Statistics of reduced words in locally free and braid groups: Abstract studies and applications to ballistic growth model

We study numerically and analytically the average length of reduced (primitive) words in the so-called locally free (LFn(d)) and braid (B-n) groups. We consider the situations when the letters in the initial words are drawn either without or with correlations. In the latter case, we show that the average length of the reduced word can be increased or lowered depending on the type of correlation. The ideas developed are used for analytical computation of the average number of peaks of the surface appearing in some specific ballistic growth model.