I live on a node in a countably infinite connected graph where of finite maximum degree. I want to take a trip. The roads are given by independently and identically selecting edges from the graph. I happen not live on an island; there are roads taking me to the rest of the mainland. I randomly choose which road to take at each crossing. I will stop when I realise I am lost. I will realise I am lost when I return to a node I have already visited. Formally, there is a countably infinite connected graph with finite maximum degree and a chosen node. Its edges are chosen i.i.d. randomly. Percolation occurs on this graph, and we condition on the event that the chosen node is in an infinite directed cluster. A trail begins from a the chosen node, and randomly selects an edge to be its next. It stops when it returns to any node it has already visited. Let's restrict our attention to say, a square lattice, because surely a general answer would be too much to ask. What is the expected length of the trail? Clearly I can do better using a strategy: for example, never taking a one-way road that I have driven a circle around and know no edge leads out of the area enclosed by this circle. Is there an optimal strategy? How well does it do? Have any similar topics been researched?
2026-03-25 09:44:44.1774431884
Expected Trail Length
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I think the closest beautiful object to what you are discussing is the Random interlacement model, invented by Alain-Sol Snitzman. It describes a model of doubly infinite self-avoiding walk. It wont answer your question directly though.