$c=1$, $R=1$ and $N\gg 1$: ZZ instantons in 2D String Theory and Matrix Integrals

We explore the non-perturbative aspects of $c=1$ string with compactified Euclidean time, its $0+0$ dimensional matrix model duals (at self-dual radius), and $0+1$ dimensional Matrix Quantum Mechanics (free fermion) description. We calculate the instanton normalizations, disk two-point function, and annulus one-point function in worldsheet formalism using string field theory insights. We further match them with the corresponding predictions from the matrix model descriptions. We also have some results and remarks regarding the multi ZZ instanton normalizations and the general $(1,n)$ ZZ instanton normalization for $c=1$ string at both self-dual and generic radius.