On uniqueness for a rough transport-diffusion equation

In this Note, we study a transport-diffusion equation with rough coefficients and we prove that solutions are unique in a low-regularity class

Schur-Weyl duality and the heat kernel measure on the unitary group

We establish a convergent power series expansion for the expectation of a product of traces of powers of a random unitary matrix under the heat kernel measure. These expectations turn out to be the generating series of certain paths in the Cayley graph of the symmetric group. We then compute the asymptotic distribution of a random unitary matrix under the heat kernel measure on the unitary group U(N) as N tends to infinity, and prove a result of asymptotic freeness result for independent large unitary matrices, thus recovering results obtained previously by Xu and Biane. We give an interpretation of our main expansion in terms of random ramified coverings of a disk. Our approach is based on the Schur-Weyl duality and we extend some of our results to the orthogonal and symplectic cases

Constrained dynamics of a polymer ring enclosing a constant area

The dynamics of a polymer ring enclosing a constant {\sl algebraic} area is studied. The constraint of a constant area is found to couple the dynamics of the two Cartesian components of the position vector of the polymer ring through the Lagrange multiplier function which is time dependent. The time dependence of the Lagrange multiplier is evaluated in a closed form both at short and long times. At long times, the time dependence is weak, and is mainly governed by the inverse of the first mode of the area. The presence of the constraint changes the nature of the relaxation of the internal modes. The time correlation of the position vectors of the ring is found to be dominated by the first Rouse mode which does not relax even at very long times. The mean square displacement of the radius vector is found to be diffusive, which is associated with the rotational diffusion of the ring.Comment: 6 page

Low-order continuous finite element spaces on hybrid non-conforming hexahedral-tetrahedral meshes

This article deals with solving partial differential equations with the finite element method on hybrid non-conforming hexahedral-tetrahedral meshes. By non-conforming, we mean that a quadrangular face of a hexahedron can be connected to two triangular faces of tetrahedra. We introduce a set of low-order continuous (C0) finite element spaces defined on these meshes. They are built from standard tri-linear and quadratic Lagrange finite elements with an extra set of constraints at non-conforming hexahedra-tetrahedra junctions to recover continuity. We consider both the continuity of the geometry and the continuity of the function basis as follows: the continuity of the geometry is achieved by using quadratic mappings for tetrahedra connected to tri-affine hexahedra and the continuity of interpolating functions is enforced in a similar manner by using quadratic Lagrange basis on tetrahedra with constraints at non-conforming junctions to match tri-linear hexahedra. The so-defined function spaces are validated numerically on simple Poisson and linear elasticity problems for which an analytical solution is known. We observe that using a hybrid mesh with the proposed function spaces results in an accuracy significantly better than when using linear tetrahedra and slightly worse than when solely using tri-linear hexahedra. As a consequence, the proposed function spaces may be a promising alternative for complex geometries that are out of reach of existing full hexahedral meshing methods

Covariant Symanzik identities

Classical isomorphism theorems due to Dynkin, Eisenbaum, Le Jan, and Sznitman establish equalities between the correlation functions or distributions of occupation times of random paths or ensembles of paths and Markovian fields, such as the discrete Gaussian free field. We extend these results to the case of real, complex, or quaternionic vector bundles of arbitrary rank over graphs endowed with a connection, by providing distributional identities between functionals of the Gaussian free vector field and holonomies of random paths. As an application, we give a formula for computing moments of a large class of random, in general non-Gaussian, fields in terms of holonomies of random paths with respect to an annealed random gauge field, in the spirit of Symanzik's foundational work on the subject.Comment: 51 pages, 10 figures. This version contains a new introduction, an additional Section (6.8) detailing an important example (the case of trace-positive holonomies), and a treatment of the quaternionic case. The introductory material on continuous time random walks on multigraphs in Section 1 was also simplifie

On The Douglas-Kazakov Phase Transition

We give a rigorous proof of the fact that a phase transition discovered by Douglas and Kazakov in 1993 in the context of two-dimensional gauge theories occurs. This phase transition can be formulated in terms of the Brownian bridge on the unitary group U(N) when N tends to infinity. We explain how it can be understood by considering the asymptotic behaviour of the eigenvalues of the unitary Brownian bridge, and how it can be technically approached by means of Fourier analysis on the unitary group. Moreover, we advertise some more or less classical methods for solving certain minimisation problems which play a fundamental role in the study of the phase transition