Asymptotic bijection and scaling limits for factorizations of a long cycle

par Paul Thévenin (Angers)

jeudi 9 avr. 2026, 14:00 → 15:00 Europe/Paris

Salle de Séminaires (Orléans)

Consider factorizations of the long cycle $(1,2 \ldots,n)$ into a product of transpositions, in the symmetric group $S_n$. It is known that one needs at least $n-1$ transpositions to generate this cycle, and that the number of such factorizations in $n^{n-2}$. We will consider here factorizations into $n-1+2g$ transpositions, for some fixed $g \geq 0$, which we call factorizations of genus $g$. 
Through bijections with a family of maps, I will present a combinatorial construction of a random (almost) uniform factorization of fixed genus, and an explicit algorithm to sample it. From this, I will deduce the scaling limit of a uniform factorization of genus $g$.

Joint work with Valentin Féray and Baptiste Louf.