Blobbed topological recursion of the $\lambda\Phi^4$ matrix model

Apparaît dans la collection : 2023 - T1A - WS1 - Quantum gravity and random geometry

We consider an $N\times N$ Hermitian matrix model with measure $d\mu_{E,\lambda}(\Phi)=\frac{1}{Z} \exp(-\frac{\lambda N}{4} \mathrm{tr}(\Phi^4)) d\mu_{E,0}(\Phi)$ where $d\mu_{E,0}$ is the Gaussian measure with covariance $\langle \Phi_{kl}\Phi_{mn}\rangle =\frac{\delta_{kn}\delta_{lm}}{N(E_k+E_l)}$ for given $E_1,...,E_N>0$. We explain how this setting gives rise to two ramified coverings $x,y$ of the Riemann sphere strongly tied by $y(z)=-x(-z)$ and a family $\omega_{g,n}$ of meromorphic differentials. We provide strong evidence that the $\omega_{g,n}$ obey blobbed topological recursion due to Borot and Shadrin. A key step is to extract from the matrix model a system of six meromorphic functions which satisfy interwoven Dyson-Schwinger equations. Two of these functions are symmetric in the preimages of $x$ and can be determined from their consistency relations. Their expansion at $\infty$ gives global linear and quadratic loop equations for the $\omega_{g,n}$. These global equations provide the $\omega_{g,n}$ not only in the vicinity of the ramification points of $x$ but also in the vicinity of all other poles located at opposite diagonals $z_i+z_j=0$ and at $z_i=0$.