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Let $K({\bf r}_1,{\bf r}_2)= e^{-\frac12|{\bf r}_1-{\bf r}_2|^2}$ be the translation and rotation invariant Gaussian kernel. I want to compute the trace Tr$\hspace{1pt}K^3$ over the $n$-dimensional ball $B_n$ ($n=2$ is the disc) of radius $R$, that is $${\rm Tr}\hspace{1pt}K^3 = \int_{B_n^3} d^n{\bf r}_1 d^n{\bf r}_2 d^n{\bf r}_3\, K({\bf r}_1,{\bf r}_2)K({\bf r}_2,{\bf r}_3)K({\bf r}_3,{\bf r}_1)\,.$$ In particular, I want to obtain the expansion at $R\rightarrow\infty$ up to the $O(1)$ term included.

I was wondering if a closed formula for the Gaussian kernel existed? I can perform one radial integral but after that it seems difficult, especially in $n>2$ dimensions. Symmetry allows to reduce the number of angular integrals, but it still remain quite complicated.

Kaio
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  • At least you can get a upperbound terms with https://en.wikipedia.org/wiki/Volume_of_an_n-ball by getting an up bound. – EDX Feb 17 '23 at 03:46
  • What norm do you take on $B_n$ ? – EDX Feb 17 '23 at 03:51
  • You should look at this answer https://math.stackexchange.com/questions/126227/reference-for-multidimensional-gaussian-integral . – EDX Feb 17 '23 at 04:09
  • I take the Euclidean norm, and the integration is, e.g., for $n=2$, $\int_{B_2} d^2{\bf r}=\int_0^{2\pi}\int_0^R r dr d\theta$. – Kaio Feb 17 '23 at 12:11

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