Charge Fluctuations Close to Phase Separation in the Two-Dimensional $\mathit{t}-\mathit{J}$ Model

We have studied the $t-J$ model using the Green function Monte Carlo technique. We have obtained accurate energies well converged in the thermodynamic limit by performing simulations for up to $242$ lattice sites. By studying the energy as a function of hole doping we conclude that there is no phase separation in the physical region relevant for high- $T_c$ superconductors. This finding is further supported by the hole-hole correlation function calculation. Remarkably, near the phase separation instability, for $J_c/t\sim 0.5$ this function displays enhanced fluctuations at incommensurate wave vectors that scale linearly with the doping, in agreement with experimental findings.