Combinatorial Nature of the Ground-State Vector of the O(1) Loop Model

Abstract

Hanging about a hypothetical connections between the ground state vector for some special spin systems and the alternating-sign matrices, we have found a numerical evidence for the fact that the numbers of the states of the fully packed loop model with fixed link-patterns coincide with the components of the ground state vector of the dense O(1) loop model considered by Batchelor, de Gier and Nienhuis. Our conjecture generalizes in a sense the conjecture of Bosley and Fidkowski, refined by Cohn and Propp, and proved by Wieland. I would say that imagination is a form of memory.(V. Nabokov)In paper [1] we made some conjectures related to combinatorial properties of the ground state vector of the XXZ spin chain for the asymmetry parameter ∆ = −1/2 and an odd number of cites. In the subsequent paper [2] Batchelor, de Gier and Nienhuis considered two variations of this model along with the corresponding O(n) loop model at n = 1 and notably increased the number of models and related combinatorial objects (see also [3]).During last months we have accumulated a lot of data on relation between the spin systems and alternating sign matrices (ASMs). In this note we limit ourselves to one qualitative conjecture which relates some classes of the states of the fully packed loop (FPL) model with the ground state vector of the dense O(n) loop model (see [4] and references therein). The states the FPL model is in bijective correspondence with the ASMs. Therefore if our conjecture is true then one can investigate the FPL model, and the ASMs, using the methods elaborated in mathematical physics for integrable model (see, for example, [5]).The background information on the ASMs and their different combinatorial forms can be found in the recent review by Propp [6] and in references therein. For more details on enumerative problems related to the states of the FPL model see the article by Wieland [7].Following the review paper by Propp [6] we define the "generalized tic-tac-toe" graph as the graph formed by n horizontal lines and n vertical lines meeting n 2 intersections of degree 4, with 4n vertices of degree 1 at the boundary. Then we number the vertices of degree 1. We start with the left top vertex and number clockwise every other vertex. Now 1

On Polynomials Interpolating Between the Stationary State of a O(n) Model and a Q.H.E. Ground State

“…It can be viewed as a finite size version of the modular invariant partition function of conformal the field theories (CFT). 4 If the heights defining the paths of section 6.1 are integer, we can restrict the paths to have a strictly positive height. Similarly, when t is a root of unity, S p = 0 in (73) and we can restrict the height to be strictly less than p. The paths obeying these restrictions are called restricted solid on solid (RSOS) paths [42].…”

“…The positivity conjectures are motivated by the R-S conjecture [4] which states that at τ = 1, the evaluations of the polynomials considered in the preceding section at z i = 1 count certain classes of alternating sign matrices. We claim that the evaluation of their deformations are positive polynomials in the deformation parameter τ (are in N[τ ]).…”

“…We observe on examples and conjecture in general that these integers are positive. When n = n ′ = 1, the disc and the cylinder polynomials become respectively the integers previously conjectured to count alternating sign matrices and half-turn alternating sign matrices in different topological sectors [4] [22].…”

“…The components of the O(1) model Hamiltonian stationary state can be normalized to be positive integers, and it is conjectured (the Razumov-Stroganov (R-S) conjecture [4]) that these integers are in bijective correspondence with the states of a square-ice-model with domain wall boundary conditions, or equivalently with certain classes of alternatingsign matrices.…”

Loop model with mixed boundary conditions,qKZ equation and alternating sign matrices

“…The present work is yet another chapter in the continuing story, inspired by the seminal papers [1,2], of the interrelation between quantum integrability and combinatorics.…”