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This study investigates cutoff phenomena in weighted random walks on graphs. We establish conditions on edge weights and graph properties necessary and sufficient for cutoff, particularly for graphs with polynomial growth and vertex-transitivity.

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Area of Science:

  • Graph Theory
  • Probability Theory
  • Computer Science

Background:

  • Random walks on graphs are fundamental in analyzing graph structure and dynamics.
  • Cutoff phenomena, a sharp transition in mixing times, are crucial for understanding random walk behavior.
  • Weighted graphs introduce complexities in analyzing random walk convergence.

Purpose of the Study:

  • To determine conditions for the existence of cutoff in weighted random walks on sequences of graphs.
  • To analyze the impact of edge weights (ε) and graph properties (degree, growth, transitivity) on cutoff.
  • To provide a comprehensive understanding of cutoff phenomena in both general and specific graph classes (e.g., expanders).

Main Methods:

  • Introduction of a modified graph G* by adding a random perfect matching with weight ε.
  • Analysis of the relationship between log(1/ε) and log|Vn| for graphs with polynomial growth.
  • Investigation of vertex-transitive graphs and their impact on necessary and sufficient conditions for cutoff.
  • Examination of graphs with linearly growing entropy in simple random walks.
  • Study of expander graphs and their complementary regime for cutoff.

Main Results:

  • For graphs with polynomial growth, log(1/ε) ≪ log|Vn| is a sufficient condition for cutoff.
  • For vertex-transitive graphs, this condition is also necessary.
  • For graphs with linearly growing entropy, 1/ε ≪ log|Vn| is sufficient for cutoff.
  • A complete analysis is provided for expander graphs in the regime 1/ε ≳ log|Vn|.

Conclusions:

  • The existence of cutoff in weighted random walks is strongly dependent on the relationship between edge weights and graph size/structure.
  • Specific graph properties like polynomial growth and vertex-transitivity significantly influence the conditions for cutoff.
  • The study offers precise mathematical criteria for predicting cutoff phenomena in various graph settings.