Standard setup. Consider a group and choose generators. Word-metric (or in the other words - distance on the Cayley graph of the group+generators) - converts a group into a metric space, which is topic of numerous research.
Consider the case of FINITE groups. Define growth polynomial $g(t) = \sum_i g_i t^{i} $, where $g_i$ - number of elements on distance strictly $i$ from the identity (sizes of spheres).
Question 1: For FINITE nilpotent groups is it true that numbers $g_i$ can be well approximated by the Gaussian normal distribution ? (A kind of central limit theorem for FINITE groups).
More precise/specific version. Motivated by simulations below and by Bass–Guivarc'h formula. Consider finite nilpotent groups such that $G_k/G_{k+1} = (Z/n)^{d_k} $ , where $G_k$ are lower central series. (Example: upper triangular matrices over $Z/n$). Assume image of the generating set forms a basis in the linear space $\oplus_k (Z/n)^{d_k} $.
Question 2: Under conditions above - is it true that the error term between $g_i$ and Gaussian can be estimated similarly to the classical Berry–Esseen theorem, for $n$ is sum of $d_i$ (i.e. rank of abelian counterpart) ?
(Two things are essential here: we want periods of $G_k/G_{k+1}$ will be more or less of the same magnitude - like in standard probability - for simplicity we required all equal to $n$, second: generators should form a basis - that guess from the simulations below).
The approximation by the Gaussian works NOT only for nilpotent groups, but sometimes for other groups e.g. for standard generators of $S_n$ by transpositions $(i,i+1)$ (result goes back to Mann,Whitney, Wilcoxon and especially Kendall (to get p-values for their tests/correlations we need it), in modern language to Diaconis (see MO320497) ). However it seems it is more like an exception : for random choice of generators for non-nilpotent groups the distribution of $g_i$ seems to be quite far from the Gaussian - it will be growing exponentially at initial period (see MO). More examples link. So it is natural to ask:
Question 3: If fit by Gaussian fails miserably (in some sense) - can we deduce that group is NOT nilpotent ? (For finite groups).
PS
Googling on these questions was not helpful, previous (less-focused) questions on MO are unanswered for years. Growth functions of finite group - computation, typical behaviour, surveys? , Metrics on finite groups and generalizations of central limit theorems for balls volumes (à la Diaconis-Graham)
Motivations. Examples. Etc.
Abelian example. For the abelian groups $\oplus (Z/n_i)$ and the standard choice the symmetric generators (i.e. $e_i$ and $-e_i$). The growth polynomial is the product of polynomials for each subgroup and for $Z/n$ it is given by $(1+2t+2t^2 +... +2t^{[n/2]-1}+ (1or2)t^{[n/2]} $, for even $n$ the last term is 1, for odd: 2. E.g. n=2 $1+t$, n=3 $1+2t$, n=4 $1+2t+t^2$, n=5 $1+2t+2t^2$. For $\oplus (Z/n_i)$ polynomials are multiplied, multiplication of polynomials is convolution in $Z\subset R$, convolution in $R$ is exactly the text-book standard limit theorem and so we are more less done and all the estimates seems follow from textbooks. See figures here: https://www.kaggle.com/code/alexandervc/growth-commutative-groups-z-p-d
The next to abelian example - is the Heisenberg group - surprisingly it is not clear is the growth polynomial known even for standard generators of H3 or not - see MO Polynomials of growth for finite Heisenberg groups . But numerical fit by Gaussian is quite good for large ranks - figures below.
Geometric perspective.
(Finitization of Gromov-like ideas and "embeddings" in machine learning). It is well-known by Bass, Guivarc'h, Gromov et.al. that for INIFINITE groups the "polynomial growth" for the word-metric spheres - means that group is "close to nilpotent" (see Gromov's theorem). Geometrically - the polynomial growth is exactly the growth of balls in $R^n$, and the whole bunch of activity teach us that we should think of these groups as kind of lattices in some manifolds (at least informally). That is close to recent machine learning activity where finding "embeddings" (i.e. latent/internal representation of graphs/texts/audio/videos/...". ) is a key to the most of the successes of artificial intelligence.
Now if groups are FINITE we may expect their natural "embeddings" into COMPACT manifolds. We may expect that for nilpotent groups the growth should be like $r^n$ (polynomial) for small radius and since everything is compact should become a "bell-shaped" (Gaussian) for larger "r". And so the questions above are related to - what are the growth patterns of sphere volumes for compact manifolds ? It seems it again follows the Guassian - thus confirming the expectations above. (See separate MO-question).
Topological perspective.
To some extent polynomials $P(t)$ might be thought as analogues of the Poincare polynomials (and P(t=-1) as Euler characteristic - see separate MO question).
Nilpotent group can be thought as multi-fibered manifold due existance of that central series $G_k/G_{k+1}$. If the fibration would be the trivial one - then $P(t) = \prod_i P_i(t)$ and we are done - so essentially the Gaussian arises similar to the classical probability due to that product structure.
So essentially the question is about the following - assume the fibration is not trivial and so Poincare polynomial is NOT a product. However we might hope that cental limit theorem will survive even in that case. It is analougues to classical probability - we do not need strictly INDEPENDENT components - just we need dependence is not that much big. Small dependence a bit decreases the convergence rate, but does not affect the final outcome - Gaussian. (See separate MO-question).
As an example take $SU(n)$ it is a kind of multiple-fibered by spheres $SU(n-1)->SU(n)->S^{n-1}$ (and iterate many times). Poincare polynomial is $\prod (1-t^{2i+1}) $ - and its coefficients are well approximated by the Gaussian - see https://www.kaggle.com/code/eugenedurymanov/poincare-polynomial-coeffs-matched-with-norm-distr (thanks to Eugene Durymanov). (May be the case is it too simple since the Poincare polynomial is product - do not feel non-triviality of fibration - but still).
(Some analogy between group/graphs quotients and fibrations is discussed in T.Tao's blog).
Numeric simulations
Here are some results of the numeric simulations. More can be found https://www.kaggle.com/code/alexandervc/growth-in-finite-groups-heisenberg-uptriag-etc Thanks to Aleksei Rozanov for performing simulations https://gist.github.com/hacklex/ and discussions.
Quite good approximations.
Group is nilpotent, but conditions of question 2 are NOT fulfilled (we take only 7 generators for 8x8 upper-triangular matrices) - approximation is worse, but may be for higher rank groups it will still be true - just the convergence is not fast. Moreover misfit is not that much seen for cdf, and we need to take LOG (log-scale for y axis) to see misfit better:
