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Avalanche frontiers in Abelian Sandpile Model (ASM) are random simple curveswhose continuum limit is known to be a Schramm-Loewner Evolution (SLE) withdiffusivity parameter $\kappa = 2$. In this paper we consider the dissipativeASM and study the statistics of the avalanche and wave frontiers for variousrates of dissipation. We examine the scaling behavior of a number of functionssuch as the correlation length, the exponent of distribution function of looplengths and gyration radius defined for waves and avalanches. We find that theydo scale with the rate of dissipation. Two significant length scales areobserved. For length scales much smaller than the correlation length, thesecurves show properties close to the critical curves and the correspondingdiffusivity parameter is nearly the same as the critical limit. We interpretthis as the ultra violet (UV) limit where $\kappa = 2$ corresponding to $c=-2$.For length scales much larger than the correlation length we find that theavalanche frontiers tend to Self-Avoiding Walk, the corresponding drivingfunction is proportional to the Brownian motion with the diffusion parameter$\kappa =8/3$ corresponding to a field theory with $c = 0$. This is the infrared (IR) limit. Correspondingly the central charge decreases from the IR to theUV point.

Avalanche frontiers in dissipative abelian sandpile model as off-critical SLE(2)