Explicit formulas for hook walks on continual Young diagrams

We consider, following the work of S. Kerov, random walks which are continuous-space generalizations of the Hook Walks defined by Greene-Nijenhuis-Wilf, performed under the graph of a continual Young diagram. The limiting point of these walks is a point on the graph of the diagram. We present several explicit formulas giving the probability densities of these limiting points in terms of the shape of the diagram. This partially resolves a conjecture of Kerov concerning an explicit formula for the so-called Markov transform. We also present two inverse formulas, reconstructing the shape of the diagram in terms of the densities of the limiting point of the walks. One of these two formulas can be interepreted as an inverse formula for the Markov transform. As a corollary, some new integration identities are derived.Comment: to appear in Adv. Appl. Mat

Differential equations and exact solutions in the moving sofa problem

The moving sofa problem, posed by L. Moser in 1966, asks for the planar shape of maximal area that can move around a right-angled corner in a hallway of unit width, and is conjectured to have as its solution a complicated shape derived by Gerver in 1992. We extend Gerver's techniques by deriving a family of six differential equations arising from the area-maximization property. We then use this result to derive a new shape that we propose as a possible solution to the "ambidextrous moving sofa problem," a variant of the problem previously studied by Conway and others in which the shape is required to be able to negotiate a right-angle turn both to the left and to the right. Unlike Gerver's construction, our new shape can be expressed in closed form, and its boundary is a piecewise algebraic curve. Its area is equal to $X+\arctan Y$, where $X$ and $Y$ are solutions to the cubic equations $x^2(x+3)=8$ and $x(4x^2+3)=1$, respectively.Comment: Version 2 update: added figures and expanded discussion in section 6. Version 3 update: simplified algebraic formulas in section

On the number of n-dimensional representations of SU(3), the Bernoulli numbers, and the Witten zeta function

We derive new results about properties of the Witten zeta function associated with the group SU(3), and use them to prove an asymptotic formula for the number of n-dimensional representations of SU(3) counted up to equivalence. Our analysis also relates the Witten zeta function of SU(3) to a summation identity for Bernoulli numbers discovered in 2008 by Agoh and Dilcher. We give a new proof of that identity and show that it is a special case of a stronger identity involving the Eisenstein series.Comment: To appear in Acta Arithmetic