The "true'' self-avoiding walk with bond repulsion on Z: limit theorems

The true self-avoiding walk with bond repulsion is a nearest neighbor random walk on Z, for which the probability of jumping along a bond of the lattice is proportional to exp(-g. number of previous jumps along that bond). First we prove a limit theorem for the distribution of the local time process of this walk. Using this result, later we prove a local limit theorem, as A → ∞, for the distribution of A -2/3 Xθ s/A , where θ s/A is a random time distributed geometrically with mean e -s/A (1 - e -s/A ) -1 = A/s + 0(1). As a by-product we also obtain an apparently new identity related to Brownian excursions and Bessel bridges.

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