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We introduce partial loop-erasing operators. We show that by applying arefinement sequence of partial loop-erasing operators to a finite Markov chain,we get a process equivalent to the chronological loop-erased Markov chain. Asan application, we construct loop-erased random paths on bounded domains ofresistance spaces as the weak limit of the loop erasure of the Markov chains ona sequence of finite sets approximating the space, and the limit is independentof the approximating sequences. The random paths we constructed are simplepaths almost surely, and they can be viewed as the loop-erasure of the paths ofthe diffusion process. Finally, we show that the scaling limit of theloop-erased random walks on the Sierpi\'nski carpet graphs exists, and isequivalent to the loop-erased random paths on the Sierpi\'nksi carpet.

Scaling limits of loop-erased Markov chains on resistance spaces via a partial loop-erasing procedure