Paul Zinn-Justin

Thermodynamic limit of the Six-Vertex Model with Domain Wall Boundary Conditions

We address the question of the dependence of the bulk free energy on boundary conditions for the six-vertex model. Here we compare the bulk free energy for periodic and domain wall boundary conditions. Using a determinant representation for the partition function with domain wall boundary conditions, we derive Toda differential equations and solve them asymptotically in order to extract the bulk free energy. We find that it is different and bears no simple relation to the free energy for periodic boundary conditions. The six-vertex model with domain wall boundary conditions is closely related to algebraic combinatorics (alternating sign matrices). This implies new results for the weighted counting for large-size alternating sign matrices. Finally, we comment on the interpretation of our results, in particular in connection with domino tilings (dimers on a square lattice).

Universality of Correlation Functions of Hermitian Random Matrices in an External Field

Abstract:The behavior of correlation functions is studied in a class of matrix models characterized by a measure exp(−S) containing a potential term and an external source term: S=N tr(V(M) −MA). In the large N limit, the short-distance behavior is found to be identical to the one obtained in previously studied matrix models, thus extending the universality of the level-spacing distribution. The calculation of correlation functions involves (finite N) determinant formulae, reducing the problem to the large N asymptotic analysis of a single kernel K. This is performed by an appropriate matrix integral formulation of K. Multi-matrix generalizations of these results are discussed.

Six-vertex model with domain wall boundary conditions and one-matrix model

The partition function of the six-vertex model on a square lattice with domain wall boundary conditions (DWBC) is rewritten as a Hermitian one-matrix model or a discretized version of it (similar to sums over Young diagrams), depending on the phase. The expression is exact for finite lattice size, which is equal to the size of the corresponding matrix. In the thermodynamic limit, the matrix integral is computed using traditional matrix model techniques, thus providing a complete treatment of the bulk free energy of the six-vertex model with DWBC in the different phases. In particular, in the antiferroelectric phase, the bulk free energy and a subdominant correction are given exactly in terms of elliptic theta functions.

The quantum Knizhnik–Zamolodchikov equation, generalized Razumov–Stroganov sum rules and extended Joseph polynomials

We prove higher rank analogues of the Razumov–Stroganov sum rule for the ground state of the O(1) loop model on a semi-infinite cylinder: we show that a weighted sum of components of the ground state of the Ak−1 IRF model yields integers that generalize the numbers of alternating sign matrices. This is done by constructing minimal polynomial solutions of the level 1 quantum Knizhnik–Zamolodchikov equations, which may also be interpreted as quantum incompressible q-deformations of quantum Hall effect wavefunctions at filling fraction ν = k. In addition to the generalized Razumov–Stroganov point q = −eiπ/k+1, another combinatorially interesting point is reached in the rational limit q → −1, where we identify the solution with extended Joseph polynomials associated with the geometry of upper triangular matrices with vanishing kth power.

On some integrals over the U(N) unitary group and their large N limit

The integral over the U(N) unitary group I = ? DU exp Tr AU BU? is re-examined. Various approaches and extensions are first reviewed. The second half of the paper deals with more recent developments: relation with integrable Toda lattice hierarchy, diagrammatic expansion and combinatorics, and what they teach us on the large N limit of log I.

Random Hermitian matrices in an external field

Abstract In this article, a model of random Hermitian matrices is considered, in which the measure exp(−S) contains a general U(N)-invariant potential and an external source term: S = Ntr(V(M) + MA). The generalization of known determinant formulae leads to compact expressions for the correlation functions of the energy levels. These expressions, exact at finite N, are potentially useful for asymptotic analysis.

POLYNOMIAL SOLUTIONS OF qKZ EQUATION AND GROUND STATE OF XXZ SPIN CHAIN AT = −1/2

Integral formulae for polynomial solutions of the quantum Knizhnik–Zamolodchikov equations associated with the R-matrix of the six-vertex model are considered. It is proved that when the deformation parameter q is equal to and the number of vertical lines of the lattice is odd, the solution under consideration is an eigenvector of the inhomogeneous transfer matrix of the six-vertex model. In the homogeneous limit, it is a ground-state eigenvector of the antiferromagnetic XXZ spin chain with the anisotropy parameter Δ equal to −1/2 and an odd number of sites. The obtained integral representations for the components of this eigenvector allow us to prove some conjectures on its properties formulated earlier. A new statement relating the ground-state components of XXZ spin chains and Temperley–Lieb loop models is formulated and proved.

Two-matrix model with ABAB interaction

Using recently developed methods of character expansions we solve exactly in the large N limit a new two-matrix model of hermitian matrices A and B with the action S = 12 (tr A2 + tr B2) − α4 (trA4 + trB4) −β2 tr(AB)2. This model can be mapped onto a special case of the 8-vertex model on dynamical planar graphs. The solution is parametrized in terms of elliptic functions. A phase transition is found: the critical point is a conformal field theory with central charge c = 1 coupled to 2D quantum gravity.

Quantum Knizhnik-Zamolodchikov Equation: Reflecting boundary conditions and Combinatorics

We consider the level 1 solution of the quantum Knizhnik–Zamolodchikov equation with reflecting boundary conditions which is relevant to the Temperley–Lieb model of loops on a strip. By use of integral formulae we prove conjectures relating it to the weighted enumeration of cyclically symmetric transpose complement plane partitions and related combinatorial objects.

Sum rules for the ground states of the O(1) loop model on a cylinder and the XXZ spin chain

The sums of components of the ground states of the O(1) loop model on a cylinder or of the XXZ quantum spin chain at Δ = −1/2 of size L are expressed in terms of combinatorial numbers. The methods include the introduction of spectral parameters and the use of integrability, a mapping from size L to L+1, and knot-theoretic skein relations.