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In this article, we establish solid foundations for the study of MaximalEntropy Random Walks (MERWs) on infinite graphs. We introduce a generalizeddefinition that extends the original concept, along with rigorous tools forhandling this generalization. Unlike conventional simple random walks, whichmaximize entropy locally, MERWs maximize entropy globally along their paths,marking a significant paradigm shift and presenting substantial computationalchallenges. Originally introduced by physicists and computer scientists in [1],MERWs have connections to concepts such as Parry measures and Doobh-transforms. Our approach addresses the challenges of existence, uniqueness,and approximation, illustrated through examples and counterexamples. Even inthe infinite setting, MERWs continue to maximize the entropy rate, albeit in aless direct manner. Additionally, we conduct an in-depth analysis of weightedspider networks, including scaling limits, revealing various phenomenacharacteristic of the infinite framework, notably a phase transition. A unifiedproof of scaling limits based on submartingale problems is presented.Furthermore, we consider some extended models, where the spider latticeprovides valuable insights, highlighting the complexity of studying these walksfor general infinite weighted graphs.

Maximum Entropy Random Walks: the Infinite Setting and the Example of Spider Networks with their Scaling Limits