A real Lipschitz height function on a graph assigns a real number to each graph vertex with height differences of absolute value at most 1 across every graph edge. If feasible boundary heights are fixed on a finite graph, with at least one fixed height in every connected component, the admissible free heights form a nonempty convex polytope. Normalized Lebesgue measure on its affine hull defines the uniform real-height model. It is separate from uniformly sampling integer-valued height functions.
Real Lipschitz Height Function
See also
Gaussian Free Field, Integer Lipschitz Height Function, Lipschitz Function, Schramm-Loewner Evolution, Triangular LatticeExplore with Wolfram|Alpha
References
Peled, R.; Samotij, W.; and Yehudayoff, A. "Grounded Lipschitz Functions on Trees Are Typically Flat." Electron. Commun. Probab. 18, Article 55, 1-9, 2013. https://doi.org/10.1214/ECP.v18-2796.Cite this as:
Weisstein, Eric W. "Real Lipschitz Height Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RealLipschitzHeightFunction.html