Spatial Mixing and Non-local Markov chains

We consider spin systems with nearest-neighbor interactions on an $n$-vertex $d$-dimensional cube of the integer lattice graph $\mathbb{Z}^d$. We study the effects that exponential decay with distance of spin correlations, specifically the strong spatial mixing condition (SSM), has on the rate of convergence to equilibrium distribution of non-local Markov chains. We prove that SSM implies $O(\log n)$ mixing of a block dynamics whose steps can be implemented efficiently. We then develop a methodology, consisting of several new comparison inequalities concerning various block dynamics, that allow us to extend this result to other non-local dynamics. As a first application of our method we prove that, if SSM holds, then the relaxation time (i.e., the inverse spectral gap) of general block dynamics is $O(r)$, where $r$ is the number of blocks. A second application of our technology concerns the Swendsen-Wang dynamics for the ferromagnetic Ising and Potts models. We show that SSM implies an $O(1)$ bound for the relaxation time. As a by-product of this implication we observe that the relaxation time of the Swendsen-Wang dynamics in square boxes of $\mathbb{Z}^2$ is $O(1)$ throughout the subcritical regime of the $q$-state Potts model, for all $q \ge 2$. We also prove that for monotone spin systems SSM implies that the mixing time of systematic scan dynamics is $O(\log n (\log \log n)^2)$. Systematic scan dynamics are widely employed in practice but have proved hard to analyze. Our proofs use a variety of techniques for the analysis of Markov chains including coupling, functional analysis and linear algebra

Uniform bounds on complexity and transfer of global properties of Nash functions

We show that the complexity of semialgebraic sets and mappings can be used to parametrize Nash sets and mappings by Nash families. From this we deduce uniform bounds on the complexity of Nash functions that lead to first-order descriptions of many properties of Nash functions and a good behaviour under real closed field extension (e.g. primary decomposition). As a distinguished application, we derive the solution of the extension and global equations problems over arbitrary real closed fields, in particular over the field of real algebraic numbers. This last fact and a technique of change of base are used to prove that the Artin-Mazur description holds for abstract Nash functions on the real spectrum of any commutative ring, and solve extension and global equations in that abstract setting. To complete the view, we prove the idempotency of the real spectrum and an abstract version of the separation problem. We also discuss the conditions for the rings of abstract Nash functions to be noetherian

Heat Kernel Bounds for the Laplacian on Metric Graphs of Polygonal Tilings

We obtain an upper heat kernel bound for the Laplacian on metric graphs arising as one skeletons of certain polygonal tilings of the plane, which reflects the one dimensional as well as the two dimensional nature of these graphs.Comment: 8 page

Tame Functions with strongly isolated singularities at infinity: a tame version of a Parusinski's Theorem

Let f be a definable function, enough differentiable. Under the condition of having strongly isolated singularities at infinity at a regular value c we give a sufficient condition expressed in terms of the total absolute curvature function to ensure the local triviality of the function f over a neighbourhood of c and doing so providing the tame version of Parusinski's Theorem on complex polynomials with isolated singularities at infinity.Comment: 20 page

Uncertainty inequalities on groups and homogeneous spaces via isoperimetric inequalities

We prove a family of $L^p$ uncertainty inequalities on fairly general groups and homogeneous spaces, both in the smooth and in the discrete setting. The crucial point is the proof of the $L^1$ endpoint, which is derived from a general weak isoperimetric inequality.Comment: 17 page

On the regularization scheme and gauge choice ambiguities in topologically massive gauge theories

It is demonstrated that in the (2+1)-dimensional topologically massive gauge theories an agreement of the Pauli-Villars regularization scheme with the other schemes can be achieved by employing pairs of auxiliary fermions with the opposite sign masses. This approach does not introduce additional violation of discrete (P and T) symmetries. Although it breaks the local gauge symmetry only in the regulator fields' sector, its trace disappears completely after removing the regularization as a result of superrenormalizability of the model. It is shown also that analogous extension of the Pauli-Villars regularization in the vector particle sector can be used to agree the arbitrary covariant gauge results with the Landau ones. The source of ambiguities in the covariant gauges is studied in detail. It is demonstrated that in gauges that are softer in the infrared region (e.g. Coulomb or axial) nonphysical ambiguities inherent to the covariant gauges do not arise.Comment: Latex, 13 pages. Replaced mainly to change preprint references to journal one

Quantitative Timed Analysis of Interactive Markov Chains

Abstract This paper presents new algorithms and accompanying tool support for analyzing interactive Markov chains (IMCs), a stochastic timed 1 1 2-player game in which delays are exponentially distributed. IMCs are compositional and act as semantic model for engineering for-malisms such as AADL and dynamic fault trees. We provide algorithms for determining the extremal expected time of reaching a set of states, and the long-run average of time spent in a set of states. The prototypical tool Imca supports these algorithms as well as the synthesis of ε-optimal piecewise constant timed policies for timed reachability objectives. Two case studies show the feasibility and scalability of the algorithms.

Curvature-dimension inequalities and Li-Yau inequalities in sub-Riemannian spaces

In this paper we present a survey of the joint program with Fabrice Baudoin originated with the paper \cite{BG1}, and continued with the works \cite{BG2}, \cite{BBG}, \cite{BG3} and \cite{BBGM}, joint with Baudoin, Michel Bonnefont and Isidro Munive.Comment: arXiv admin note: substantial text overlap with arXiv:1101.359

Kaplan-Narayanan-Neuberger lattice fermions pass a perturbative test

We test perturbatively a recent scheme for implementing chiral fermions on the lattice, proposed by Kaplan and modified by Narayanan and Neuberger, using as our testing ground the chiral Schwinger model. The scheme is found to reproduce the desired form of the effective action, whose real part is gauge invariant and whose imaginary part gives the correct anomaly in the continuum limit, once technical problems relating to the necessary infinite extent of the extra dimension are properly addressed. The indications from this study are that the Kaplan--Narayanan--Neuberger (KNN) scheme has a good chance at being a correct lattice regularization of chiral gauge theories.Comment: LaTeX 18 pages, 3 figure