Polyharmonic Fields and Liouville Quantum Gravity Measures on Tori of Arbitrary Dimension: from Discrete to Continuous

For an arbitrary dimension $n$, we study: (a) the Polyharmonic Gaussian Field $h_L$ on the discrete torus ${\mathbb{T}}_L^n=\frac{1}{L}{\mathbb{Z}}^n\mathrm{/}{\mathbb{Z}}^n$, that is the random field whose law on ${\mathbb{R}}^{{\mathbb{T}}_L^n}$ given by \begin{equation*} c_n, e^{-b_n|(-Δ_L)^{n/4}h|^2} dh, \end{equation*} where $dh$ is the Lebesgue measure and ${\mathrm{\Delta }}_L$ is the discrete Laplacian; (b) the associated discrete Liouville Quantum Gravity measure associated with it, that is the random measure on ${\mathbb{T}}_L^n$ \begin{equation*}μ_{L}(dz) = \exp \Big( γh_L(z) - \frac{γ^{2}}{2} $\mathbf{E}$ h_{L}(z) \Big) dz,\end{equation*} where $\gamma$ is a regularity parameter. As $L\to \mathrm{\infty }$, we prove convergence of the fields $h_L$ to the Polyharmonic Gaussian Field $h$ on the continuous torus ${\mathbb{T}}^n={\mathbb{R}}^n\mathrm{/}{\mathbb{Z}}^n$, as well as convergence of the random measures ${\mu }_L$ to the LQG measure $\mu$ on ${\mathbb{T}}^n$, for all $\mathrm{\mid }\gamma \mathrm{\mid }<\sqrt{2n}$.