Search for: All records

Abstract It is well recognized that interpreting transport experiment results can be challenging when the samples being measured are spatially nonuniform. However, quantitative understanding on the differences between measured and actual transport coefficients, especially the Hall effects, in inhomogeneous systems is lacking. In this work we use homogenization theory to find exact bounds of the measured or homogenized anomalous Hall conductivity (AHC) in inhomogeneous conductors under minimal assumptions. In particular, we prove that the homogenized AHC cannot exceed the bounds of the local AHC. However, in common experimental setups, anomalies that appear to violate the above bounds can occur, with a popular example being the “humps” or “dips” of the Hall hysteresis curves usually ascribed to the topological Hall effect (THE). We give two examples showing how such apparent anomalies could be caused by different types of inhomogeneities and discuss their relevance in experiments. 

We initiate the study of approximately counting the number of list packings of a graph. The analogous problem for usual vertex coloring and list coloring has attracted substantial attention. For list packing the setup is similar, but we seek a full decomposition of the lists of colors into pairwise-disjoint proper list colorings. The existence of a list packing implies the existence of a list coloring, but the converse is false. Recent works on list packing have focused on existence or extremal results of on the number of list packings, but here we turn to the algorithmic aspects of counting and sampling. In graphs of maximum degree Δ and when the number of colors is at least Ω(Δ²), we give a fully polynomial-time randomized approximation scheme (FPRAS) based on rapid mixing of a natural Markov chain (the Glauber dynamics) which we analyze with the path coupling technique. Some motivation for our work is the investigation of an atypical spin system, one where the number of spins for each vertex is much larger than the graph degree. 

Abstract Convergence to equilibrium of underdamped Langevin dynamics is studied under general assumptions on the potential U allowing for singularities. By modifying the direct approach to convergence in L 2 pioneered by Hérau and developed by Dolbeault et al , we show that the dynamics converges exponentially fast to equilibrium in the topologies L 2 (d μ ) and L 2 ( W * d μ ), where μ denotes the invariant probability measure and W * is a suitable Lyapunov weight. In both norms, we make precise how the exponential convergence rate depends on the friction parameter γ in Langevin dynamics, by providing a lower bound scaling as min( γ , γ −1 ). The results hold for usual polynomial-type potentials as well as potentials with singularities such as those arising from pairwise Lennard-Jones interactions between particles. 

Abstract Cantor sets are constructed from iteratively removing sections of intervals. This process yields a cumulative distribution function (CDF), constructed from the invariant Borel probability measure associated with their iterated function systems. Under appropriate assumptions, we identify sampling schemes of such CDFs, meaning that the underlying Cantor set can be reconstructed from sufficiently many samples of its CDF. To this end, we prove that two Cantor sets have almost-nowhere intersection with respect to their corresponding invariant measures.