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Optimal self-avoiding paths in dilute random medium

By a new type of finite size scaling analysis on the square lattice, and by renormalization group calculations on hierarchical lattices we investigate the effects of dilution on optimal undirected self-avoiding paths in a random environment. The behaviour of the optimal paths remains the same as for directed paths in undiluted medium, as long as forbidden bonds are not exceeding the percolation threshold. Thus, overhanging configurations do not alter the standard self-affine directed polymer scaling regime, even above the directed threshold, when they become unavoidable. When dilution reaches the undirected threshold, the optimal path becomes fractal, with fractal dimension equal to $D_{\rm min}$, the dimension of the minimal length path on percolation cluster backbone. In this regime the optimal path energy fluctuation, $\overline{ΔE}$, can be ascribed entirely to minimal length fluctuations, and satisfies $\overline{ΔE} \propto L^ω$, with $ω=1.02 \pm 0.06$ in $2d$, $L$ being the Euclidean distance. Hierarchical lattice calculations confirm that $ω$ is also the exponent of the leading scaling correction to $\overline E \propto L^{D_{\rm min}}$. Upon approaching threshold, the probability, ${\cal R}$, that the optimal path does not stick entirely on the minimal length one, obeys ${\cal R} \sim \left( Δp\right) ^ρ$, with $ρ\sim 1.0 \pm 0.05$ on hierarchical lattices. Such behaviour could be characteristic of the crossover to fractal regime. Transfer matrix results on square lattice show that a similar full sticking does not occur for directed paths at the directed percolation threshold.