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The probability that a $n\times n$ real matrix (with elements that are independent random variables with standard normal distributions) has only real eigenvalues is given by $$2^{-n(n-1)/4}$$ Reference: A. Edelman, The Probability that a Random Real Gaussian Matrix has $k$ Real Eigenvalues, Related Distributions, and the Circular Law. Journal of ...

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The axioms don't tell you what theory you constructed. For that you need to go beyond the construction of correlation functions of the elementary field $\phi$ (the basic chapter on renormalization in QFT textbooks) and produce, e.g., by a point-splitting procedure, correlations with insertion of composite fields like $\phi^3$. You should then identify your ...

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Your probability measure is a product measure, so by $$\text{Var}_y(\langle z,X\rangle) = \sum_{i=1}^nz_i^2\text{Var}_{y_i}(X_i)$$ everything reduces to the 1d case. Let $q_y(dx)=e^{xy-\varphi(x)-C(y)}dx$ be one of the marginals, where $y\in\mathbb{R}$ and $C(y)$ is chosen such that $q_y$ is normalized, and denote by $U_y$ the 1d r.v. with distributon $q_y$. ...

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$\newcommand\Ga\Gamma$ Without loss of generality $a=1$. Let then $Q:=\mathcal Q$, $k:=\kappa>0$, and $t:=\theta\sqrt b>0$, so that $\sqrt b\,\gamma$ has the gamma distribution with parameters $k,t$. Let also $c:=\Ga(k)t^k$. Then, letting $f$ denote the standard normal pdf, we have $$c\,EaQ(\sqrt b\,\gamma)=\int_0^\infty dy\,y^{k-1} e^{-y/t}Q(y) =\... 1 Mathematica gives an answer in terms of a hypergeometric function:$$\frac{a 2^{-\frac{\kappa}{2}-\frac{3}{2}} b^{-\frac{\kappa}{2}} {\theta}^{-k} \, _2F_2\left(\frac{\kappa}{2}+\frac{1}{2},\frac{\kappa}{2};\frac{1}{2},\frac{\kappa}{2}+1;\frac{1}{2 b {\theta}^2}\right)}{\sqrt{\pi } \Gamma \left(\frac{\kappa}{2}+1\right)}-\frac{a 2^{-\frac{\kappa}{2}-2} \...

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How about the following reference T. Gneiting, Criteria of Po´lya type for radial positive definite functions, Proc. Am. Math. Soc. 129 (2001), 2309–2318. I have plotted the results for 6th derivative all numeric but it seems working,

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This is equation (4.12) in the cited paper (also at arXiv:1004.4389), which follows from theorem (4.1). The $\xi_i$'s may be independent equally probable random signs, or they may be independent standard normal variables.

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I think Timothy Chow's comment is right that there is no result about planar graphs with your lemma as an explicit corollary. I believe the following 2007 research paper by Guido Helden might be of use to you: http://publications.rwth-aachen.de/record/62349/ It is about hamiltonicity of maximal planar graphs and planar triangulations, and starts with a very ...

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Such a sequence $a_n$ does not exist even for a well studied example like returns to the origin of simple random walk in one dimension. If $X_i$ denotes the number of steps from the $i-1$ time the walk returned to the origin to the $i$'th time, then $X_i$ are i.i.d. and their sum $S_n$ is the number of steps until the $n$'th return time to the origin. In [1] ...

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