We consider the trajectories of points on under sequences of certain folding maps associated with reflections. The main result characterizes collections of folding maps that produce dense trajectories. The minimal number of maps in such a collection is d+1.
There are many known results analogous to Theorem for other collections of maps associated with linear isometries of spheres. For example, Crouch and Silva Leite @cite_1 found pairs of one-parameter subgroups of @math which generate all of @math , and produced upper bounds on the number of elements from each subgroup required to generate any element (see also Levitt and Sussmann @cite_13 for related results). An example of this phenomenon is the Euler angles decomposition, a formula for writing any element of @math as a product of three rotations about the @math and @math axes. Rosenthal @cite_5 has established bounds on the rate of convergence to the steady state for random walks generated by conjugacy classes of planar rotations on @math . Porod @cite_9 has obtained analogous results for random walks on @math , @math and @math generated by the conjugacy class of reflections.
@cite_13 @cite_5 @cite_9 @cite_1
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We prove a conjecture raised by the work of Diaconis and Shahshahani (1981) about the mixing time of random walks on the permutation group induced by a given conjugacy class. To do this we exploit a connection with coalescence and fragmentation processes and control the Kantorovitch distance by using a variant of a coupling due to Oded Schramm. Recasting our proof in the language of Ricci curvature, our proof establishes the occurrence of a phase transition, which takes the following form in the case of random transpositions: at time @math , the curvature is asymptotically zero for @math and is strictly positive for @math . We consider an analogue of the Kac random walk on the special orthogonal group SO(N)SO(N), in which at each step a random rotation is performed in a randomly chosen 2-plane of RNRN. We obtain sharp asymptotics for the rate of convergence in total variance distance, establishing a cut-off phenomenon in the large NN limit. In the special case where the angle of rotation is deterministic, this confirms a conjecture of Rosenthal [Ann. Probab. 22 (1994) 398–423]. Under mild conditions, we also establish a cut-off for convergence of the walk to stationarity under the L2L2 norm. Depending on the distribution of the randomly chosen angle of rotation, several surprising features emerge. For instance, it is sometimes the case that the mixing times differ in the total variation and L2L2 norms. Our estimates use an integral representation of the characters of the special orthogonal group together with saddle point analysis. In this article we prove global rigidity results for hyperbolic actions of higher-rank lattices. Suppose @math is a lattice in semisimple Lie group, all of whose factors have rank @math or higher. Let @math be a smooth @math -action on a compact nilmanifold @math that lifts to an action on the universal cover. If the linear data @math of @math contains a hyperbolic element, then there is a continuous semiconjugacy intertwining the actions of @math and @math , on a finite-index subgroup of @math . If @math is a @math action and contains an Anosov element, then the semiconjugacy is a @math conjugacy. As a corollary, we obtain @math global rigidity for Anosov actions by cocompact lattices in semisimple Lie group with all factors rank @math or higher. We also obtain global rigidity of Anosov actions of @math on @math for @math and probability-preserving Anosov actions of arbitrary higher-rank lattices on nilmanifolds. We study the pioneer points of the simple random walk on the uniform infinite planar quadrangulation (UIPQ) using an adaptation of the peeling procedure of Angel (Geom Funct Anal 13:935–974, 2003) to the quadrangulation case. Our main result is that, up to polylogarithmic factors, n 3 pioneer points have been discovered before the walk exits the ball of radius n in the UIPQ. As a result we verify the KPZ relation (Modern Phys Lett A 3:819–826, 1988) in the particular case of the pioneer exponent and prove that the walk is subdiffusive with exponent less than 1 3. Along the way, new geometric controls on the UIPQ are established.