Exact solution of the ${\mathrm{\Phi }}_2^3$ finite matrix model

We find the exact solutions of the ${\mathrm{\Phi }}_2^3$ finite matrix model (Grosse-Wulkenhaar model). In the ${\mathrm{\Phi }}_2^3$ finite matrix model, multipoint correlation functions are expressed as $G_{|a_{1}^{1}\ldots a_{N_{1}}^{1}|\ldots|a_{1}^{B}\ldots a_{N_{B}}^{B}|}$\mathrm{.}The$\displaystyle \sum_{i=1}^{B}N_{i}$-pointfunctiondenotedby$G_{|a_{1}^{1}\ldots a_{N_{1}}^{1}|\ldots|a_{1}^{B}\ldots a_{N_{B}}^{B}|}$ is given by the sum over all Feynman diagrams (ribbon graphs) on Riemann surfaces with $B$-boundaries, and each $\mathrm{\mid }{a}_1^i\cdots a_{N_i}^i\mathrm{\mid }$ corresponds to the Feynman diagrams having $N_i$-external lines from the $i$-th boundary. It is known that any $G_{\mathrm{\mid }{a}_1^1\dots a_{N_1}^1\mathrm{\mid }\dots \mathrm{\mid }{a}_1^B\dots a_{N_B}^B\mathrm{\mid }}$ can be expressed using $G_{\mathrm{\mid }{a}^1\mathrm{\mid }\dots \mathrm{\mid }{a}^n\mathrm{\mid }}$ type $n$-point functions. Thus we focus on rigorous calculations of $G_{\mathrm{\mid }{a}^1\mathrm{\mid }\dots \mathrm{\mid }{a}^n\mathrm{\mid }}$. The formula for $G_{\mathrm{\mid }{a}^1\mathrm{\mid }\dots \mathrm{\mid }{a}^n\mathrm{\mid }}$ is obtained, and it is achieved by using the partition function $\mathcal{Z}[J]$ calculated by the Harish-Chandra-Itzykson-Zuber integral. We give $G_{\mathrm{\mid }a\mathrm{\mid }}$, $G_{\mathrm{\mid }ab\mathrm{\mid }}$, $G_{\mathrm{\mid }a\mathrm{\mid }b\mathrm{\mid }}$, and $G_{\mathrm{\mid }a\mathrm{\mid }b\mathrm{\mid }c\mathrm{\mid }}$ as the specific simple examples. All of them are described by using the Airy functions.