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Exact partition functions and correlation functions of multiple Hamiltonian walks on the Manhattan lattice. (English) Zbl 1086.82501
Summary: This is a general and exact study of multiple Hamiltonian walks (HAW) filling the two-dimensional (2D) Manhattan lattice. We generalize the original exact solution for a single HAW by Kasteleyn to a system of multiple closed walks, aimed at modeling a polymer melt. In 2D, two basic nonequivalent topological situations are distinguished. (1) the Hamiltonian loops are all rooted and contractible to a point: adjacent one to another, and, on a torus, homotopic to zero. (2) the loops can encircle one another and, on a torus, can wind around it. For case 1, the grand canonical partition function and multiple correlation functions are calculated exactly as those of multiple rooted spanning trees or of a massive 2D free field, critical at zero mass (zero fugacity). The conformally invariant continuum limit on a Manhattan torus is studied in detail. The melt entropy is calculated exactly. We also consider the relevant effect of free boundary conditions. The number of single HAWs on Manhattan lattices with other perimeter shapes (rectangular, Kagomé, triangular, and arbitrary) is studied and related to the spectral theory of the Dirichlet Laplacian. This allows the calculation of exact shape-dependent configuration exponents y. An exact surface critical exponent is obtained. Forcase 2, nested and winding Hamiltonian circuits are allowed. An exact equivalence to the critical \(Q\)-state Potts model exists, where \(Q^{1/2}\) is the walk fugacity. The Hamiltonian system is then always critical (for \(Q<-4\)). The exact critical exponents, in infinite numbers, are universal and identical to those of the \(O(n=Q^{1/2})\) model in its low-temperature phase, i.e. are those of dense polymers. The exact critical partition functions on the torus are given from conformai invariance theory. These models 1 and 2 yield the two first exactly solved models of polymer melts.

MSC:
82B05 Classical equilibrium statistical mechanics (general)
05C20 Directed graphs (digraphs), tournaments
05C45 Eulerian and Hamiltonian graphs
60G50 Sums of independent random variables; random walks
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
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