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1	On Weak Limits and Unimodular Measures Artemenko, Igor 14 January 2014 (has links) In this thesis, the main objects of study are probability measures on the isomorphism classes of countable, connected rooted graphs. An important class of such measures is formed by unimodular measures, which satisfy a certain equation, sometimes referred to as the intrinsic mass transport principle. The so-called law of a finite graph is an example of a unimodular measure. We say that a measure is sustained by a countable graph if the set of rooted connected components of the graph has full measure. We demonstrate several new results involving sustained unimodular measures, and provide thorough arguments for known ones. In particular, we give a criterion for unimodularity on connected graphs, deduce that connected graphs sustain at most one unimodular measure, and prove that unimodular measures sustained by disconnected graphs are convex combinations. Furthermore, we discuss weak limits of laws of finite graphs, and construct counterexamples to seemingly reasonable conjectures. tree automorphism group ball path connected component rooted birooted graph Cayley converges weakly extreme point mass transport principle involution invariance law neighbourhood orbit rigid subgraph unimodular unimodularity vertex-transitive walk weak limit weak convergence probability measure barred binary tree bi-infinite path first ancestor judicial lawless negligible stabilizer sustained strictly sustained random graph
2	On Weak Limits and Unimodular Measures Artemenko, Igor January 2014 (has links) In this thesis, the main objects of study are probability measures on the isomorphism classes of countable, connected rooted graphs. An important class of such measures is formed by unimodular measures, which satisfy a certain equation, sometimes referred to as the intrinsic mass transport principle. The so-called law of a finite graph is an example of a unimodular measure. We say that a measure is sustained by a countable graph if the set of rooted connected components of the graph has full measure. We demonstrate several new results involving sustained unimodular measures, and provide thorough arguments for known ones. In particular, we give a criterion for unimodularity on connected graphs, deduce that connected graphs sustain at most one unimodular measure, and prove that unimodular measures sustained by disconnected graphs are convex combinations. Furthermore, we discuss weak limits of laws of finite graphs, and construct counterexamples to seemingly reasonable conjectures. tree automorphism group ball path connected component rooted birooted graph Cayley converges weakly extreme point mass transport principle involution invariance law neighbourhood orbit rigid subgraph unimodular unimodularity vertex-transitive walk weak limit weak convergence probability measure barred binary tree bi-infinite path first ancestor judicial lawless negligible stabilizer sustained strictly sustained random graph

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On Weak Limits and Unimodular Measures

On Weak Limits and Unimodular Measures