Random Walks with Local Memory

We prove a quenched invariance principle for class of random walks in environment on $${\mathbb {Z}}^d$$ , where the walker alters its own environment. The consists an outgoing edge from each vertex. updates e at current location to new $$e'$$ (whose law depends e) and then steps other endpoint . show that native these (i.e., is stationary time perspective walker) wired uniform spanning forest oriented toward walker, plus independent walker.