Properties of the wormhole-dominant phase in two-dimensional quantum gravity

We study the $N\times N$ Hermitian one-matrix model modified by the double-trace interaction. It is known that the coupling for the double-trace interaction can control the weight for the microscopic wormholes if interpreting the matrix model as the lattice model of random surface; tuning the coupling to its critical value, the effect of wormholes become substantial to change the critical behavior of the pure $2$D quantum gravity, which is characterized by a certain positive value of the string susceptibility. In the large-$N$ limit, we calculate the continuum limit of the disk amplitude in which the wormhole effects are important. The resulting continuum disk amplitude is the same as that of the pure $2$D quantum gravity. We also introduce the renormalized coupling for the double-trace interaction, and show that the newly introduced renormalized coupling can alter the renormalized bulk cosmological constant effectively.