Random generation and scaling limits of fixed genus factorizations into transpositions

Abstract We study the asymptotic behaviour of random factorizations n -cycle into transpositions fixed genus $$g>0$$ g > 0 . They have a geometric interpretation as branched covers sphere and their enumeration Hurwitz numbers was extensively studied in algebraic combinatorics enumerative geometry. On probabilistic side, several models properties permutation were previous works, particular minimal cycles (which corresponds to case $$g=0$$ = this work). Using representation via unicellular maps, we first exhibit an algorithm which samples asymptotically uniform factorization g linear time. In second time, code process chords appearing one by unit disk, prove convergence (as $$n\rightarrow \infty $$ n → ∞ ) associated with -cycle. The limit can be explicitly constructed from Brownian excursion. Finally, establish natural process, coding appearance successive genera factorization.