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We develop a geometric representation for the ground state of the spin-1/2quantum XXZ ferromagnetic chain in terms of suitably weighted random walks in atwo-dimensional lattice. The path integral model so obtained admits a genuineclassical statistical mechanics interpretation with a translation invariantHamiltonian. This new representation is used to study the interface groundstates of the XXZ model. We prove that the probability of having a number ofdown spins in the up phase decays exponentially with the sum of their distancesto the interface plus the square of the number of down spins. As an applicationof this bound, we prove that the total third component of the spin in a largeinterval of even length centered on the interface does not fluctuate, i.e., haszero variance. We also show how to construct a path integral representation inhigher dimensions and obtain a reduction formula for the partition functions intwo dimensions in terms of the partition function of the one-dimensional model.Comment: 23 pages, 9 figure

PATH INTEGRAL REPRESENTATION FOR INTERFACE STATES OF THE ANISOTROPIC HEISENBERG MODEL