2. Edgeflip chains

Edgeflip chain on dyadic tilings and rectangular dissections (Sarah Cannon)
Consider rectangular dissections of an $n\times n$ lattice into $n$ rectangles of area $n$, where $n=2^k$. The edgeflip chain proceeds by picking an edge bordering two rectangles and replacing it by its bisector. In the dyadic case, the edge is flipped provided the resulting tiling remains dyadic. In the weighted setting, a parameter $\lambda>0$ is fixed, and the weight of a dissection is given by $$ \pi(\sigma)=\frac{\lambda^{\sigma }}{Z} , $$ where $\sigma$ is the total edgelength. The chains (in the dyadic and general cases) then correspond to the Glauber dynamics.Problem 2.1.
[Sarah Cannon] For $\lambda=1$, can we establish a polynomial upperbound (fastmixing) for the mixing time of the edgeflip chain on dyadic tilings or on rectangular dissections?
References:
 Sarah Cannon, Sarah Miracle, and Dana Randall, "Phase Transitions in Random Dyadic Tilings and Rectangular Dissections".
 Svante Janson, Dana Randall, and Joel Spencer, "Random dyadic tilings of the unit square".
 Mike Korm, PhD thesis, Chapter 7. 
Edgeflip chain on triangulations
Problem 2.2.
[Alexandre Stauffer] Consider the chain over triangulations of $[0,n]^2$, in which, at each step, an edge is randomly chosen and, if it is the diagonal of convex quadrilateral, flipped to the opposite diagonal. Does this chain have polynomial mixing time?
References:
 Pietro Caputo, Fabio Martinelli, Alistair Sinclair, Alexandre Stauffer, "Random lattice triangulations".
 Pietro Caputo, Fabio Martinelli, Alistair Sinclair, Alexandre Stauffer, "Dynamics of Lattice Triangulations on Thin Rectangles".
 Alexandre Stauffer, "A Lyapunov function for Glauber dynamics on lattice triangulations". 
Edgeflip on triangulations of a convex polygon
Problem 2.3.
[Emma Cohen] Consider the random walk on triangulations of a convex polygon, driven by flips of a randomly chosen diagonal. What is the mixing time? 
Random walks on Dyck’s paths
Problem 2.4.
[Emma Cohen] Consider the Markov chain on the set of Dyck’s paths, which moves by choosing uniformly at random two coordinates and exchanging their value ($+$ or $1$) provided the resulting path remains a Dyck’s path. What is the mixing time?
Cite this as: AimPL: Markov chain mixing times, available at http://aimpl.org/markovmixing.