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 $k<k_s$ and $r<r_s$, the transition is continuous and belongs to the universality class of standard two-dimensional (2D) percolation. For $k>k_s$ 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 $1/\ln L$, 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.