Polyharmonic fields and Liouville quantum gravity measures on tori of arbitrary dimension: From discrete to continuous

For an arbitrary dimension n$n$, we study: the polyharmonic Gaussian field hL$h_L$ on the discrete torus TLn=1LZn/Zn$\mathbb {T}^n_L = \frac{1}{L} \mathbb {Z}^{n} / \mathbb {Z}^{n}$, that is the random field whose law on RTLn$\mathbb {R}^{\mathbb {T}^{n}_{L}}$ given by cne−bn(−ΔL)n/4h2dh,$$\begin{equation*} \hspace*{-4.5pc}c_n\, \text{e}^{-b_n{\left\Vert (-\Delta _L)^{n/4}h\right\Vert} ^2} dh, \end{equation*}$$where dh$dh$ is the Lebesgue measure and ΔL$\Delta _{L}$ is the discrete Laplacian; the associated discrete Liouville quantum gravity (LQG) measure associated with it, that is, the random measure on TLn$\mathbb {T}^{n}_{L}$ μL(dz)=expγhL(z)−γ22EhL(z)dz,$$\begin{equation*} \hspace*{-7.5pc}\mu _{L}(dz) = \exp {\left(\gamma h_L(z) - \frac{\gamma ^{2}}{2} \mathbf {E} h_{L}(z) \right)} dz, \end{equation*}$$where γ$\gamma$ is a regularity parameter. As L→∞$L\rightarrow \infty$, we prove convergence of the fields hL$h_L$ to the polyharmonic Gaussian field h$h$ on the continuous torus Tn=Rn/Zn$\mathbb {T}^n = \mathbb {R}^{n} / \mathbb {Z}^{n}$, as well as convergence of the random measures μL$\mu _L$ to the LQG measure μ$\mu$ on Tn$\mathbb {T}^n$, for all γ