arXiv subjects

Lucas Teyssier

Publications and source records attributed to Lucas Teyssier.

9 recordsLinked to original sources

arXiv2026

Cutoff profiles for conjugacy invariant random walks on symmetric groups

We prove asymptotic equivalents of characters for finite-level representations of symmetric groups, that is, for Young diagrams which have all but finitely many boxes on their first row. The proofs rely on computing the number of ribbon tableaux of different types, which allows us to estimate characters via the Murnaghan--Nakayama rule. We deduce that random walks on symmetric groups generated by conjugacy classes with a macroscopic number of fixed points have a Poissonian cutoff profile. We also prove that the random involution walk exhibits cutoff and find its cutoff profile. Finally, we obtain numerics for the random transposition walk on a deck of 52 cards, giving concrete estimates on the question that originally motivated Diaconis and Shahshahani.

math.PR↗

arXiv2026

Stationary hitting times on vertex-transitive graphs

We prove a refined version of the Aldous and Brown's exponential approximation of stationary hitting times. These are valid for all reversible Markov chains. We then specialise our estimates for vertex-transitive graphs, where we obtain improved bounds which depend on the growth of the graphs. The most delicate cases are when the diameter is comparable to that of low-dimensional tori. In particular, in "dimensions" less than four (up to logarithmic factors) our error terms are the square of those of Aldous and Brown. These improved bounds play a crucial role in the companion work arXiv:2202.02255 characterising the fluctuations of the cover time on vertex-transitive graphs.

math.PR↗

arXiv2025

Every cutoff profile is possible

We introduce fruit-inosculated-tree Markov chains. These chains have easily tunable parameters and are a good source of examples. In particular, we prove that every cutoff profile is possible, with any cutoff time and window size.

math.PR↗

arXiv2025

Bounds on skew dimensions and characters of symmetric groups via thick hook decompositions

We bound the number of standard tableaux of skew shapes via thick hook decompositions in the Naruse hook length formula. Combining this with elementary counting arguments in the Murnaghan--Nakayama rule, we establish a uniform bound on characters of symmetric groups $\mathfrak{S}_n$. In the case of balanced representations, this improves on the character bounds of F\'eray and \'Sniady for permutations with support size at least $n^{2/3}$, and is sharp for permutations with support size of order $n$.

math.CO↗

arXiv2025

Sharp character bounds and cutoff for symmetric groups

We develop a flexible technique to bound the characters of symmetric groups, via the Naruse hook length formula, the Larsen--Shalev character bounds, and appropriate diagram slicings. It allows us to prove a uniform exponential character bound with optimal constant $1/2$. We furthermore prove sharp character bounds for conjugacy classes having a macroscopic number of fixed points, and deduce that the random walks on the associated Cayley graphs exhibit a total variation and $L^2$ cutoff.

math.RT↗

arXiv2025

Characters of symmetric groups: sharp bounds on virtual degrees and the Witten zeta function

We prove sharp bounds on the virtual degrees introduced by Larsen and Shalev. This leads to improved bounds on characters of symmetric groups. We then sharpen bounds of Liebeck and Shalev concerning the Witten zeta function. Our main application is a characterization of the fixed-point free conjugacy classes whose associated random walk mixes in 2 steps.

math.RT↗

arXiv2022

On the universality of fluctuations for the cover time

We consider random walks on finite vertex-transitive graphs $\Gamma$ of bounded degree. We find a simple geometric condition which characterises the cover time fluctuations: the suitably normalised cover time converges to a standard Gumbel variable if and only if $\mathrm{Diam}(\Gamma)^2 = o(n/\log n)$, where $n = |\Gamma|$. We prove that this condition is furthermore equivalent to the decorrelation of the uncovered set. The arguments rely on recent breakthroughs by Tessera and Tointon on finitary versions of Gromov's theorem on groups of polynomial growth, which we leverage into strong heat kernel bounds, and refined quantitative estimates on Aldous and Brown's exponential approximation of hitting times, which are of independent interest.

math.PR↗

arXiv2021

Cutoff profiles for quantum Lévy processes and quantum random transpositions

We consider a natural analogue of Brownian motion on free orthogonal quantum groups and prove that it exhibits a cutoff at time $N\ln(N)$. Then, we study the induced classical process on the real line and compute its atoms and density. This enables us to find the cutoff profile, which involves free Poisson distributions and the semi-circle law. We prove similar results for quantum permutations and quantum random transpositions.

math.PR↗

arXiv2019

Limit profile for random transpositions

We present an improved version of Diaconis' upper bound lemma, which is used to compute the limiting value of the distance to stationarity. We then apply it to random transpositions studied by Diaconis and Shahshahani.

math.PR↗