Rigidity percolation and boundary conditions. (English) Zbl 1062.60098

The author discusses rigidity properties of components in the bond percolation on the two-dimensional triangular lattice T. While finite rigid components here are straightforward to define, the notion of infinite rigid components is not so obvious. In this paper, the latter are defined as limits of rigid components in a finite region of the lattice with specified boundary conditions (i.e., with fixed states of edges outside the region) as the region approaches T.
Among various boundary conditions there are two “extremal” choices – the “free” and “wired” boundary conditions – with all edges outside the region being closed and open, respectively. While theoretically such choices might give rise to different definitions of rigidity, the main result of the paper claims that, in two dimensions, the corresponding critical probabilities coincide. Moreover, it is shown that for all values of the tuning parameter \(p\) (with possible exception of the common critical value) the sets of “free” and “wired” rigid components are identical, almost surely w.r.t. the percolation measure \({\mathbf P}_p\).


60K35 Interacting random processes; statistical mechanics type models; percolation theory
52C25 Rigidity and flexibility of structures (aspects of discrete geometry)
82B43 Percolation
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