Abstract
By a well known calculation on Gaussian measure, if X is finite, for any \({X}{\varepsilon}{\mathbb{R}}^{X}_{+}\) \(\frac{\sqrt{\det(M_{\lambda}-C)}}{(2\pi)^{\left| X\right| /2}}\int_{\mathbb{R}^{X}}e^{-\frac{1}{2}\sum\chi_{u}(v^{u})^{2}}e^{-\frac{1} {2}e(v)}\Pi_{u\in X}dv^{u}=\sqrt{\frac{\det(G_{\chi})}{\det(G)}}\) and \(\frac{\sqrt{\det(M_{\lambda}-C)}}{(2\pi)^{\left| X\right| /2}}\int_{\mathbb{R}^{X}}v^xv^ye^{-\frac{1}{2}\sum\chi_{u}(v^{u})^{2}}e^{-\frac{1} {2}e(v)}\Pi_{u\in X}dv^{u}=(G_{\chi})^{x,y}\sqrt{\frac{\det(G_{\chi})}{\det(G)}} \)
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© 2011 Springer-Verlag Berlin Heidelberg
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Jan, Y.L. (2011). The Gaussian Free Field. In: Markov Paths, Loops and Fields. Lecture Notes in Mathematics(), vol 2026. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21216-1_5
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DOI: https://doi.org/10.1007/978-3-642-21216-1_5
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Online ISBN: 978-3-642-21216-1
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