Titlebook: Random Walks on Disordered Media and their Scaling Limits; École d‘Été de Proba Takashi Kumagai Book 2014 Springer International Publishing

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Introduction,Around mid-1960s, mathematical physicists began to analyze properties of disordered media such as structures of polymers and networks, growth of molds and crystals (see for example, [44]). They observed anomalous behavior for random walks/diffusions on the media.

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Heat Kernel Estimates: General Theory,In this chapter, we will consider various inequalities which imply (or which are equivalent to) the Nash-type heat kernel upper bound, i.e. . for some . > 0. We will also discuss Poincaré inequalities and their relations to heat kernel estimates in Sect. 3.3. We will prefer to discuss them under a general framework including weighted graphs.

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Heat Kernel Estimates Using Effective Resistance,In this chapter, we will study detailed asymptotic properties of Green functions and heat kernels using the effective resistance. Let (., .) be a weighted graph. We say that (., .) is a . if for any . ≥ 3, there is no set of distinct points . such that .. ∼ .. for 1 ≤ . ≤ . where we set ..

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,Alexander–Orbach Conjecture Holds When Two-Point Functions Behave Nicely,This chapter is based on the paper by Kozma-Nachmias [162] with some simplification by [197]. Our framework here is unimodular transitive graphs that contains . as a typical example.

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Random Conductance Model,In this chapter, we will discuss recent developments on the random conductance model (RCM). We will mainly discuss the quenched invariance principle, i.e. the functional central limit theorem which is almost sure with respect to the randomness of the environments.

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Takashi KumagaiStarts from basics on discrete potential theory.Contains many interesting examples of disordered media with anomalous heat conduction.Anomalous behavior of random walk at criticality on random media.C

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uth, marrying theory with lived experience, and policies witYoung people who are considered ‘vulnerable’ or ‘at risk’ are a particular target of various policies, schemes and interventions. But what does vulnerability mean? Interrogating Conceptions of “Vulnerable Youth” explores this question in re