Equivalent-neighbor k-core percolation in two dimensions

We perform large-scale numerical simulations to investigate the critical behavior of k-core percolation in two dimensions with an extended interaction range r. By systematically varying both the core index k and the interaction range r, we construct a comprehensive phase diagram in the (k,r) plane. In contrast to k-core percolation in infinite dimensions, no hybrid transition is observed in two dimensions: The phase diagram contains only a continuous transition regime and a strictly first-order regime, separated by a tricritical or critical end point (k_{s},r_{s}). For kk_{s} and finite r>r_{s}, the transition is discontinuous, with no hybrid features or critical singularities. In this first-order regime, the pseudocritical point approaches the critical point as 1/lnL, where L is the linear system size, distinct from the L^{-d} scaling typical of conventional thermodynamic first-order transitions in d dimensions. This logarithmic finite-size drift is consistent with a nucleation-driven mechanism, in which rare voids trigger the collapse of the finite-range k-core. These results demonstrate that geometric constraints can fundamentally alter the nature of k-core percolation found in finite dimensions.