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Vincent Jugé

Publications and source records attributed to Vincent Jugé.

22 records · Page 2Linked to original sources
arXiv2016

Uniform measures on braid monoids and dual braid monoids

We aim at studying the asymptotic properties of typical positive braids, respectively positive dual braids. Denoting by $μ_k$ the uniform distribution on positive (dual) braids of length $k$, we prove that the sequence $(μ_k)_k$ converges to a unique probability measure $μ_{\infty}$ on infinite positive (dual) braids. The key point is that the limiting measure $μ_{\infty}$ has a Markovian structure which can be described explicitly using the combinatorial properties of braids encapsulated in the Möbius polynomial. As a by-product, we settle a conjecture by Gebhardt and Tawn (J. Algebra, 2014) on the shape of the Garside normal form of large uniform braids.

math.GR↗
arXiv2016

Dynamic Complexity of Parity Games with Bounded Tree-Width

Dynamic complexity is concerned with updating the output of a problem when the input is slightly changed. We study the dynamic complexity of two-player parity games over graphs of bounded tree-width, where updates may add or delete edges, or change the owner or color of states. We show that this problem is in DynFO (with LOGSPACE precomputation); this is achieved by a reduction to a Dyck-path problem on an acyclic automaton.

cs.CC↗
arXiv2015

Curve Diagrams, Laminations, and the Geometric Complexity of Braids

Braids can be represented geometrically as curve diagrams. The geometric complexity of a braid is the minimal complexity of a curve diagram representing it. We introduce and study the corresponding notion of geometric generating function. We compute explicitly the geometric generating function for the group of braids on three strands and prove that it is neither rational nor algebraic, nor even holonomic. This result may appear as counterintuitive. Indeed, the standard complexity (due to the Artin presentation of braid groups) is algorithmically harder to compute than the geometric complexity, yet the associated generating function for the group of braids on three strands is rational.

math.GT↗