, 0]) fn 1([c, 0] [ 1, c]) − × − ⊂ fn 2([c, 0] [c, 0]) = fn 2(Y ) Y , ≥ (cid:3) for all n 2. ∈ − × − \{ } ⊂ ∈ N Proposition 5. If z [ 1, c[ [ 1, c[ p R , then there exists N ∈ 2 such that fN(z) Y . ∈ − × − \{ } Proof. Let z = (x, y) [ 1, c[ [ 1, c[ p . { } − Case 1: min x, y > 1. ∈ N ∈ Supposes that there not exists N such that fN(z) Y . Thus, from Lemma 2.5 and Lemma 2.2, we have that (1) ∈ ∈ ∈ ∈ N f3n(x, y) R , f3n+1(x, y) R and f3n+2(x, y) R , for all integer n . 2 3 1 Let (x , y )√= fn(x, y)√. Observe that if there exists n ∈ N suc√h that n∈ n− − × − − ∈ × − − (x , y ) [ c, c] [ c, c], then (x , y ) [c2+c, 0] [ c, c] ⊂ 3n 3×n − 3n+1 3n+1 ∈ [c, 0] [ 1, c], and from Lemma 2.6 it follows that (x , y ) Y , for ≥ 3n+k 3n+k all k 3. Therefore, we need to suppose that √ { } − − ∈ N (2) min x , y < c, for all n . 3n 3n √ { } − − Let z = min x, y . Thus, from relation (2), we have z < c, which 0 ∈ 0 × implies that z2 + c > 0. Therefore, it follows that (x, y) [z , c] [z , c] and ∈ 0 × 0 0 (x , y ) [c2 + c, z2 + c] [z , c]. Hence, from relation (1), it follows that 1 1 0 0 ∈ × ∈ × (x , y ) [0, z2 + c] [z , c], (x , y ) [z (z2 + c) + c, c] [0, z2 + c] and 1 1 ∈0 0 2 2 0 0 × 0 (x , y ) [(z (z2 + c) + c)(z2 + c) + c, c] [z (z2 + c) + c, c]. 3 3 0 0 0 0 0 ≤ It is not hard to prove that z (z2 + c) + c (z (z2 + c) + c)(z2 + c) + c, ∈ ×0 0 0 0 0 i.e. (x , y ) [z (z2 + c) + c, c] [z (z2 + c) + c, c]. 3 3R −→0R 0 0 0 Let g : deﬁned by g(x) = x(x2 + c) + c and deﬁne z = g(z ), ∈ N n+1 n for all n . ≥ ≥ It is easy to check that g(z ) z . Therefore, we have that z = g(z ) z 0 0 ∈ × 1 0 0 and by deﬁnition of g, it follows that (x , y ) [z , c] [z , c]. 3 3 1 ∈ 1 × √ √ In the same way, we can prove that (x , y ) [z , c] [z , c], with − − − 3k {3k }k − −k 1 < z < z < . . . < z < c and min x , y < c, for all ∈ N 0 1 k 3k 3k k . Hence, (z ) is an increasing and bounded sequence. Therefore, n n ∈ {− } √ (z ) admits limit l. Thus, we have that g(l) = l, i.e. l 1, a , a , which n n − ≤ − − 1 2 is absurd, since 1 < z < z < . . . < z < . . . < l c < c < a < a . 0 ∈ N1 k ∈ 1 2 Therefore, there exists N such that fN(z) Y . ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: R

text: DYNAMICS

text: FOR

text: A

text: CLASS

text: OF

text: POLYNOMIAL

text: MAPS

text: IN

text: fn([

text: c]

text: [c,

text: fn

text: c])

text: fn

text: [c,

text: fn

text: Y

text: for

text: all

text: n

text: N

text: Proposition

text: If

text: z

text: c[

text: c[

text: p

text: R

text: then

text: there

text: exists

text: N

text: such

text: that

text: fN(z)

text: Y

text: Proof.

text: Let

text: z

text: (x,

text: y)

text: c[

text: c[

text: p

text: Case

text: min

text: x,

text: y

text: N

text: Supposes

text: that

text: there

text: not

text: exists

text: N

text: such

text: that

text: fN(z)

text: Y

text: Thus,

text: from

text: Lemma

text: and

text: Lemma

text: we

text: have

text: that

text: N

text: y)

text: R

text: y)

text: R

text: and

text: y)

text: R

text: for

text: all

text: integer

text: n

text: Let

text: (x

text: y

text: fn(x,

text: y)√.

text: Observe

text: that

text: if

text: there

text: exists

text: n

text: N

text: suc√h

text: that

text: n∈

text: n−

text: (x

text: y

text: c,

text: c]

text: c,

text: c],

text: then

text: (x

text: y

text: c,

text: c]

text: [c,

text: c],

text: and

text: from

text: Lemma

text: it

text: follows

text: that

text: (x

text: y

text: Y

text: for

text: all

text: k

text: Therefore,

text: we

text: need

text: to

text: suppose

text: that

text: N

text: min

text: x

text: y

text: c,

text: for

text: all

text: n

text: Let

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text: min

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text: y

text: Thus,

text: from

text: relation

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text: have

text: z

text: c,

text: which

text: implies

text: that

text: c

text: Therefore,

text: it

text: follows

text: that

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text: y)

text: [z

text: c]

text: [z

text: c]

text: and

text: (x

text: y

text: c,

text: c]

text: [z

text: c].

text: Hence,

text: from

text: relation

text: it

text: follows

text: that

text: (x

text: y

text: c]

text: [z

text: c],

text: (x

text: y

text: [z

text: c)

text: c,

text: c]

text: c]

text: and

text: ∈0

text: (x

text: y

text: [(z

text: c)

text: c)

text: c,

text: c]

text: [z

text: c)

text: c,

text: c].

text: It

text: is

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text: c

text: (z

text: c)

text: c)

text: c,

text: ×0

text: i.e.

text: (x

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text: [z

text: c)

text: c,

text: c]

text: [z

text: c)

text: c,

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text: −→0R

text: Let

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text: g(x)

text: c)

text: c

text: and

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text: z

text: g(z

text: N

text: n

text: for

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text: n

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text: easy

text: to

text: check

text: that

text: g(z

text: z

text: Therefore,

text: we

text: have

text: that

text: z

text: g(z

text: z

text: and

text: by

text: deﬁnition

text: of

text: g,

text: it

text: follows

text: that

text: (x

text: y

text: [z

text: c]

text: [z

text: c].

text: In

text: the

text: same

text: way,

text: we

text: can

text: prove

text: that

text: (x

text: y

text: [z

text: c]

text: [z

text: c],

text: with

text: }k

text: −k

text: z

text: z

text: z

text: c

text: and

text: min

text: x

text: y

text: c,

text: for

text: all

text: N

text: k

text: k

text: Hence,

text: (z

text: is

text: an

text: increasing

text: and

text: bounded

text: sequence.

text: Therefore,

text: n

text: n

text: (z

text: admits

text: limit

text: l.

text: Thus,

text: we

text: have

text: that

text: g(l)

text: l,

text: i.e.

text: l

text: a

text: a

text: which

text: n

text: n

text: is

text: absurd,

text: since

text: z

text: z

text: z

text: l

text: c

text: c

text: a

text: a

text: k

text: Therefore,

text: there

text: exists

text: N

text: such

text: that

text: fN(z)

text: Y

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: [FW70] C. Fan and F. Y. Wu. General lattice model of phase transitions. Phys. Rev. B, 2:723–733, 1970. [Kas61] P. W. Kasteleyn. The statistics of dimers on a lattice : I. the number of dimer arrangements on a quadratic lattice. Physica, 27:1209–1225, December 1961. [Kas67] P. W. Kasteleyn. Graph theory and crystal physics. In Graph Theory and Theoretical Physics, pages 43–110. Academic Press, London, 1967. [KB79] L. P. Kadanoﬀ and A. C. Brown. Correlation functions on the critical lines of the Baxter and Ashkin-Teller models. Ann. Phys., 121(1–2):318 – 342, 1979. [KC71] L. P. Kadanoﬀ and H. Ceva. Determination of an operator algebra for the two- dimensional Ising model. Phys. Rev. B, 3:3918–3939, Jun 1971. [Ken99] A. E. Kennelly. The equivalence of triangles and three-pointed stars in conducting networks. Electrical World and Engineer, 34:413–414, 1899. [Ken97] R. Kenyon. Local statistics of lattice dimers. Ann. Inst. H. Poincar´e Probab. Statist., 33(5):591–618, 1997. [Ken02] R. Kenyon. The Laplacian and Dirac operators on critical planar graphs. Invent. Math., 150(2):409–439, 2002. [Ken17] R. Kenyon. Determinantal spanning forests on planar graphs. ArXiv:1702.03802, 2017. [Kir47] G. Kirchhoﬀ. Ueber die auﬂ¨osung der gleichungen, auf welche man bei der un- tersuchung der linearen vertheilung galvanischer str¨ome gefu¨hrt wird. Annalen der Physik, 148:497–508, 1847. [KOS06] R. Kenyon, A. Okounkov, and S. Sheﬃeld. Dimers and amoebae. Ann. of Math. (2), 163(3):1019–1056, 2006. [KPW00] R. W. Kenyon, J. G. Propp, and D. B. Wilson. Trees and matchings. Electron. J. Combin, 7(1):R25, 2000. [KS05] R. Kenyon and J.-M. Schlenker. Rhombic embeddings of planar quad-graphs. Trans. Amer. Math. Soc., 357(9):3443–3458 (electronic), 2005. [Kup98] G. Kuperberg. An exploration of the permanent-determinant method. Electron. J. Combin., 5(1):R46, 1998. [KW41a] H. A. Kramers and G. H. Wannier. Statistics of the two-dimensional ferromagnet. Part I. Phys. Rev., 60(3):252–262, Aug 1941. [KW41b] H. A. Kramers and G. H. Wannier. Statistics of the two-dimensional ferromagnet. Part II. Phys. Rev., 60(3):263–276, Aug 1941. 103

label: reference

text: C.

text: Fan

text: and

text: F.

text: Y.

text: Wu.

text: General

text: lattice

text: model

text: of

text: phase

text: transitions.

text: Phys.

text: Rev.

text: B,

text: P.

text: W.

text: Kasteleyn.

text: The

text: statistics

text: of

text: dimers

text: on

text: a

text: lattice

text: I.

text: the

text: number

text: of

text: dimer

text: arrangements

text: on

text: a

text: quadratic

text: lattice.

text: Physica,

text: December

text: P.

text: W.

text: Kasteleyn.

text: Graph

text: theory

text: and

text: crystal

text: physics.

text: In

text: Graph

text: Theory

text: and

text: Theoretical

text: Physics,

text: pages

text: Academic

text: Press,

text: London,

text: L.

text: P.

text: Kadanoﬀ

text: and

text: A.

text: C.

text: Brown.

text: Correlation

text: functions

text: on

text: the

text: critical

text: lines

text: of

text: the

text: Baxter

text: and

text: Ashkin-Teller

text: models.

text: Ann.

text: Phys.,

text: L.

text: P.

text: Kadanoﬀ

text: and

text: H.

text: Ceva.

text: Determination

text: of

text: an

text: operator

text: algebra

text: for

text: the

text: two-

text: dimensional

text: Ising

text: model.

text: Phys.

text: Rev.

text: B,

text: Jun

text: A.

text: E.

text: Kennelly.

text: The

text: equivalence

text: of

text: triangles

text: and

text: three-pointed

text: stars

text: in

text: conducting

text: networks.

text: Electrical

text: World

text: and

text: Engineer,

text: R.

text: Kenyon.

text: Local

text: statistics

text: of

text: lattice

text: dimers.

text: Ann.

text: Inst.

text: H.

text: Poincar´e

text: Probab.

text: Statist.,

text: R.

text: Kenyon.

text: The

text: Laplacian

text: and

text: Dirac

text: operators

text: on

text: critical

text: planar

text: graphs.

text: Invent.

text: Math.,

text: R.

text: Kenyon.

text: Determinantal

text: spanning

text: forests

text: on

text: planar

text: graphs.

text: G.

text: Kirchhoﬀ.

text: Ueber

text: die

text: auﬂ¨osung

text: der

text: gleichungen,

text: auf

text: welche

text: man

text: bei

text: der

text: un-

text: tersuchung

text: der

text: linearen

text: vertheilung

text: galvanischer

text: str¨ome

text: gefu¨hrt

text: wird.

text: Annalen

text: der

text: Physik,

text: R.

text: Kenyon,

text: A.

text: Okounkov,

text: and

text: S.

text: Sheﬃeld.

text: Dimers

text: and

text: amoebae.

text: Ann.

text: of

text: Math.

text: R.

text: W.

text: Kenyon,

text: J.

text: G.

text: Propp,

text: and

text: D.

text: B.

text: Wilson.

text: Trees

text: and

text: matchings.

text: Electron.

text: J.

text: Combin,

text: R.

text: Kenyon

text: and

text: J.-M.

text: Schlenker.

text: Rhombic

text: embeddings

text: of

text: planar

text: quad-graphs.

text: Trans.

text: Amer.

text: Math.

text: Soc.,

text: (electronic),

text: G.

text: Kuperberg.

text: An

text: exploration

text: of

text: the

text: permanent-determinant

text: method.

text: Electron.

text: J.

text: Combin.,

text: H.

text: A.

text: Kramers

text: and

text: G.

text: H.

text: Wannier.

text: Statistics

text: of

text: the

text: two-dimensional

text: ferromagnet.

text: Part

text: I.

text: Phys.

text: Rev.,

text: Aug

text: H.

text: A.

text: Kramers

text: and

text: G.

text: H.

text: Wannier.

text: Statistics

text: of

text: the

text: two-dimensional

text: ferromagnet.

text: Part

text: II.

text: Phys.

text: Rev.,

text: Aug


text: [Sun16] W. Sun. Toroidal dimer model and temperley’s bijection. ArXiv:1603.00690, 2016. [Tem74] H. N. V. Temperley. In Combinatorics: Proceedings of the British combinatorial conference 1973. page 202–204, 1974. [TF61] H. N. V. Temperley and M. E. Fisher. Dimer problem in statistical mechanics-an exact result. Philosophical Magazine, 6(68):1061–1063, 1961. [Tut48] W. T. Tutte. The dissection of equilateral triangles into equilateral triangles. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 44, pages 463–482. Cambridge Univ Press, 1948. [Wil11] D. B. Wilson. XOR-Ising loops and the Gaussian free ﬁeld. ArXiv: 1102.3782, 2011. [WL75] F. Y. Wu and K. Y. Lin. Staggered ice-rule vertex model - The Pfaﬃan solution. Phys. Rev. B, 12:419–428, Jul 1975. [Wu71] F. W. Wu. Ising model with four-spin interactions. Phys. Rev. B, 4:2312–2314, Oct 1971. 105

text: W.

text: Sun.

text: Toroidal

text: dimer

text: model

text: and

text: temperley’s

text: bijection.

text: H.

text: N.

text: V.

text: Temperley.

text: In

text: Combinatorics:

text: Proceedings

text: of

text: the

text: British

text: combinatorial

text: conference

text: page

text: H.

text: N.

text: V.

text: Temperley

text: and

text: M.

text: E.

text: Fisher.

text: Dimer

text: problem

text: in

text: statistical

text: mechanics-an

text: exact

text: result.

text: Philosophical

text: Magazine,

text: W.

text: T.

text: Tutte.

text: The

text: dissection

text: of

text: equilateral

text: triangles

text: into

text: equilateral

text: triangles.

text: In

text: Mathematical

text: Proceedings

text: of

text: the

text: Cambridge

text: Philosophical

text: Society,

text: volume

text: pages

text: Cambridge

text: Univ

text: Press,

text: D.

text: B.

text: Wilson.

text: XOR-Ising

text: loops

text: and

text: the

text: Gaussian

text: free

text: ﬁeld.

text: ArXiv:

text: F.

text: Y.

text: Wu

text: and

text: K.

text: Y.

text: Lin.

text: Staggered

text: ice-rule

text: vertex

text: model

text: The

text: Pfaﬃan

text: solution.

text: Phys.

text: Rev.

text: B,

text: Jul

text: F.

text: W.

text: Wu.

text: Ising

text: model

text: with

text: four-spin

text: interactions.

text: Phys.

text: Rev.

text: B,

text: Oct


text: operator and also the Z-massive Laplacian of [BdTR17b]. The strength of these identities is that they allow to reach inverse operators and also, after some extra work, determinants. − As a consequence, in the ﬁnite and inﬁnite cases, we express coeﬃcients of (KF) 1 using the inverse Zu-Dirac operator and also using the Z-massive Green function; this is the subject of Section , see also the corresponding part of the introduction. In essence, this proves that the contour Ising Boltzmann/Gibbs measures can be computed from (massive) random walks (with speciﬁc boundary conditions in the ﬁnite case). In the ﬁnite case we also prove that the squared Ising partition function is equal, up to an explicit constant, to the determinant of the massive Laplacian, that is to the partition function of rooted spanning forests, see Corol- lary 38 and also Theorem 7 of the introduction. Comments on how these results connect to our “inspiration” and other papers are given at the end of this section. Section contains preliminaries. Section contains the main intermediate step: we consider the dimer model on the bipartite graph GQ arising from the XOR-Ising model [Wil11] constructed from two independent Z-invariant Ising models [Dub11, BdT14]. We prove matrix identities relating its Kasteleyn matrix/operator KQ and the Zu-Dirac operator. In Section , building − on the work of Dub´edat [Dub11], we express coeﬃcients of the inverse operator (KF) 1 using − coeﬃcients of the inverse operator (KQ) 1; this result holds for the dimer model on the Fisher graph GF arising from any 2d-Ising model, not necessarily Z-invariant; combining this with the results of Section then allows us to deduce the Ising results. In Section , we specify some of our results in two important cases: the Z-invariant critical case, and the full Z-invariant Z case when the underlying graph is 2. To give detailed statements, let us be more precise about Z-invariant models [Ons44, Ken99], fully developed by Baxter [Bax78, Bax86, Bax89], see also [PAY06, AP87, AP02]. A Z- invariant model is naturally deﬁned on an isoradial graph G = (V, E); parameters are chosen so that the partition function only changes by a constant when performing a star-triangle transformation of the underlying graph, i.e., they are required to satisfy the Yang-Baxter equations. The solution to this set of equations for the Ising model has, given the embedding (cid:48) − ∈ ∞ of the graph, a free elliptic parameter k, such th(cid:16)at (k )2 := 1 (cid:17)k2 (0, ), and the coupling constants J are [Bax89]: | ∀ ∈ 1 1 + sn(θ k) e E, J = ln | e , e 2 cn(θ k) e where sn, cn are two of the Jacobi elliptic trigonometric functions, and θ¯ = θ π is an e e 2K(cid:48) angle associated to the edge e in the isoradial embedding. When k = 0, i.e. k = 1, the elliptic functions sn, cn, are the trigonometric functions sin, cos, and the Ising model is (cid:48) ∞ critical [Li12, CD13, Lis14b]. As (k )2 varies from 0 to , the coupling constants range ∞ from to 0 [BdTR17a] thus covering the whole range of inverse temperatures. In the paper [BdTR17b], we introduce Z-invariant rooted spanning forests with associated operator the massive Laplacian ∆m; when k = 0, we recover the critical Laplacian of [Ken02]. We also prove an explicit local expression for its inverse, the Z-massive Green function Gm, using the discrete massive exponential function [BdTR17b]. We are now ready to give a detailed overview of this paper. 3 ##LTLine## ##LTLine## ##LTLine##

text: operator

text: and

text: also

text: the

text: Z-massive

text: Laplacian

text: of

text: The

text: strength

text: of

text: these

text: identities

text: is

text: that

text: they

text: allow

text: to

text: reach

text: inverse

text: operators

text: and

text: also,

text: after

text: some

text: extra

text: work,

text: determinants.

text: As

text: a

text: consequence,

text: in

text: the

text: ﬁnite

text: and

text: inﬁnite

text: cases,

text: we

text: express

text: coeﬃcients

text: of

text: (KF)

text: using

text: the

text: inverse

text: Zu-Dirac

text: operator

text: and

text: also

text: using

text: the

text: Z-massive

text: Green

text: function;

text: this

text: is

text: the

text: subject

text: of

text: Section

text: see

text: also

text: the

text: corresponding

text: part

text: of

text: the

text: introduction.

text: In

text: essence,

text: this

text: proves

text: that

text: the

text: contour

text: Ising

text: Boltzmann/Gibbs

text: measures

text: can

text: be

text: computed

text: from

text: (massive)

text: random

text: walks

text: (with

text: speciﬁc

text: boundary

text: conditions

text: in

text: the

text: ﬁnite

text: case).

text: In

text: the

text: ﬁnite

text: case

text: we

text: also

text: prove

text: that

text: the

text: squared

text: Ising

text: partition

text: function

text: is

text: equal,

text: up

text: to

text: an

text: explicit

text: constant,

text: to

text: the

text: determinant

text: of

text: the

text: massive

text: Laplacian,

text: that

text: is

text: to

text: the

text: partition

text: function

text: of

text: rooted

text: spanning

text: forests,

text: see

text: Corol-

text: lary

text: and

text: also

text: Theorem

text: of

text: the

text: introduction.

text: Comments

text: on

text: how

text: these

text: results

text: connect

text: to

text: our

text: “inspiration”

text: and

text: other

text: papers

text: are

text: given

text: at

text: the

text: end

text: of

text: this

text: section.

text: Section

text: contains

text: preliminaries.

text: Section

text: contains

text: the

text: main

text: intermediate

text: step:

text: we

text: consider

text: the

text: dimer

text: model

text: on

text: the

text: bipartite

text: graph

text: GQ

text: arising

text: from

text: the

text: XOR-Ising

text: model

text: constructed

text: from

text: two

text: independent

text: Z-invariant

text: Ising

text: models

text: We

text: prove

text: matrix

text: identities

text: relating

text: its

text: Kasteleyn

text: matrix/operator

text: KQ

text: and

text: the

text: Zu-Dirac

text: operator.

text: In

text: Section

text: building

text: on

text: the

text: work

text: of

text: Dub´edat

text: we

text: express

text: coeﬃcients

text: of

text: the

text: inverse

text: operator

text: (KF)

text: using

text: coeﬃcients

text: of

text: the

text: inverse

text: operator

text: (KQ)

text: this

text: result

text: holds

text: for

text: the

text: dimer

text: model

text: on

text: the

text: Fisher

text: graph

text: GF

text: arising

text: from

text: any

text: model,

text: not

text: necessarily

text: Z-invariant;

text: combining

text: this

text: with

text: the

text: results

text: of

text: Section

text: then

text: allows

text: us

text: to

text: deduce

text: the

text: Ising

text: results.

text: In

text: Section

text: we

text: specify

text: some

text: of

text: our

text: results

text: in

text: two

text: important

text: cases:

text: the

text: Z-invariant

text: critical

text: case,

text: and

text: the

text: full

text: Z-invariant

text: Z

text: case

text: when

text: the

text: underlying

text: graph

text: is

text: To

text: give

text: detailed

text: statements,

text: let

text: us

text: be

text: more

text: precise

text: about

text: Z-invariant

text: models

text: fully

text: developed

text: by

text: Baxter

text: see

text: also

text: A

text: Z-

text: invariant

text: model

text: is

text: naturally

text: deﬁned

text: on

text: an

text: isoradial

text: graph

text: G

text: (V,

text: E);

text: parameters

text: are

text: chosen

text: so

text: that

text: the

text: partition

text: function

text: only

text: changes

text: by

text: a

text: constant

text: when

text: performing

text: a

text: star-triangle

text: transformation

text: of

text: the

text: underlying

text: graph,

text: i.e.,

text: they

text: are

text: required

text: to

text: satisfy

text: the

text: Yang-Baxter

text: equations.

text: The

text: solution

text: to

text: this

text: set

text: of

text: equations

text: for

text: the

text: Ising

text: model

text: has,

text: given

text: the

text: embedding

text: of

text: the

text: graph,

text: a

text: free

text: elliptic

text: parameter

text: k,

text: such

text: (k

text: and

text: the

text: coupling

text: constants

text: J

text: are

text: sn(θ

text: k)

text: e

text: E,

text: J

text: ln

text: e

text: e

text: cn(θ

text: k)

text: e

text: where

text: sn,

text: cn

text: are

text: two

text: of

text: the

text: Jacobi

text: elliptic

text: trigonometric

text: functions,

text: and

text: θ¯

text: θ

text: π

text: is

text: an

text: e

text: e

text: angle

text: associated

text: to

text: the

text: edge

text: e

text: in

text: the

text: isoradial

text: embedding.

text: When

text: k

text: i.e.

text: k

text: the

text: elliptic

text: functions

text: sn,

text: cn,

text: are

text: the

text: trigonometric

text: functions

text: sin,

text: cos,

text: and

text: the

text: Ising

text: model

text: is

text: critical

text: As

text: (k

text: varies

text: from

text: to

text: the

text: coupling

text: constants

text: range

text: from

text: to

text: thus

text: covering

text: the

text: whole

text: range

text: of

text: inverse

text: temperatures.

text: In

text: the

text: paper

text: we

text: introduce

text: Z-invariant

text: rooted

text: spanning

text: forests

text: with

text: associated

text: operator

text: the

text: massive

text: Laplacian

text: ∆m;

text: when

text: k

text: we

text: recover

text: the

text: critical

text: Laplacian

text: of

text: We

text: also

text: prove

text: an

text: explicit

text: local

text: expression

text: for

text: its

text: inverse,

text: the

text: Z-massive

text: Green

text: function

text: Gm,

text: using

text: the

text: discrete

text: massive

text: exponential

text: function

text: We

text: are

text: now

text: ready

text: to

text: give

text: a

text: detailed

text: overview

text: of

text: this

text: paper.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:48) 2eiβ¯e v θ¯e e v 2eiα¯e Figure 5: Left: original isoradial graph (black lines) with circumcircles (grey) and dual vertices embedded as circumcenters (diamonds). Right: isoradial graph G (black lines) with (cid:5) the additional boundary vertices (grey bullets); diamond graph G (grey lines); rhombus (cid:48) (yellow) assigned to an edge e = (v, v ) with the corresponding half-angle θ¯ and rhombus e vectors 2eiα¯e, 2eiβ¯e; a boundary rhombus pair of R∂ (light grey). dual vertex is joined to all vertices bounding the face it corresponds to. Since the graph G is (cid:5) isoradial, faces of the diamond graph G are side-length-2 rhombi. (cid:5) ∗ There is a bijection between rhombi of G and pairs e, e of primal and dual edges, the latter ∈ being the two diagonals of the rhombi. To every edge e, one assigns an angle θ¯ (0, π) e 2 deﬁned to be the half-angle of the corresponding rhombus at the edge e. We furthermore ask ∈ − ∗ − ∗ that θ¯ (ε, π ε), for some ε > 0. The rhombus angle of the dual edge e is θ¯ ∗ = π θ¯ := θ¯ . e 2 (cid:48) e 2 e e(cid:5) To a directed edge e = (v, v ) of G we further assign two rhombus vectors 2eiα¯e, 2eiβ¯e of G , (cid:48) such that 2eiα¯e is on t−he right of the edge (v, v ), see Figure 5 (right). The angles α¯e and β¯e are deﬁned so that β¯e α¯e = θ¯ . Whenever no confusion occurs, we remove the subscript e 2 e (cid:5) from the notation. In the ﬁnite case, the diamond graph, also denoted G , is constructed in a ∗ similar way from G and its restricted dual G . One then adds the missing half-rhombi along the boundary; they may overlap but this causes no problem, see Figure 5 (right). Angles and rhombus vectors assigned to edges are deﬁned in the same way. Because of the additional vertex on each boundary edge and because of our choice of em- (cid:5) bedding, rhombi along the boundary of G come in pairs, both having the same rhombus half-angle; let us denote by R∂ the set of boundary rhombus pairs, an instance is highlighted 21 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: v

text: θ¯e

text: e

text: v

text: Figure

text: Left:

text: original

text: isoradial

text: graph

text: (black

text: lines)

text: with

text: circumcircles

text: (grey)

text: and

text: dual

text: vertices

text: embedded

text: as

text: circumcenters

text: (diamonds).

text: Right:

text: isoradial

text: graph

text: G

text: (black

text: lines)

text: with

text: the

text: additional

text: boundary

text: vertices

text: (grey

text: bullets);

text: diamond

text: graph

text: G

text: (grey

text: lines);

text: rhombus

text: (yellow)

text: assigned

text: to

text: an

text: edge

text: e

text: (v,

text: v

text: with

text: the

text: corresponding

text: half-angle

text: θ¯

text: and

text: rhombus

text: e

text: vectors

text: a

text: boundary

text: rhombus

text: pair

text: of

text: R∂

text: (light

text: grey).

text: dual

text: vertex

text: is

text: joined

text: to

text: all

text: vertices

text: bounding

text: the

text: face

text: it

text: corresponds

text: to.

text: Since

text: the

text: graph

text: G

text: is

text: isoradial,

text: faces

text: of

text: the

text: diamond

text: graph

text: G

text: are

text: rhombi.

text: There

text: is

text: a

text: bijection

text: between

text: rhombi

text: of

text: G

text: and

text: pairs

text: e,

text: e

text: of

text: primal

text: and

text: dual

text: edges,

text: the

text: latter

text: being

text: the

text: two

text: diagonals

text: of

text: the

text: rhombi.

text: To

text: every

text: edge

text: e,

text: one

text: assigns

text: an

text: angle

text: θ¯

text: π)

text: e

text: deﬁned

text: to

text: be

text: the

text: half-angle

text: of

text: the

text: corresponding

text: rhombus

text: at

text: the

text: edge

text: e.

text: We

text: furthermore

text: ask

text: that

text: θ¯

text: (ε,

text: π

text: ε),

text: for

text: some

text: ε

text: The

text: rhombus

text: angle

text: of

text: the

text: dual

text: edge

text: e

text: is

text: θ¯

text: π

text: θ¯

text: θ¯

text: e

text: e

text: e

text: To

text: a

text: directed

text: edge

text: e

text: (v,

text: v

text: of

text: G

text: we

text: further

text: assign

text: two

text: rhombus

text: vectors

text: of

text: G

text: such

text: that

text: is

text: on

text: t−he

text: right

text: of

text: the

text: edge

text: (v,

text: v

text: see

text: Figure

text: (right).

text: The

text: angles

text: α¯e

text: and

text: β¯e

text: are

text: deﬁned

text: so

text: that

text: β¯e

text: α¯e

text: θ¯

text: Whenever

text: no

text: confusion

text: occurs,

text: we

text: remove

text: the

text: subscript

text: e

text: e

text: from

text: the

text: notation.

text: In

text: the

text: ﬁnite

text: case,

text: the

text: diamond

text: graph,

text: also

text: denoted

text: G

text: is

text: constructed

text: in

text: a

text: similar

text: way

text: from

text: G

text: and

text: its

text: restricted

text: dual

text: G

text: One

text: then

text: adds

text: the

text: missing

text: half-rhombi

text: along

text: the

text: boundary;

text: they

text: may

text: overlap

text: but

text: this

text: causes

text: no

text: problem,

text: see

text: Figure

text: (right).

text: Angles

text: and

text: rhombus

text: vectors

text: assigned

text: to

text: edges

text: are

text: deﬁned

text: in

text: the

text: same

text: way.

text: Because

text: of

text: the

text: additional

text: vertex

text: on

text: each

text: boundary

text: edge

text: and

text: because

text: of

text: our

text: choice

text: of

text: em-

text: bedding,

text: rhombi

text: along

text: the

text: boundary

text: of

text: G

text: come

text: in

text: pairs,

text: both

text: having

text: the

text: same

text: rhombus

text: half-angle;

text: let

text: us

text: denote

text: by

text: R∂

text: the

text: set

text: of

text: boundary

text: rhombus

text: pairs,

text: an

text: instance

text: is

text: highlighted

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##


text: Lemma 23. Consider a perfect matching M of GD,r. The bipartite, weighted adjacency 0 matrices Kg(u) and K(u)(cid:16)a(cid:89)re gauge equ(cid:17)i(cid:16)valen(cid:89)t and we have, (cid:17) | | | | det K(u) = sc(θw)12 [dn(uαe) dn(uβe)]12 det Kg(u) . (21) ∈ ∈ w W e=wx M0 Proof. Details on gauge equivalences for bipartite, weighted adjacency matrices are given in Section of Appendix . To prove that K and Kg are gauge equivalent, it suﬃces to show that the alternating products of K and Kg around inner face-cycles of GD,r are equal. Inner face-cycles of GD,r are quadrangles; using the notation of Figure 11 we have, Kgw,vKgw(cid:48),f Kw,vKw(cid:48),f ⇔ gw,v gw,f = = 1. Kg Kg (cid:48) K K (cid:48) g (cid:48) g (cid:48) w,f w ,v w,f w ,v w ,f w ,v By deﬁnition of g we have, (cid:48) g g (cid:48) cs(θ) nd(u ) nd(u ) cs(θ ) nd(u (cid:48)) nd(u (cid:48) ) w,v w ,f α β β α +2K = (cid:48) = 1, g g (cid:48) cs(θ) nd(u − ) nd(u ) cs(θ ) nd(u (cid:48)) nd(u (cid:48)) w,f w ,v (cid:81) β 2K α (cid:81) α β (cid:48) since α¯ = β¯[2π]. The equality between determinants comes from [Kup98], see also Corol- lary 67, and from the fact that, ∈ sc(θ ) = ∈ sc(θ ), since a dimer conﬁguration wx M0 w w W w covers all white vertices W r of GD,r, and W r = W . Now, the matrix Kg(u) is a co(cid:40)mplex, bipartite Kasteleyn matrix of the graph GD,r, where edges are assigned the positive weight function cg(u) given by: ∈ 1 if x Vr cg(u)w,x = (cid:48) 1 ∈ ∗ (k )2 cs(θw) nd(uαe) nd(uβe) if x V . Returning to Section , the weight function cg(u) determines a weight function c˜g(u) on ∗ (cid:48) directed edges of G, G¯ , see (3). Then, c˜g(u) (cid:48) = 1, for all directed edges (v, v ) of G such v,v that v is a vertex of Gr. Let us now express the weight function c˜g(u) on directed edges of ∗ G¯ using the rhombus vectors and half-angles assigned to those edges. Recall that an edge (cid:48) ∗ (f, f ) of the restricted dual G is assigned two rhombus vectors 2eiα¯, 2eiβ¯ and a half-angle ∗ (cid:5) ∗ θ¯ of G . An edge (f, o) of G¯ corresponds to an edge fw of GD, and one assigns to (f, o) ∗ (cid:5) the rhombus vectors eiα¯, eiβ¯ and half-angle θ¯ of G /2 associated to the edge (f, w). Then, (cid:48) ∗ ∗ for every directed edge (f, f ) of G¯ such that f is a vertex of G we have, using the notation of (3): (cid:48) c˜g(u)f,f(cid:48) = cg(u)w,f = (k )12 cs(θw) nd(uαe) nd(uβe) (cid:48) − ∗ 1 = (k ) 2 sc(θ ) dn(uα) dn(uβ), (22) 39 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Lemma

text: Consider

text: a

text: perfect

text: matching

text: M

text: of

text: GD,r.

text: The

text: bipartite,

text: weighted

text: adjacency

text: matrices

text: Kg(u)

text: and

label: equation

text: gauge

text: and

text: we

text: have,

text: det

text: K(u)

text: [dn(uαe)

text: det

text: Kg(u)

text: w

text: W

text: e=wx

text: Proof.

text: Details

text: on

text: gauge

text: equivalences

text: for

text: bipartite,

text: weighted

text: adjacency

text: matrices

text: are

text: given

text: in

text: Section

text: of

text: Appendix

text: To

text: prove

text: that

text: K

text: and

text: Kg

text: are

text: gauge

text: equivalent,

text: it

text: suﬃces

text: to

text: show

text: that

text: the

text: alternating

text: products

text: of

text: K

text: and

text: Kg

text: around

text: inner

text: face-cycles

text: of

text: GD,r

text: are

text: equal.

text: Inner

text: face-cycles

text: of

text: GD,r

text: are

text: quadrangles;

text: using

text: the

text: notation

text: of

text: Figure

text: we

text: have,

text: gw,v

text: gw,f

text: Kg

text: Kg

text: K

text: K

text: g

text: g

text: w,f

text: w

text: ,v

text: w,f

text: w

text: ,v

text: w

text: ,f

text: w

text: ,v

text: By

text: deﬁnition

text: of

text: g

text: we

text: have,

text: g

text: g

text: cs(θ)

text: nd(u

text: nd(u

text: cs(θ

text: nd(u

text: nd(u

text: w,v

text: w

text: ,f

text: α

text: β

text: β

text: α

text: g

text: g

text: cs(θ)

text: nd(u

text: nd(u

text: cs(θ

text: nd(u

text: nd(u

text: w,f

text: w

text: ,v

text: β

text: α

text: α

text: β

text: since

text: α¯

text: The

text: equality

text: between

text: determinants

text: comes

text: from

text: see

text: also

text: Corol-

text: lary

text: and

text: from

text: the

text: fact

text: that,

text: sc(θ

text: sc(θ

text: since

text: a

text: dimer

text: conﬁguration

text: wx

text: w

text: w

text: W

text: w

text: covers

text: all

text: white

text: vertices

text: W

text: r

text: of

text: GD,r,

text: and

text: W

text: r

text: W

text: Now,

text: the

text: matrix

text: Kg(u)

text: is

text: a

text: bipartite

text: Kasteleyn

text: matrix

text: of

text: the

text: graph

text: GD,r,

text: where

text: edges

text: are

text: assigned

text: the

text: positive

text: weight

text: function

text: cg(u)

text: given

text: by:

text: if

text: x

text: Vr

text: cg(u)w,x

text: (k

text: cs(θw)

text: nd(uαe)

text: nd(uβe)

text: if

text: x

text: V

text: Returning

text: to

text: Section

text: the

text: weight

text: function

text: cg(u)

text: determines

text: a

text: weight

text: function

text: c˜g(u)

text: on

text: directed

text: edges

text: of

text: G,

text: G¯

text: see

text: Then,

text: c˜g(u)

text: for

text: all

text: directed

text: edges

text: (v,

text: v

text: of

text: G

text: such

text: v,v

text: that

text: v

text: is

text: a

text: vertex

text: of

text: Gr.

text: Let

text: us

text: now

text: express

text: the

text: weight

text: function

text: c˜g(u)

text: on

text: directed

text: edges

text: of

text: G¯

text: using

text: the

text: rhombus

text: vectors

text: and

text: half-angles

text: assigned

text: to

text: those

text: edges.

text: Recall

text: that

text: an

text: edge

text: (f,

text: f

text: of

text: the

text: restricted

text: dual

text: G

text: is

text: assigned

text: two

text: rhombus

text: vectors

text: and

text: a

text: half-angle

text: θ¯

text: of

text: G

text: An

text: edge

text: (f,

text: o)

text: of

text: G¯

text: corresponds

text: to

text: an

text: edge

text: fw

text: of

text: GD,

text: and

text: one

text: assigns

text: to

text: (f,

text: o)

text: the

text: rhombus

text: vectors

text: eiα¯,

text: eiβ¯

text: and

text: half-angle

text: θ¯

text: of

text: G

text: associated

text: to

text: the

text: edge

text: (f,

text: w).

text: Then,

text: for

text: every

text: directed

text: edge

text: (f,

text: f

text: of

text: G¯

text: such

text: that

text: f

text: is

text: a

text: vertex

text: of

text: G

text: we

text: have,

text: using

text: the

text: notation

text: of

text: cg(u)w,f

text: (k

text: cs(θw)

text: nd(uαe)

text: nd(uβe)

text: (k

text: sc(θ

text: dn(uα)

text: dn(uβ),

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: This gives, in Theorem 31, an explicit local expression for a Gibbs measure of the dimer model on the double graph GD, where the locality property is inherited from that of the Z-massive Green functions of [BdTR17b]. Using the KPW-Temperley bijection [Tem74, KPW00], probabilities of pairs of dual directed spanning trees are computed using the Green function of a massive, non-directed random walk. Apart from the locality property which is Z speciﬁc, a similar result is obtained by Chhita [Chh12] in the case of 2 with a speciﬁc choice of weights. In Theorem 20 and Corollary 22, we restrict to the ﬁnite case and prove relations on deter- minants; we show, (cid:16) (cid:89) (cid:17)(cid:16) (cid:89) (cid:17) ∈ (cid:60) T (cid:48) Theorem 3. Let M be a dimer conﬁguration of GD,r. Then, for every u ( (k)) , we have 0 | | (cid:48) | ∗| ∗ det K(u) = (k ) V2 (cid:16) (cid:89)sc(θw)12 (cid:17)(cid:16) (cid:89) [dn(uαe) dn(uβe)]12 de(cid:17)t ∆m, , ∈ ∈ w W e=wx M0 | | (cid:48) | |− (cid:48) det K∂(u) = (k ) V2 1 cs(θw)12 [k nd(uαe) nd(uβe)]12 det ∆m,∂(u). ∈ ∈ w W e=wx M0 As a consequence, the partition function of pairs of dual directed spanning trees is equal, up to a constant, to the partition function of rooted spanning forests. In the critical case, k = 0, this is an easy consequence of Temperley’s bijection [Tem74], but the correspondence (cid:54) does not extend when k = 0. The main tools of the proof are gauge equivalences on bipartite adjacency matrices and on adjacency matrices of digraphs, see also Appendix . The Zu-Dirac operator is equivalent to a model of directed spanning trees. In Proposition 25, we prove that the latter is Z-invariant, thus explaining the part “Zu” of the terminology “Zu- Dirac operator”. Section : Kasteleyn operator of the graph GQ and Zu-Dirac operator. We con- sider the dimer model on the graph GQ arising from the Z-invariant XOR-Ising model, with Kasteleyn matrix KQ. The main result of this section is Theorem 35 proving the following relations between the matrix KQ and the Zu-Dirac operator. The matrices S(u) and T (u) are deﬁned in Section and the statement is as follows. Theorem 4. • ∈ (cid:60) T Inﬁnite case. Let u ( (k)), then the Kasteleyn matrix KQ, the Zu-Dirac operator K(u) and the matrices S(u), T (u) are related by the following identity: KQ T (u) = S(u) K(u). • ∈ (cid:60) T (cid:48) Finite case. Let u ( (k)) , then the Kasteleyn matrix KQ, the Zu-Dirac operator K∂(u) and the matrices S(u), T (u) are related by the following identity: KQ T (u) = S(u) K∂(u). 5 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: This

text: gives,

text: in

text: Theorem

text: an

text: explicit

text: local

text: expression

text: for

text: a

text: Gibbs

text: measure

text: of

text: the

text: dimer

text: model

text: on

text: the

text: double

text: graph

text: GD,

text: where

text: the

text: locality

text: property

text: is

text: inherited

text: from

text: that

text: of

text: the

text: Z-massive

text: Green

text: functions

text: of

text: Using

text: the

text: KPW-Temperley

text: bijection

text: probabilities

text: of

text: pairs

text: of

text: dual

text: directed

text: spanning

text: trees

text: are

text: computed

text: using

text: the

text: Green

text: function

text: of

text: a

text: massive,

text: non-directed

text: random

text: walk.

text: Apart

text: from

text: the

text: locality

text: property

text: which

text: is

text: Z

text: speciﬁc,

text: a

text: similar

text: result

text: is

text: obtained

text: by

text: Chhita

text: in

text: the

text: case

text: of

text: with

text: a

text: speciﬁc

text: choice

text: of

text: weights.

text: In

text: Theorem

text: and

text: Corollary

text: we

text: restrict

text: to

text: the

text: ﬁnite

text: case

text: and

text: prove

text: relations

text: on

text: deter-

text: minants;

text: we

text: show,

text: T

text: Theorem

text: Let

text: M

text: be

text: a

text: dimer

text: conﬁguration

text: of

text: GD,r.

text: Then,

text: for

text: every

text: u

text: (k))

text: we

text: have

text: det

text: K(u)

text: (k

text: [dn(uαe)

text: ∆m,

text: w

text: W

text: e=wx

text: det

text: K∂(u)

text: (k

text: [k

text: nd(uαe)

text: det

text: ∆m,∂(u).

text: w

text: W

text: e=wx

text: As

text: a

text: consequence,

text: the

text: partition

text: function

text: of

text: pairs

text: of

text: dual

text: directed

text: spanning

text: trees

text: is

text: equal,

text: up

text: to

text: a

text: constant,

text: to

text: the

text: partition

text: function

text: of

text: rooted

text: spanning

text: forests.

text: In

text: the

text: critical

text: case,

text: k

text: this

text: is

text: an

text: easy

text: consequence

text: of

text: Temperley’s

text: bijection

text: but

text: the

text: correspondence

text: does

text: not

text: extend

text: when

text: k

text: The

text: main

text: tools

text: of

text: the

text: proof

text: are

text: gauge

text: equivalences

text: on

text: bipartite

text: adjacency

text: matrices

text: and

text: on

text: adjacency

text: matrices

text: of

text: digraphs,

text: see

text: also

text: Appendix

text: The

text: Zu-Dirac

text: operator

text: is

text: equivalent

text: to

text: a

text: model

text: of

text: directed

text: spanning

text: trees.

text: In

text: Proposition

text: we

text: prove

text: that

text: the

text: latter

text: is

text: Z-invariant,

text: thus

text: explaining

text: the

text: part

text: “Zu”

text: of

text: the

text: terminology

text: “Zu-

text: Dirac

text: operator”.

text: Section

text: Kasteleyn

text: operator

text: of

text: the

text: graph

text: GQ

text: and

text: Zu-Dirac

text: operator.

text: We

text: con-

text: sider

text: the

text: dimer

text: model

text: on

text: the

text: graph

text: GQ

text: arising

text: from

text: the

text: Z-invariant

text: XOR-Ising

text: model,

text: with

text: Kasteleyn

text: matrix

text: KQ.

text: The

text: main

text: result

text: of

text: this

text: section

text: is

text: Theorem

text: proving

text: the

text: following

text: relations

text: between

text: the

text: matrix

text: KQ

text: and

text: the

text: Zu-Dirac

text: operator.

text: The

text: matrices

text: S(u)

text: and

text: T

text: (u)

text: are

text: deﬁned

text: in

text: Section

text: and

text: the

text: statement

text: is

text: as

text: follows.

text: Theorem

text: T

text: Inﬁnite

text: case.

text: Let

text: u

text: (k)),

text: then

text: the

text: Kasteleyn

text: matrix

text: KQ,

text: the

text: Zu-Dirac

text: operator

text: K(u)

text: and

text: the

text: matrices

text: S(u),

text: T

text: (u)

text: are

text: related

text: by

text: the

text: following

text: identity:

text: KQ

text: T

text: (u)

text: S(u)

text: K(u).

text: T

text: Finite

text: case.

text: Let

text: u

text: (k))

text: then

text: the

text: Kasteleyn

text: matrix

text: KQ,

text: the

text: Zu-Dirac

text: operator

text: K∂(u)

text: and

text: the

text: matrices

text: S(u),

text: T

text: (u)

text: are

text: related

text: by

text: the

text: following

text: identity:

text: KQ

text: T

text: (u)

text: S(u)

text: K∂(u).

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Remark 26. • By Theorem 20, we have: (cid:48) · ∗ ∗ ∗ Z (GD,r, c(u)) = C (u) Z (G , ρ , m ), dimer rSF and in the paper [BdTR17b], the model of rooted spanning forests with these weights is shown to be Z-invariant. But, since the proof of Theorem 20 does not provide a bijection between directed spanning trees and rooted spanning forests, the two decompositions of the partition functions are not directly comparable; they should nevertheless be compatible. Note that the computations of the proof of Proposition 25 are reminiscent but much simpler than those of [BdTR17b] (the latter have been removed from the published version). • The critical spanning tree model of [Ken02] with conductances tan(θ) is Z-invariant [Ken99]. The result of [BdTR17b] extends this to rooted spanning forest while Proposition 25 extends it to directed spanning trees. Proof. Let GY and G∆ be two ﬁnite isoradial graphs diﬀering by a star-triangle transformation, and let r be a ﬁxed root on the boundary of the graph, outside of the hexagon deﬁning the star/triangle. Let γ∆(u), resp. γY(u), be the weight function on r-dST of G∆, resp. GY. We use the notation of Figure 13, and write γ∆ /γY for the weight of the edge (v , v ). Using the (cid:48) ∗ i,j i,−j (cid:48) − i j identities dn(u + K) = k nd(u), sc(θ ) = sc(K θ) = (k ) 1 cs(θ), we have the following: for ∈ { } every j 1, 2, 3 , with cyclic notation for indices: (cid:48) − (cid:48) − γ0Y,j = (k(cid:48)) 12 sc(θj) dn(uαj) dn(uαj+1), γj∆,j+1 = (k(cid:48))−12 cs(θj−1) nd(uαj) dn(uαj−1), γjY,0 = (k )32 sc(θj) nd(uαj) nd(uαj+1), γj∆+1,j = (k ) 12 cs(θj−1) dn(uαj) nd(uαj−1). v2 v2 v3 eiα¯3 v3 θ¯1∗ eiα¯1 θ¯3∗ θ¯3 eiα¯2 θ¯2 eiα¯2 eiα¯3 eiα¯1 θ¯1 v0 ∗ θ¯ 2 v1 v1 Figure 13: Notation for vertices, angles and rhombus vectors in a star-triangle transformation. Consider a directed edge conﬁguration T outside of the hexagon which can be extended to an r-dST of GY/G∆. Then given T, the conﬁgurations inside the Y-∆ only depend on connection properties of T outside. As in the non-directed case [Ken99], we thus have three kinds of conﬁgurations to consider. 43 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Remark

label: list

text: By

text: Theorem

text: we

text: have:

text: Z

text: (GD,r,

text: c(u))

text: C

text: (u)

text: Z

text: (G

text: ρ

text: m

text: dimer

text: rSF

text: and

text: in

text: the

text: paper

text: the

text: model

text: of

text: rooted

text: spanning

text: forests

text: with

text: these

text: weights

text: is

text: shown

text: to

text: be

text: Z-invariant.

text: But,

text: since

text: the

text: proof

text: of

text: Theorem

text: does

text: not

text: provide

text: a

text: bijection

text: between

text: directed

text: spanning

text: trees

text: and

text: rooted

text: spanning

text: forests,

text: the

text: two

text: decompositions

text: of

text: the

text: partition

text: functions

text: are

text: not

text: directly

text: comparable;

text: they

text: should

text: nevertheless

text: be

text: compatible.

text: Note

text: that

text: the

text: computations

text: of

text: the

text: proof

text: of

text: Proposition

text: are

text: reminiscent

text: but

text: much

text: simpler

text: than

text: those

text: of

text: (the

text: latter

text: have

text: been

text: removed

text: from

text: the

text: published

text: version).

text: The

text: critical

text: spanning

text: tree

text: model

text: of

text: with

text: conductances

text: tan(θ)

text: is

text: Z-invariant

text: The

text: result

text: of

text: extends

text: this

text: to

text: rooted

text: spanning

text: forest

text: while

text: Proposition

text: extends

text: it

text: to

text: directed

text: spanning

text: trees.

text: Proof.

text: Let

text: GY

text: and

text: G∆

text: be

text: two

text: ﬁnite

text: isoradial

text: graphs

text: diﬀering

text: by

text: a

text: star-triangle

text: transformation,

text: and

text: let

text: r

text: be

text: a

text: ﬁxed

text: root

text: on

text: the

text: boundary

text: of

text: the

text: graph,

text: outside

text: of

text: the

text: hexagon

text: deﬁning

text: the

text: star/triangle.

text: Let

text: γ∆(u),

text: resp.

text: γY(u),

text: be

text: the

text: weight

text: function

text: on

text: r-dST

text: of

text: G∆,

text: resp.

text: GY.

text: We

text: use

text: the

text: notation

text: of

text: Figure

text: and

text: write

text: γ∆

text: /γY

text: for

text: the

text: weight

text: of

text: the

text: edge

text: (v

text: v

text: Using

text: the

text: i,j

text: i,−j

text: i

text: j

text: identities

text: dn(u

text: K)

text: k

text: nd(u),

text: sc(θ

text: sc(K

text: θ)

text: (k

text: cs(θ),

text: we

text: have

text: the

text: following:

text: for

text: every

text: j

text: with

text: cyclic

text: notation

text: for

text: indices:

text: sc(θj)

text: dn(uαj)

text: nd(uαj)

text: (k

text: sc(θj)

text: nd(uαj)

text: (k

text: dn(uαj)

text: θ¯1∗

text: θ¯3∗

text: θ¯3

text: θ¯2

text: θ¯1

text: θ¯

text: Figure

text: Notation

text: for

text: vertices,

text: angles

text: and

text: rhombus

text: vectors

text: in

text: a

text: star-triangle

text: transformation.

text: Consider

text: a

text: directed

text: edge

text: conﬁguration

text: T

text: outside

text: of

text: the

text: hexagon

text: which

text: can

text: be

text: extended

text: to

text: an

text: r-dST

text: of

text: GY/G∆.

text: Then

text: given

text: T,

text: the

text: conﬁgurations

text: inside

text: the

text: Y-∆

text: only

text: depend

text: on

text: connection

text: properties

text: of

text: T

text: outside.

text: As

text: in

text: the

text: non-directed

text: case

text: we

text: thus

text: have

text: three

text: kinds

text: of

text: conﬁgurations

text: to

text: consider.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ∀ ∈ { } Restriction III. We have, j 1, 2, 3 , | Z(GY III ) = γY γY γY j+1,0 j+2,0 0,j (cid:48) × × 5 = (k )2 sc(θj+1) nd(uαj+1) nd(uαj+2) × sc(θj+2) nd(uαj+2) nd(uαj) sc(θ ) dn(u ) dn(u ) j αj αj+1 (cid:48) = (k )52 sc(θj) sc(θj+1) sc(θj+2) nd2(uαj+2), | Z(G∆ III ) = γ∆ γ∆ + γ∆ γ∆ + γ∆ γ∆ j+1,j+2 j+2,j j+2,j+1 j+1,j j+1,j j+2,j (cid:48) − = (k ) 1[cs(θ ) nd(u ) dn(u ) cs(θ ) nd(u ) dn(u )+ j αj+1 αj j+1 αj+2 αj+1 + cs(θ ) dn(u ) nd(u ) cs(θ ) dn(u ) nd(u )+ j αj+1 αj j+2 αj αj+2 (cid:81) + cs(θ ) dn(u ) nd(u ) cs(θ ) nd(u ) dn(u )] (cid:48) − j+2 αj αj+2 j+1 αj+2 αj+1 (k ) 1 = nd2(u )[sc(θ ) dn(u ) dn(u )+ 3 sc(θ ) αj+2 j+2 αj+2 αj j=1 j + sc(θ ) dn(u ) dn(u ) + sc(θ ) dn(u ) dn(u )] (cid:48) j+1 αj+1 αj+2 j αj αj+1 = k nd2(u ), by the proof of Restriction I. αj+2 | | The proof that Z(GY III ) = C Z(G∆ III ) is concluded by using Equation (29) again. 3.6 Inverse Zu-Dirac operator and dimer model on the double graph − Using Theorem 18, in Corollaries 27 and 29, we express the inverse Zu-Dirac operators K(u) 1 − and K∂(u) 1 using the Z-massive and dual Z-massive Green functions. In the inﬁnite case, this allows to prove in Theorem 31 an explicit local expression for a Gibbs measure for the dimer model on the double graph with weight function c(u). 3.6.1 Inverse Zu-Dirac operator and Z-massive Green functions Inﬁnite case. Consider an inﬁnite isoradial graph G, and the Zu-Dirac operator K(u). Con- ∗ sider also the Z-massive Laplacian ∆m on G, the dual Z-massive Laplacian ∆m, of the dual ∗ (cid:54) ∗ G [BdTR17b]. When k = 0, let Gm and Gm, be the Z-massive and dual massive Green ∗ functions of G and G of [BdTR17b], whose deﬁnition is recalled in Section . When k = 0, ∗ the mass is 0 and we let G0 and G0, be the Green and dual Green functions of [Ken02]. ∗ Notation for coeﬃcients of Corollary 27. Let v, resp. f, be a vertex of G, resp. G . Let w be a white vertex of GD, its neig(cid:5)hbors in GD are¯v1, f1, v2¯, f2, and let eiα¯f, eiβ¯f, θ¯f be the rhombus vectors and half-angle of G /2 associated to the edge (w, v ), where the subscript 2 “f” stands for “ﬁnal”, see Figure 14. As a consequence of Theorem 18 (inﬁnite case) we obtain, 45 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Restriction

text: III.

text: We

text: have,

text: j

text: Z(GY

text: III

text: γY

text: γY

text: γY

text: (k

text: nd(uαj)

text: sc(θ

text: dn(u

text: dn(u

text: j

text: αj

text: (k

text: sc(θj)

text: Z(G∆

text: III

text: γ∆

text: γ∆

text: γ∆

text: γ∆

text: γ∆

text: γ∆

text: (k

text: nd(u

text: dn(u

text: cs(θ

text: nd(u

text: dn(u

text: j

text: αj

text: cs(θ

text: dn(u

text: nd(u

text: cs(θ

text: dn(u

text: nd(u

text: j

text: αj

text: αj

text: cs(θ

text: dn(u

text: nd(u

text: cs(θ

text: nd(u

text: dn(u

text: αj

text: (k

text: )[sc(θ

text: dn(u

text: dn(u

text: sc(θ

text: αj

text: j

text: sc(θ

text: dn(u

text: dn(u

text: sc(θ

text: dn(u

text: dn(u

text: j

text: αj

text: k

text: by

text: the

text: proof

text: of

text: Restriction

text: I.

text: The

text: proof

text: that

text: Z(GY

text: III

text: C

text: Z(G∆

text: III

text: is

text: concluded

text: by

text: using

text: Equation

text: again.

text: Inverse

text: Zu-Dirac

text: operator

text: and

text: dimer

text: model

text: on

text: the

text: double

text: graph

text: Using

text: Theorem

text: in

text: Corollaries

text: and

text: we

text: express

text: the

text: inverse

text: Zu-Dirac

text: operators

text: K(u)

text: and

text: K∂(u)

text: using

text: the

text: Z-massive

text: and

text: dual

text: Z-massive

text: Green

text: functions.

text: In

text: the

text: inﬁnite

text: case,

text: this

text: allows

text: to

text: prove

text: in

text: Theorem

text: an

text: explicit

text: local

text: expression

text: for

text: a

text: Gibbs

text: measure

text: for

text: the

text: dimer

text: model

text: on

text: the

text: double

text: graph

text: with

text: weight

text: function

text: c(u).

text: Inverse

text: Zu-Dirac

text: operator

text: and

text: Z-massive

text: Green

text: functions

text: Inﬁnite

text: case.

text: Consider

text: an

text: inﬁnite

text: isoradial

text: graph

text: G,

text: and

text: the

text: Zu-Dirac

text: operator

text: K(u).

text: Con-

text: sider

text: also

text: the

text: Z-massive

text: Laplacian

text: ∆m

text: on

text: G,

text: the

text: dual

text: Z-massive

text: Laplacian

text: ∆m,

text: of

text: the

text: dual

text: G

text: When

text: k

text: let

text: Gm

text: and

text: Gm,

text: be

text: the

text: Z-massive

text: and

text: dual

text: massive

text: Green

text: functions

text: of

text: G

text: and

text: G

text: of

text: whose

text: deﬁnition

text: is

text: recalled

text: in

text: Section

text: When

text: k

text: the

text: mass

text: is

text: and

text: we

text: let

text: and

text: be

text: the

text: Green

text: and

text: dual

text: Green

text: functions

text: of

text: Notation

text: for

text: coeﬃcients

text: of

text: Corollary

text: Let

text: v,

text: resp.

text: f,

text: be

text: a

text: vertex

text: of

text: G,

text: resp.

text: G

text: Let

text: w

text: be

text: a

text: white

text: vertex

text: of

text: GD,

text: its

text: in

text: GD

text: and

text: let

text: eiα¯f,

text: eiβ¯f,

text: θ¯f

text: be

text: the

text: rhombus

text: vectors

text: and

text: half-angle

text: of

text: G

text: associated

text: to

text: the

text: edge

text: (w,

text: v

text: where

text: the

text: subscript

text: “f”

text: stands

text: for

text: “ﬁnal”,

text: see

text: Figure

text: As

text: a

text: consequence

text: of

text: Theorem

text: (inﬁnite

text: case)

text: we

text: obtain,

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: In the inﬁnite case, Theorem 4 yields the following relations on inverse matrices, see also Corollary 45; the statement in the ﬁnite case is the subject of Corollary 46. ∈ (cid:60) T (cid:48) Corollary 5 (Inﬁnite case). For every u ( (k)), for every w, v, f, and every w, b, b as (cid:104) ¯ ¯ ¯ (cid:105) in Figure 20, (KQ)−1 cn(u )−i(KQ)−1 (cid:48) sn(u ) dn(u ) = eiβ¯f−2β¯i cn(u )K(u)−1 − i sn(u )K(u)−1 , w,b βf w,b βf αf Λ(u , u ) βi v,w βi f,w ¯ ¯ αf βf ¯ ¯ − 1 where Λ(uα, uβ) = [sn θ cn θ nd(uα) nd(uβ)]2 , and θ = uα uβ. Specifying the value of the parameter u allows to express coeﬃcients of the inverse Kaste- − leyn operator (KQ) 1 using the inverse Zu-Dirac operator; combining this with Theorem 2 yields an expression using the Z-massive Green function of [BdTR17b]. We obtain, see also Corollary 48 and Corollary 50 for the ﬁnite case, (cid:16) (cid:17) (cid:0) (cid:1) (cid:0) (cid:1) Corollary 6 (Inﬁnite case). For every w, b of GQ as in Figure 22, ¯ (KQ)−1 = eiβ¯f−2β¯i (cid:16) (cid:0) cn β(cid:1)f(cid:2)−βi K(β )−1 − i sn βf−(cid:3) βi K(cid:0)(β )−1(cid:1)(cid:2) (cid:3)(cid:17) w,b 1 2 f v,w 2 f f,w ¯ [cn(θf)sn(θf)nd(θf)]2 ¯ ¯ − − (KQ)−w¯,1b = e−iβ¯i+2α¯f (k(cid:48))−1 cncnβ(fθ2fβ)i dn(θf)G¯mv,v2 − k(cid:48)G¯mv,v1 − snsnβ(fθ2fβ)i dn(θf)G¯mf,f,∗2 − G¯mf,f,∗1 . This proves, in an alternative way, an explicit local expression for a Gibbs measure of the dimer model on the graph GQ [BdTR17a], where the locality property is seen as directly inherited from that of the Z-massive Green function. Using Theorem 4 and additional combinatorial arguments, we prove in Theorem 36 that the determinants of KQ and of the Zu-Dirac operator are equal, up to an explicit constant. By [Dub11], the determinant of KQ is equal up to a constant, to the squared partition function of the Ising model. Combining this with Theorem 3 gives, see also Corollary 38 for the explicit value of C(u), ∈ (cid:60) T (cid:48)(cid:48) Theorem 7. For every u ( (k)) , | | [Z+ (G, J)]2 = C(u) det ∆m,∂(u) . Ising When k = 0, we essentially recover the result of [dT16]; see Section . Section : Dimer model on the Fisher graph GF and the Kasteleyn matrix KQ. Con- sider the dimer model on GF with Kasteleyn matrix KF arising from an Ising model with cou- pling constants J, not necessarily Z-invariant, and the corresponding dimer model on the bi- partite graph GQ, with (real) Kasteleyn matrix K˜Q. Building on the work of Dub´edat [Dub11] − and proving additional matrix relations, we express coeﬃcients of the inverse operator (KF) 1 − ∪ using coeﬃcients of the inverse operator (K˜Q) 1. Partitioning vertices of GF as A B as in [Dub11], we obtain, see also Theorem 55, 6 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: In

text: the

text: inﬁnite

text: case,

text: Theorem

text: yields

text: the

text: following

text: relations

text: on

text: inverse

text: matrices,

text: see

text: also

text: Corollary

text: the

text: statement

text: in

text: the

text: ﬁnite

text: case

text: is

text: the

text: subject

text: of

text: Corollary

text: T

text: Corollary

text: (Inﬁnite

text: case).

text: For

text: every

text: u

text: (k)),

text: for

text: every

text: w,

text: v,

text: f,

text: and

text: every

text: w,

text: b,

text: b

text: as

text: in

text: Figure

text: cn(u

text: sn(u

text: dn(u

text: eiβ¯f−2β¯i

text: cn(u

text: i

text: sn(u

text: w,b

text: βf

text: w,b

text: βf

text: αf

text: Λ(u

text: u

text: βi

text: v,w

text: βi

text: f,w

text: αf

text: βf

text: where

text: Λ(uα,

text: uβ)

text: [sn

text: θ

text: cn

text: θ

text: nd(uα)

text: and

text: θ

text: uα

text: uβ.

text: Specifying

text: the

text: value

text: of

text: the

text: parameter

text: u

text: allows

text: to

text: express

text: coeﬃcients

text: of

text: the

text: inverse

text: Kaste-

text: leyn

text: operator

text: (KQ)

text: using

text: the

text: inverse

text: Zu-Dirac

text: operator;

text: combining

text: this

text: with

text: Theorem

text: yields

text: an

text: expression

text: using

text: the

text: Z-massive

text: Green

text: function

text: of

text: We

text: obtain,

text: see

text: also

text: Corollary

text: and

text: Corollary

text: for

text: the

text: ﬁnite

text: case,

text: Corollary

text: (Inﬁnite

text: case).

text: For

text: every

text: w,

text: b

text: of

text: GQ

text: as

text: in

text: Figure

text: eiβ¯f−2β¯i

text: cn

text: K(β

text: i

text: sn

text: βi

text: w,b

text: f

text: v,w

text: f

text: f,w

text: This

text: proves,

text: in

text: an

text: alternative

text: way,

text: an

text: explicit

text: local

text: expression

text: for

text: a

text: Gibbs

text: measure

text: of

text: the

text: dimer

text: model

text: on

text: the

text: graph

text: GQ

text: where

text: the

text: locality

text: property

text: is

text: seen

text: as

text: directly

text: inherited

text: from

text: that

text: of

text: the

text: Z-massive

text: Green

text: function.

text: Using

text: Theorem

text: and

text: additional

text: combinatorial

text: arguments,

text: we

text: prove

text: in

text: Theorem

text: that

text: the

text: determinants

text: of

text: KQ

text: and

text: of

text: the

text: Zu-Dirac

text: operator

text: are

text: equal,

text: up

text: to

text: an

text: explicit

text: constant.

text: By

text: the

text: determinant

text: of

text: KQ

text: is

text: equal

text: up

text: to

text: a

text: constant,

text: to

text: the

text: squared

text: partition

text: function

text: of

text: the

text: Ising

text: model.

text: Combining

text: this

text: with

text: Theorem

text: gives,

text: see

text: also

text: Corollary

text: for

text: the

text: explicit

text: value

text: of

text: C(u),

text: T

text: Theorem

text: For

text: every

text: u

text: (k))

text: [Z+

text: (G,

text: C(u)

text: det

text: ∆m,∂(u)

text: Ising

text: When

text: k

text: we

text: essentially

text: recover

text: the

text: result

text: of

text: see

text: Section

text: Section

text: Dimer

text: model

text: on

text: the

text: Fisher

text: graph

text: GF

text: and

text: the

text: Kasteleyn

text: matrix

text: KQ.

text: Con-

text: sider

text: the

text: dimer

text: model

text: on

text: GF

text: with

text: Kasteleyn

text: matrix

text: KF

text: arising

text: from

text: an

text: Ising

text: model

text: with

text: cou-

text: pling

text: constants

text: J,

text: not

text: necessarily

text: Z-invariant,

text: and

text: the

text: corresponding

text: dimer

text: model

text: on

text: the

text: bi-

text: partite

text: graph

text: GQ,

text: with

text: (real)

text: Kasteleyn

text: matrix

text: K˜Q.

text: Building

text: on

text: the

text: work

text: of

text: Dub´edat

text: and

text: proving

text: additional

text: matrix

text: relations,

text: we

text: express

text: coeﬃcients

text: of

text: the

text: inverse

text: operator

text: (KF)

text: using

text: coeﬃcients

text: of

text: the

text: inverse

text: operator

text: (K˜Q)

text: Partitioning

text: vertices

text: of

text: GF

text: as

text: A

text: B

text: as

text: in

text: we

text: obtain,

text: see

text: also

text: Theorem

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: where eiβ¯ is the rhombus vector (w, v) and ei(β¯+π) is the rhombus vector (w, f). Suppose that GQ is ﬁnite, and let w = wc for some boundary rhombus pair of R∂, then wc (cid:5) is on a rhombus edge vcfc of G , see Figure 15 (right). As long as the rhombus pair is not the root one, i.e., the one where vc = r, the row of T (u) corresponding to wc has non-zero coeﬃcients given by, t(u)wc,vc = −ik(cid:48)e−iα¯2r sn(θ∂) nd(uαr) cd(uβr), (36) t(u)wc,fc = e−iβ¯2(cid:96) sn((uβ(cid:96))∗) = e−iβ¯2(cid:96) cd(uβ(cid:96)). When the boundary rhombus pair is the root pair, then only the term t(u) is deﬁned. wc,fc Note that the deﬁnition is speciﬁc for vc, and coherent with the non-boundary case for fc since the rhombus vector (wc, fc) is eiβ¯(cid:96). Remark 34. Rhombus angles are well deﬁned mod 2π implying that half-angles are well deﬁned mod 4π, but the coeﬃcients of S(u) and T (u) are nevertheless well deﬁned. Indeed, − ∈ − keeping in mind that by deﬁnition β¯ α¯ (ε, π ε), one has for example, see Equation (33): 2 2 (cid:0) (cid:1)(cid:0) (cid:1) e−iβ¯+22π cn(uβ+4K)[sn(θ) cn(θ) nd(uα+4K) nd(uβ+4K)]12 = = −e−iβ2¯ − cn(uβ) [sn(θ) cn(θ) nd(uα) nd(uβ)]12 = sb,w, − − − − using that u = u 2K and cn(u 2K) = cn(u), nd(u 2K) = nd(u). Similar β+4K β arguments hold for other coeﬃcients. We are now ready to state our main result. Theorem 35. • ∈ (cid:60) T Inﬁnite case. Let u ( (k)), then the Kasteleyn matrix KQ, the Zu-Dirac operator K(u) and the matrices S(u), T (u) are related by the following identity: KQ T (u) = S(u) K(u). (37) • ∈ (cid:60) T (cid:48) Finite case. Let u ( (k)) , then the Kasteleyn matrix KQ, the Zu-Dirac operator K∂(u) and the matrices S(u), T (u) are related by the following identity: KQ T (u) = S(u) K∂(u). (38) 53 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: eiβ¯

text: is

text: the

text: rhombus

text: vector

text: (w,

text: v)

text: and

text: ei(β¯+π)

text: is

text: the

text: rhombus

text: vector

text: (w,

text: f).

text: Suppose

text: that

text: GQ

text: is

text: ﬁnite,

text: and

text: let

text: w

text: wc

text: for

text: some

text: boundary

text: rhombus

text: pair

text: of

text: R∂,

text: then

text: wc

text: is

text: on

text: a

text: rhombus

text: edge

text: vcfc

text: of

text: G

text: see

text: Figure

text: (right).

text: As

text: long

text: as

text: the

text: rhombus

text: pair

text: is

text: not

text: the

text: root

text: one,

text: i.e.,

text: the

text: one

text: where

text: vc

text: r,

text: the

text: row

text: of

text: T

text: (u)

text: corresponding

text: to

text: wc

text: has

text: non-zero

text: coeﬃcients

text: given

text: by,

text: t(u)wc,vc

text: sn(θ∂)

text: nd(uαr)

text: cd(uβr),

text: t(u)wc,fc

text: When

text: the

text: boundary

text: rhombus

text: pair

text: is

text: the

text: root

text: pair,

text: then

text: only

text: the

text: term

text: t(u)

text: is

text: deﬁned.

text: wc,fc

text: Note

text: that

text: the

text: deﬁnition

text: is

text: speciﬁc

text: for

text: vc,

text: and

text: coherent

text: with

text: the

text: non-boundary

text: case

text: for

text: fc

text: since

text: the

text: rhombus

text: vector

text: (wc,

text: fc)

text: is

text: Remark

text: Rhombus

text: angles

text: are

text: well

text: deﬁned

text: mod

text: 2π

text: implying

text: that

text: half-angles

text: are

text: well

text: deﬁned

text: mod

text: but

text: the

text: coeﬃcients

text: of

text: S(u)

text: and

text: T

text: (u)

text: are

text: nevertheless

text: well

text: deﬁned.

text: Indeed,

text: keeping

text: in

text: mind

text: that

text: by

text: deﬁnition

text: β¯

text: α¯

text: (ε,

text: π

text: ε),

text: one

text: has

text: for

text: example,

text: see

text: Equation

text: cn(θ)

text: −e−iβ2¯

text: cn(uβ)

text: [sn(θ)

text: cn(θ)

text: nd(uα)

text: sb,w,

text: using

text: that

text: u

text: u

text: and

text: cn(u

text: cn(u),

text: nd(u

text: nd(u).

text: Similar

text: β

text: arguments

text: hold

text: for

text: other

text: coeﬃcients.

text: We

text: are

text: now

text: ready

text: to

text: state

text: our

text: main

text: result.

text: Theorem

text: T

text: Inﬁnite

text: case.

text: Let

text: u

text: (k)),

text: then

text: the

text: Kasteleyn

text: matrix

text: KQ,

text: the

text: Zu-Dirac

text: operator

text: K(u)

text: and

text: the

text: matrices

text: S(u),

text: T

text: (u)

text: are

text: related

text: by

text: the

text: following

text: identity:

text: KQ

text: T

text: (u)

text: S(u)

text: K(u).

text: T

text: Finite

text: case.

text: Let

text: u

text: (k))

text: then

text: the

text: Kasteleyn

text: matrix

text: KQ,

text: the

text: Zu-Dirac

text: operator

text: K∂(u)

text: and

text: the

text: matrices

text: S(u),

text: T

text: (u)

text: are

text: related

text: by

text: the

text: following

text: identity:

text: KQ

text: T

text: (u)

text: S(u)

text: K∂(u).

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: b1 w1 w4 a5 a1 b a4 w1 b b b(cid:48) b w2 a2 (cid:48) a3 a1 w2 b w3 a2 Figure 28: Notation for the proof of Equation (64). We need to introduce an additional matrix, the matrix D , which has rows indexed by A,B A-vertices and columns by B-vertices of GF. It is diagonal: to an A-vertex a corresponds the unique B-vertex b such that b comes before a in the cw ordering of the triangle containing a and b. The diagonal coeﬃcient d is: a,b 1 d = ε . a,b b,a 2 For example to a of Figure 28 corresponds the vertex b, and the coeﬃcient d = 1ε . 2 a2,b 2 b,a2 Let us ﬁrst show that − κ KF = D . (65) A,A A,B A,B Consider a B-vertex b of GF. Then, the coeﬃcient (κ KF ) of the LHS of (65) is a priori A,A A,B a,b non-zero for all A-vertices a belonging to the same decoration as b. We have, (κ KF ) = κ KF + κ KF = κ ε + κ ε . A,A A,B a,b a,a1 a1,b a,a2 a2,b a,a1 a1,b a,a2 a2,b (cid:54) Returning to the deﬁnition of the matrix Λ, as long as a = a , we have that κ = κ ε 2 a,a1 a,a2 a2,a1 implying, − (κ KF ) = κ [ε ε + ε ] = 0 = d , A,A A,B a,b a,a2 a2,a1 a1,b a2,b a,b since the orientation around the triangle a , a , b is Kasteleyn and using the deﬁnition of − 1 2 D . When a = a , then κ = 1ε , κ = 1; thus A,B 2 a2,a1 4 a2,a1 a2,a2 4 −1 1 −1 − (κ KF ) = ε ε + ε = ε = d , A,A A,B a2,b 4 a2,a1 a1,b 4 a2,b 2 b,a2 b,a2 using again the Kasteleyn orientation around the triangle and the deﬁnition of D , thus A,B ending the proof of (65). Plugging (65) into (64) leaves us with showing the equivalent − 2X 1KF = D KF + 2K˜QI D . (66) B,B B,A A,B W,A A,B 85 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: b

text: b

text: b

text: b

text: b

text: Figure

text: Notation

text: for

text: the

text: proof

text: of

text: Equation

text: We

text: need

text: to

text: introduce

text: an

text: additional

text: matrix,

text: the

text: matrix

text: D

text: which

text: has

text: rows

text: indexed

text: by

text: A,B

text: A-vertices

text: and

text: columns

text: by

text: B-vertices

text: of

text: GF.

text: It

text: is

text: diagonal:

text: to

text: an

text: A-vertex

text: a

text: corresponds

text: the

text: unique

text: B-vertex

text: b

text: such

text: that

text: b

text: comes

text: before

text: a

text: in

text: the

text: cw

text: ordering

text: of

text: the

text: triangle

text: containing

text: a

text: and

text: b.

text: The

text: diagonal

text: coeﬃcient

text: d

text: is:

text: a,b

text: d

text: ε

text: a,b

text: b,a

text: For

text: example

text: to

text: a

text: of

text: Figure

text: corresponds

text: the

text: vertex

text: b,

text: and

text: the

text: coeﬃcient

text: d

text: 1ε

text: Let

text: us

text: ﬁrst

text: show

text: that

text: κ

text: KF

text: D

text: A,A

text: A,B

text: A,B

text: Consider

text: a

text: B-vertex

text: b

text: of

text: GF.

text: Then,

text: the

text: coeﬃcient

text: (κ

text: KF

text: of

text: the

text: LHS

text: of

text: is

text: a

text: priori

text: A,A

text: A,B

text: a,b

text: non-zero

text: for

text: all

text: A-vertices

text: a

text: belonging

text: to

text: the

text: same

text: decoration

text: as

text: b.

text: We

text: have,

text: (κ

text: KF

text: κ

text: KF

text: κ

text: KF

text: κ

text: ε

text: κ

text: ε

text: A,A

text: A,B

text: a,b

text: Returning

text: to

text: the

text: deﬁnition

text: of

text: the

text: matrix

text: Λ,

text: as

text: long

text: as

text: a

text: a

text: we

text: have

text: that

text: κ

text: κ

text: ε

text: implying,

text: (κ

text: KF

text: κ

text: [ε

text: ε

text: ε

text: d

text: A,A

text: A,B

text: a,b

text: a,b

text: since

text: the

text: orientation

text: around

text: the

text: triangle

text: a

text: a

text: b

text: is

text: Kasteleyn

text: and

text: using

text: the

text: deﬁnition

text: of

text: D

text: When

text: a

text: a

text: then

text: κ

text: 1ε

text: κ

text: thus

text: A,B

text: −1

text: −1

text: (κ

text: KF

text: ε

text: ε

text: ε

text: ε

text: d

text: A,A

text: A,B

text: using

text: again

text: the

text: Kasteleyn

text: orientation

text: around

text: the

text: triangle

text: and

text: the

text: deﬁnition

text: of

text: D

text: thus

text: A,B

text: ending

text: the

text: proof

text: of

text: Plugging

text: into

text: leaves

text: us

text: with

text: showing

text: the

text: equivalent

text: D

text: KF

text: D

text: B,B

text: B,A

text: A,B

text: W,A

text: A,B

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: I − − where, Γ(u) = { (cid:54) }t(u) K∂(u) 1 + t(u) K∂(u) 1 , and coeﬃcients of t(u) are given w=wc,r w,v v,w w,f f,w ¯ ¯ ¯ ¯ ¯ ¯ ¯ by (35) and (36). The mass of the Z-massive Laplacian is equal to 0 away from the bound- − ary [BdTR17b]; then Example 30 expresses coeﬃcients of K∂(uf) 1 using the Z-Green func- (cid:16) ∗ (cid:17) tion G0,∂(uf) and dual Z-Green function G0, ; we have (cid:16) (cid:17) K∂(uf)−v,1w = e−iα¯f+2β¯f tan(θf)12 G0,∂(uf)v2,v − G0,∂(uf)v1,v ¯ ¯ ¯ K∂(uf)−f,1w = −ie−iα¯f+2β¯f cot(θf)12(cid:80)I{w∈/{w(cid:96),wr∈R∂}}Gf02,∗,f − Gf01,∗,f + ¯ + i cos(ufβr)−cos(ufα(cid:96))¯K∂(uf)−¯1 · G0,∗ . ∈ cos(uf ) vc,w fc,f (vc,fc) R∂,r α(cid:96) ¯ − A similar expression holds for K∂(vf) 1. Dimer model on the Fisher graph GF(cid:0), ﬁni(cid:1)te case. Corollary 59 (ﬁnite case) simply becomes: (KF)−a,1b = eiβ¯2f sin(θi∂)I{¯w∈{wc∈R∂}} (cid:16)cos K+2θf 1 × (cid:17) ¯ [cos(θf) sin(θf)]2 × I − − { (cid:54) }t(uf) K∂(uf) 1 + t(uf) K∂(uf) 1 . w=wc,r w,v v,w w,f f,w ¯ ¯ ¯ ¯ ¯ ¯ ¯ Partition function of the Ising model with + boundary conditions. Corollary 38, ∈ (cid:60) T (cid:48)(cid:48) expressing the squared Z-invariant Ising partition function, holds for every u ( (k)) . (cid:48) (cid:48) → −∞ When k = 0, a nice expression is obtained by setting u = iu and taking the limit u . We have (cid:48) cos((iu ) ) − (cid:48) u(cid:48)→lim−∞ cos((iv)βα) = eiβ 2α , u(cid:48)→lim−∞ tan((iu )α) = 1. As a consequence,     − (cid:48) ∈ { ∈ } (cid:80)tan(θ) if (v, v ) / (vc, v(cid:96)) R∂,r ∆m,∂(−i∞)v,v(cid:48) = −ediα(cid:96)−2tβarnt(aθn)(θ∂) iiff (vv(cid:48) ,=v(cid:48)v) ∈={v(∂vc∈, v{(cid:96)v)r∈, vR(cid:96) ∈∂,r}R∂} j=1 j tan(θ∂)(eiα(cid:96)−2βr − 1) if v(cid:48) = v = vc ∈ {vc ∈ R∂,r}, where we recover the critical Lapl(cid:16)aci(cid:89)an of [Ken02] a(cid:17)way from the boundary. Corollary 38 becomes: | |− 1 + sin(θ ) | − ∞ | [Z+ (G, J)]2 = 2 V 1 w sn(θ∂,r) det ∆m,∂( i ) . Ising 2 sin(θ ) ∈ w w W∂ We essentially recover the main result of [dT16] proving that the squared critical Z-invariant Ising model partition function is equal, up to an explicit constant, to the partition function 91 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: I

text: where,

text: Γ(u)

font: default

text: }t(u)

text: K∂(u)

text: t(u)

text: K∂(u)

text: and

text: coeﬃcients

text: of

text: t(u)

text: are

text: given

text: w=wc,r

text: w,v

text: v,w

text: w,f

text: f,w

text: by

text: and

text: The

text: mass

text: of

text: the

text: Z-massive

text: Laplacian

text: is

text: equal

text: to

text: away

text: from

text: the

text: bound-

text: ary

text: then

text: Example

text: expresses

text: coeﬃcients

text: of

text: K∂(uf)

text: using

text: the

text: Z-Green

text: func-

text: tion

text: and

text: dual

text: Z-Green

text: function

text: we

text: have

text: i

text: cos(uf

text: vc,w

text: fc,f

text: (vc,fc)

text: R∂,r

text: A

text: similar

text: expression

text: holds

text: for

text: K∂(vf)

text: Dimer

text: model

text: on

text: the

text: Fisher

text: graph

text: case.

text: Corollary

text: (ﬁnite

text: case)

text: simply

text: becomes:

text: sin(θi∂)I{¯w∈{wc∈R∂}}

text: [cos(θf)

text: I

text: }t(uf)

text: K∂(uf)

text: t(uf)

text: K∂(uf)

text: w=wc,r

text: w,v

text: v,w

text: w,f

text: f,w

text: Partition

text: function

text: of

text: the

text: Ising

text: model

text: with

text: boundary

text: conditions.

text: Corollary

text: T

text: expressing

text: the

text: squared

text: Z-invariant

text: Ising

text: partition

text: function,

text: holds

text: for

text: every

text: u

text: (k))

text: When

text: k

text: a

text: nice

text: expression

text: is

text: obtained

text: by

text: setting

text: u

text: iu

text: and

text: taking

text: the

text: limit

text: u

text: We

text: have

text: cos((iu

text: cos((iv)βα)

text: eiβ

text: 2α

text: tan((iu

text: )α)

text: As

text: a

text: consequence,

text: if

text: (v,

text: v

text: (vc,

text: R∂,r

text: iiff

text: ∈={v(∂vc∈,

text: ∈∂,r}R∂}

text: j

text: if

text: v

text: vc

text: {vc

text: R∂,r},

text: where

text: we

text: recover

text: the

text: critical

text: of

text: from

text: the

text: boundary.

text: Corollary

text: becomes:

text: sin(θ

text: [Z+

text: (G,

text: V

text: w

text: sn(θ∂,r)

text: det

text: ∆m,∂(

text: i

text: Ising

text: sin(θ

text: w

text: w

text: W∂

text: We

text: essentially

text: recover

text: the

text: main

text: result

text: of

text: proving

text: that

text: the

text: squared

text: critical

text: Z-invariant

text: Ising

text: model

text: partition

text: function

text: is

text: equal,

text: up

text: to

text: an

text: explicit

text: constant,

text: to

text: the

text: partition

text: function

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