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Context

I would like to predict the connectivity of the so-called cosmic web in arbitrary dimensions.

This is the cosmic web (in a hydrodynamical simulation)

Mathematica graphics

The little wiggly things are galaxies (as traced by the gas density) while the inset shows the corresponding light produced by the stars which were born from that gas.

The connectivity $\kappa$ is defined as the number of ridges connecting a given maxima to its surrounding saddles, which in turn therefore corresponds to $2 n_{\rm max}/ n_{\rm saddle}$ where $n_{\rm max}$ and $n_{\rm saddle}$ are the total number of maxima and saddles in the field.

This is how the cosmic web looks like in 3D on larger scales (using a filament tracer algorithm):

Mathematica graphics

so the motivation is to predict how many filaments are connected to a given node from first principes.

Definitions

As a starting point I am restricting myself to a Gaussian random field. For such fields the connectivity $k^d$ is then simply

\begin{equation} \kappa^{d} =\frac{2n_{\rm saddle}}{n_{\rm max}}= \frac{2 \left\langle \Theta_{\rm H}(-\{\lambda_{i}\}_{i<d}) \Theta_{\rm H}(\lambda_{d}) \left| \prod \lambda_i \right| \right\rangle} {\left\langle \Theta_{\rm H}(-\{\lambda_{i}]\}_{i\leq d}) \left| \prod \lambda_i \right| \right\rangle}\,, \label{eq:defkapND} \end{equation}

where the expectation is to be carried over Gaussian PDFs. Technically I therefore need to evaluate in arbitrary dimensions integrals such as

\begin{equation} \int \cdots \int \prod_{i\le { d}} d \lambda_i \, G(\{\lambda_i\}) \prod_{i<j} (\lambda_j-\lambda_i) \, \prod_{i\le d}\Theta_{\rm H}(-\lambda_i) \,, \label{eq:defQprob} \end{equation} where $\Theta_{\rm H}$ is a Heaviside function and $G(\lambda_i)$ is a multi-Gaussian PDF with variance covariance given by \begin{equation} M_{i,j}= \frac{1}{d (d+2)} \quad {\rm for}\quad i\neq j\\ M_{i,i}= \frac{3}{d (d+2)} \end{equation}

For instance for $d=7$ this matrix reads

\begin{equation} \left( \begin{array}{cccccc} \frac{1}{16} & \frac{1}{48} & \frac{1}{48} & \frac{1}{48} & \frac{1}{48} & \frac{1}{48} \\ \frac{1}{48} & \frac{1}{16} & \frac{1}{48} & \frac{1}{48} & \frac{1}{48} & \frac{1}{48} \\ \frac{1}{48} & \frac{1}{48} & \frac{1}{16} & \frac{1}{48} & \frac{1}{48} & \frac{1}{48} \\ \frac{1}{48} & \frac{1}{48} & \frac{1}{48} & \frac{1}{16} & \frac{1}{48} & \frac{1}{48} \\ \frac{1}{48} & \frac{1}{48} & \frac{1}{48} & \frac{1}{48} & \frac{1}{16} & \frac{1}{48} \\ \frac{1}{48} & \frac{1}{48} & \frac{1}{48} & \frac{1}{48} & \frac{1}{48} & \frac{1}{16} \\ \end{array} \right) \end{equation}

Questions

Is is possible to compute the ratio analytically for $d>3$?

For instance

\begin{equation}\kappa^2=4 \end{equation}

for $d=2$ and

\begin{equation} \kappa^3=2 \frac{18 \sqrt{2} + 29 \sqrt{3}}{ {-18 \sqrt{2} + 29 \sqrt{3}}} \approx 6.11 \end{equation}

for $d=3$. It is strikingly close to a cubic face centred lattice, but with some level of impurity to the crystal.

Can this ratio be computed numerically for $d>11$?

So far I have

$\kappa^d$ = $4$, $6.11$, $8.35$, $10.73$, $13.23$, $15.85$, $18.7$, $21.4$, $24.4$, $27.4$

for $d=$ $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$ and $11$ resp.

From inspection of $d\le 11$ we conjecture that it is closely approximated by

\begin{equation} \kappa^d= 2d+\left(\frac{2d-4}{7}\right)^{7/4}. \end{equation}

which suggests that the $d$ cosmic web behaves like a crystal with growing impurity.

Mathematica graphics

can the large d asymptote be computed?

FYI, this question is linked to that mathematica question and this paper. FYI2: this paper may be relevant to the asymptote ?

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  • $\begingroup$ The integral you are trying to compute looks a heck of lot like the expectation values of certain matrix models (cf. mathoverflow.net/q/292318/106114 and mathoverflow.net/a/95256/106114). Perhaps you could use that information to find a closed form solution. Good luck! $\endgroup$ – AccidentalFourierTransform May 22 '18 at 14:44
  • $\begingroup$ @AccidentalFourierTransform it would be short of the Heaviside functions. It would then be closely related to Euler Characteristics. $\endgroup$ – chris May 22 '18 at 14:50
  • $\begingroup$ Based on your Mathematica code of the previous question, it seems to me that you are interested in $$ \kappa^d = 2d \, \frac{\mathbb{E}\left[|X|\mathbf{1}_{\{\lambda_1\leq 0,\cdots,\lambda_d\leq 0\}}\right]}{\mathbb{E}\left[|X|\mathbf{1}_{\{\lambda_1\leq 0,\cdots\lambda_{d-1}\leq 0\}}\mathbf{1}_{\{\lambda_d\geq 0\}}\right]},$$ where $\lambda\sim\mathcal{N}(0,M)$ is the centered gaussian field with the covariance kernel $M$ and $$X=\lambda_1\cdots\lambda_d\prod_{1\leq i<j\leq d}(\lambda_j-\lambda_i).$$ Is your definition of $n_{\text{saddle}}$ entails the appearance of the factor $d$? $\endgroup$ – Sangchul Lee May 21 at 9:19
  • $\begingroup$ Ah yes, my bad. The numerator and the denominator is swapped. As for the factor $2d$, it is based on your previous Mathematica code, where the quantity of interest was written as the 2d times the ratio of two Integrate functions. $\endgroup$ – Sangchul Lee May 21 at 11:01
  • $\begingroup$ the factor of $d$ was a typo. I have removed it. $\endgroup$ – chris May 21 at 14:03
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It can be shown that $$\kappa^4 =\frac{200 \pi }{75 \pi -114 -100 \cot ^{-1}(2)}$$ The proof relies on noting that the ratios of numbers of critical points can be equal to the ratio of the disappearance rate of critical points as one smooths the fields plus one. The latter involves expressions like

\begin{equation} \int \cdots \int \prod_{i\leq d} d \lambda_i \, G(\{\lambda_i\}) \prod_{i<j} (\lambda_j-\lambda_i) \,\big( \prod_{i\leq d}\Theta_{\rm H}(-\lambda_i)\big) \delta_{\rm D}(\lambda_k) \end{equation}

noting the extra Dirac Delta. Thanks to this Dirac the integral can still be evaluated for $d=4$.

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