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22
Symmetry classes of alternatingsign matrices under one
"... In a previous article [23], we derived the alternatingsign matrix (ASM) theorem from the IzerginKorepin determinant [12, 13, 19] for a partition function for square ice with domain wall boundary. Here we show that the same argument enumerates three other symmetry classes of alternatingsign matric ..."
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Cited by 55 (0 self)
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In a previous article [23], we derived the alternatingsign matrix (ASM) theorem from the IzerginKorepin determinant [12, 13, 19] for a partition function for square ice with domain wall boundary. Here we show that the same argument enumerates three other symmetry classes of alternatingsign matrices: VSASMs (vertically symmetric ASMs), even HTSASMs (halfturnsymmetric ASMs), and even QTSASMs (quarterturnsymmetric ASMs). The VSASM enumeration was conjectured by Mills; the others by Robbins [31]. We introduce several new types of ASMs: UASMs (ASMs with a Uturn side), UUASMs (two Uturn sides), OSASMs (offdiagonally symmetric ASMs), OOSASMs (offdiagonally, offantidiagonally symmetric), and UOSASMs (offdiagonally symmetric with Uturn sides). UASMs generalize VSASMs, while UUASMs generalize VHSASMs (vertically and horizontally symmetric ASMs) and another new class, VHPASMs (vertically and horizontally perverse). OSASMs, OOSASMs, and UOSASMs are related to the remaining symmetry classes of ASMs, namely DSASMs (diagonally symmetric), DASASMs (diagonally, antidiagonally symmetric), and TSASMs (totally symmetric ASMs). We enumerate several of these new classes, and we provide several 2enumerations
Completeness of the Bethe ansatz for the six and eightvertex models
 J. Stat. Phys
"... We discuss some of the difficulties that have been mentioned in the literature in connection with the Bethe ansatz for the sixvertex model and XXZ chain, and for the eightvertex model. In particular we discuss the “beyond the equator”, infinite momenta and exact complete string problems. We show h ..."
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Cited by 24 (1 self)
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We discuss some of the difficulties that have been mentioned in the literature in connection with the Bethe ansatz for the sixvertex model and XXZ chain, and for the eightvertex model. In particular we discuss the “beyond the equator”, infinite momenta and exact complete string problems. We show how they can be overcome and conclude that the coordinate Bethe ansatz does indeed give a complete set of states, as expected.
Exact solution of the sixvertex model with domain wall boundary conditions. Critical line between . . .
, 2008
"... This is a continuation of the papers [4] of Bleher and Fokin and [6] of Bleher and Liechty, in which the large n asymptotics is obtained for the partition function Zn of the sixvertex model with domain wall boundary conditions in the disordered and ferroelectric phases, respectively. In the presen ..."
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Cited by 12 (2 self)
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This is a continuation of the papers [4] of Bleher and Fokin and [6] of Bleher and Liechty, in which the large n asymptotics is obtained for the partition function Zn of the sixvertex model with domain wall boundary conditions in the disordered and ferroelectric phases, respectively. In the present paper we obtain the large n asymptotics of Zn on the critical line between these two phases.
Aspects of the ODE/IM correspondence
 CONTRIBUTION TO THE PROCEEDINGS “RECENT TRENDS IN EXPONENTIAL ASYMPTOTICS”, JUNE 28 JULY 2 (2004), RIMS, KYOTO
, 2004
"... We review a surprising correspondence between certain twodimensional integrable models and the spectral theory of ordinary differential equations. Particular emphasis is given to the relevance of this correspondence to certain problems in PTsymmetric quantum mechanics. ..."
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Cited by 11 (1 self)
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We review a surprising correspondence between certain twodimensional integrable models and the spectral theory of ordinary differential equations. Particular emphasis is given to the relevance of this correspondence to certain problems in PTsymmetric quantum mechanics.
The six and eightvertex models revisited
 J. Stat. Phys
"... Elliott Lieb’s icetype models opened up the whole field of solvable models in statistical mechanics. Here we discuss the “commuting transfer matrix” T,Q equations for these models, writing them in a more explicit and transparent notation that we believe offers new insights. The approach manifests t ..."
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Cited by 6 (0 self)
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Elliott Lieb’s icetype models opened up the whole field of solvable models in statistical mechanics. Here we discuss the “commuting transfer matrix” T,Q equations for these models, writing them in a more explicit and transparent notation that we believe offers new insights. The approach manifests the relationship between the sixvertex and chiral Potts models, and between the eightvertex and KashiwaraMiwa models.
Schur Polynomials and the and the YangBaxter Equation
, 2009
"... Tokuyama [31] proved a deformation of the Weyl character formula for GLn(C). A substantial generalization of Tokuyama’s deformation was found by Hamel and King [8]. The formula of Hamel and King expresses the Schur polynomial times a deformation of the Weyl denominator as a sum over states of the tw ..."
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Cited by 4 (3 self)
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Tokuyama [31] proved a deformation of the Weyl character formula for GLn(C). A substantial generalization of Tokuyama’s deformation was found by Hamel and King [8]. The formula of Hamel and King expresses the Schur polynomial times a deformation of the Weyl denominator as a sum over states of the twodimensional ice or sixvertex model in statistical mechanics. It turns out that there are two fundamentally distinct ways of doing this. We will call these Gamma ice and Delta ice. The Delta model is essentially that given by Hamel and King. In statistical physics, the partition function is the sum of certain Boltzmann weights over all states of the system. The sixvertex model is an example that is much studied in the literature. If the Boltzmann weights are invariant under sign reversal the system is called fieldfree, corresponding to the physical assumption of the absence of an external field. For fieldfree weights, the sixvertex model was solved by Lieb [19] and Sutherland [30], in the sense that the partition function can be exactly computed. A very interesting treatment based on the “startriangle relation ” or YangBaxter equation ([13], [21]) was given by Baxter [1] and [2], Chapter 9. The papers of Lieb, Sutherland and Baxter assume periodic boundary conditions, but nonperiodic boundary conditions were treated by Korepin [14] and Izergin [12]. Much of the literature assumes that the model is field free, but Baxter asserts that the sixvertex model can be solved even in the presence of fields. We do not know whether this has been carried out using the method of [1] and [2]. We will exhibit two particular choices of Boltzmann weights and boundary conditions in the sixvertex model giving systems SΓ λ and S∆λ for every partition λ of length � n. We will study the system by the method of [1] and [2]. The 1 partition functions are Z(S Γ λ) = ∏ (tizj + zi)sλ(z1, · · · , zn), i<j Z(S ∆ λ) = ∏ (tjzj + zi)sλ(z1, · · · , zn), (1) i<j where ti are deformation parameters and sλ is the Schur polynomial ([20]). The Boltzmann weights we use are not fieldfree. To justify these evaluations of the partition function define s Γ λ (z1, · · · , zn; t1, · · · , tn) = Z(S Γ λ) i<j (tizj
The nested SU(N) offshell Bethe ansatz and exact form factors
, 611
"... This work is dedicated to the 75th anniversary of H. Bethe’s foundational work on the Heisenberg chain The form factor equations are solved for an SU(N) invariant Smatrix under the assumption that the antiparticle is identified with the bound state of N − 1 particles. The solution is obtained expl ..."
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This work is dedicated to the 75th anniversary of H. Bethe’s foundational work on the Heisenberg chain The form factor equations are solved for an SU(N) invariant Smatrix under the assumption that the antiparticle is identified with the bound state of N − 1 particles. The solution is obtained explicitly in terms of the nested offshell Bethe ansatz where the contribution from each level is written in terms of multiple contour integrals. PACS: 11.10.z; 11.10.Kk; 11.55.Ds
Anomalous universality in the Anisotropic Ashkin–Teller model
, 2004
"... Abstract. The Ashkin–Teller (AT) model is a generalization of Ising 2–d to a four states spin model; it can be written in the form of two Ising layers (in general with different couplings) interacting via a four–spin interaction. It was conjectured long ago (by Kadanoff and Wegner, Wu and Lin, Baxte ..."
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Cited by 3 (1 self)
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Abstract. The Ashkin–Teller (AT) model is a generalization of Ising 2–d to a four states spin model; it can be written in the form of two Ising layers (in general with different couplings) interacting via a four–spin interaction. It was conjectured long ago (by Kadanoff and Wegner, Wu and Lin, Baxter and others) that AT has in general two critical points, and that universality holds, in the sense that the critical exponents are the same as in the Ising model, except when the couplings of the two Ising layers are equal (isotropic case). We obtain an explicit expression for the specific heat from which we prove this conjecture in the weakly interacting case and we locate precisely the critical points. We find the somewhat unexpected feature that, despite universality holds for the specific heat, nevertheless nonuniversal critical indexes appear: for instance the distance between the critical points rescales with an anomalous exponent as we let the couplings of the two Ising layers coincide (isotropic limit); and so does the constant in front of the logarithm in the specific heat. Our result also explains how the crossover from universal to nonuniversal behaviour is realized. 1.
On Valiant’s holographic algorithms
"... Leslie Valiant recently proposed a theory of holographic algorithms. These novel algorithms achieve exponential speedups for certain computational problems compared to naive algorithms for the same problems. The methodology uses Pfaffians and (planar) perfect matchings as basic computational primit ..."
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Cited by 2 (2 self)
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Leslie Valiant recently proposed a theory of holographic algorithms. These novel algorithms achieve exponential speedups for certain computational problems compared to naive algorithms for the same problems. The methodology uses Pfaffians and (planar) perfect matchings as basic computational primitives, and attempts to create exponential cancellations in computation. In this article we survey this new theory of matchgate computations and holographic algorithms.