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Resolvent for tensors

This micro post is devoted to elementary integral formulas.

Integral representation of the determinant. If $\Sigma$ is a positive-definite symmetric $n\times n$ real matrix then the normalization of the density of the multivariate Gaussian law $\mathcal{N}(0,\Sigma)$ is

\[
Z:=\int_{\mathbb{R}^n}\mathrm{e}^{-\frac{1}{2}\langle\Sigma^{-1}x,x\rangle}\mathrm{d}x
=\sqrt{(2\pi)^n\det(\Sigma)}.
\] Conversely, this can also be used as an integral representation of the determinant or equivalently its logarithm : if $S$ is a positive-definite symmetric $n\times n$ real matrix then \[
\log\det(S)
=n\log(2\pi)
-2\log\int_{\mathbb{R}^n}\mathrm{e}^{-\frac{1}{2}\langle Sx,x\rangle}\mathrm{d}x.
\]

Trace of the resolvent. If $\lambda<\min\mathrm{spec}(S)$, then $S-\lambda\mathrm{Id}$ is positive-definite symmetric and

\begin{align*}
\partial_\lambda\log\det(S-\lambda\mathrm{Id})
&=\sum_{i=1}^n\partial_\lambda\log(\lambda_i-\lambda)\\
&=\sum_{i=1}^n\frac{1}{\lambda_i-\lambda}\\
&=\mathrm{trace}((S-\lambda\mathrm{Id})^{-1}).
\end{align*}

Alternatively, with $V_\lambda(x):=\frac{1}{2}\langle(S-\lambda\mathrm{Id})x,x\rangle=\frac{1}{2}\langle\Sigma_\lambda^{-1}x,x\rangle$, and $X_\lambda\sim\mathcal{N}(0,\Sigma_\lambda)$,

\begin{align*}
-2\partial_\lambda\log\int_{\mathbb{R}^n}\mathrm{e}^{-V_\lambda(x)}\mathrm{d}x
&=-2\frac{\int_{\mathbb{R}^n}\partial_\lambda V_\lambda(x)\mathrm{e}^{-V_\lambda(x)}\mathrm{d}x}{\int_{\mathbb{R}^n}\mathrm{e}^{-V_\lambda(x)}\mathrm{d}x}\\
&=\frac{1}{Z_\lambda}\int_{\mathbb{R}^n}\|x\|^2\mathrm{e}^{-\frac{1}{2}\langle\Sigma_\lambda^{-1}x,x\rangle}\mathrm{d}x\\
&=\mathbb{E}[\|X_\lambda\|^2]\\
&=\mathrm{trace}(\Sigma_\lambda)\\
&=\mathrm{trace}((S-\lambda\mathrm{Id})^{-1}).
\end{align*}

This gives also an integral representation of the trace of the resolvent

\[
\mathrm{trace}((S-\lambda\mathrm{Id})^{-1})
=\frac{1}{Z_\lambda}
\int_{\mathbb{R}^n}\|x\|^2\mathrm{e}^{-\langle Sx,x\rangle+\lambda\|x\|^2}\mathrm{d}x.
\]

Alternatively we could use $\lambda\mathrm{Id}-S$ with $\lambda>\max\mathrm{spec}(S)$ to get similar mirror formulas seeing the spectrum from the right instead of from the left. These Gaussian integral formulas allow to define a trace of resolvent for symmetric tensors, beyond symmetric matrices, making a link with a $p$-spin model. This idea was explored notably by Rǎzvan Gurǎu. Rémi Bonnin is working on this topic for his PhD. For a $p$-fold tensor $T$, something like \[
\mathrm{TraceOfResolvent}(T)(\lambda)
:=\frac{1}{Z_\lambda}
\int_{\mathbb{R}^n}\|x\|^2\mathrm{e}^{\lambda\|x\|^2-\sum_{i_1,\ldots,i_p}T_{i_1,\ldots,i_p}x_{i_1}\cdots x_{i_p}}\mathrm{d}x,
\] allowing via series expansions to get a tensor analogue of the trace of powers.

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