Manhattan lattice Theta -point exponents from kinetic growth walks and exact results from the Nienhuis O(n) model

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Abstract
Kinetic growth walks (KGW) on the Manhattan lattice have previously been shown to be equivalent to the static problem of interacting self-avoiding walks on that lattice at the theta -temperature. Here, we illustrate how a complete set of exponents for the static problem, including the crossover exponent phi and surface exponents at the ordinary and special transition points, may be obtained from simulations of kinetic walks. In the process we find that phi m 0.430+or-0.006 which encompasses the conjectured value 3/7 approximately=0.42857 for the theta -point on an isotropic lattice. Our numerics confirm a predicted set of exponents for both the bulk and surface transitions in addition to results such as the exact internal energy at the bulk transition. Furthermore, we point out that a recently examined variant of the Nienhuis O(n) model can be mapped onto theta -point walks on the Manhattan lattice which allows identification of the scaling dimensions for that problem and thereby provides a method for proving all the numerical conjectures.