Building a path-integral calculus: a covariant discretization approach

Path integrals are a central tool when it comes to describing quantum or thermal fluctuations of particles or fields. Their success dates back to Feynman who showed how to use them within the framework of quantum mechanics. Since then, path integrals have pervaded all areas of physics where fluctuation effects, quantum and/or thermal, are of paramount importance. Their appeal is based on the fact that one converts a problem formulated in terms of operators into one of sampling classical paths with a given weight. Path integrals are the mirror image of our conventional Riemann integrals, with functions replacing the real numbers one usually sums over. However, unlike conventional integrals, path integration suffers a serious drawback: in general, one cannot make non-linear changes of variables without committing an error of some sort. Thus, no path-integral based calculus is possible. Here we identify which are the deep mathematical reasons causing this important caveat, and we come up with cures for systems described by one degree of freedom. Our main result is a construction of path integration free of this longstanding problem, through a direct time-discretization procedure.Comment: 22 pages, 2 figures, 1 table. Typos correcte

Weak disorder asymptotics in the stochastic mean-field model of distance

In the recent past, there has been a concerted effort to develop mathematical models for real-world networks and to analyze various dynamics on these models. One particular problem of significant importance is to understand the effect of random edge lengths or costs on the geometry and flow transporting properties of the network. Two different regimes are of great interest, the weak disorder regime where optimality of a path is determined by the sum of edge weights on the path and the strong disorder regime where optimality of a path is determined by the maximal edge weight on the path. In the context of the stochastic mean-field model of distance, we provide the first mathematically tractable model of weak disorder and show that no transition occurs at finite temperature. Indeed, we show that for every finite temperature, the number of edges on the minimal weight path (i.e., the hopcount) is $\Theta(\log{n})$ and satisfies a central limit theorem with asymptotic means and variances of order $\Theta(\log{n})$, with limiting constants expressible in terms of the Malthusian rate of growth and the mean of the stable-age distribution of an associated continuous-time branching process. More precisely, we take independent and identically distributed edge weights with distribution $E^s$ for some parameter $s>0$, where $E$ is an exponential random variable with mean 1. Then the asymptotic mean and variance of the central limit theorem for the hopcount are $s\log{n}$ and $s^2\log{n}$, respectively. We also find limiting distributional asymptotics for the value of the minimal weight path in terms of extreme value distributions and martingale limits of branching processes.Comment: Published in at http://dx.doi.org/10.1214/10-AAP753 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Finding a point in the relative interior of a polyhedron

A new initialization or `Phase I' strategy for feasible interior point methods for linear programming is proposed that computes a point on the primal-dual central path associated with the linear program. Provided there exist primal-dual strictly feasible points - an all-pervasive assumption in interior point method theory that implies the existence of the central path - our initial method (Algorithm 1) is globally Q-linearly and asymptotically Q-quadratically convergent, with a provable worst-case iteration complexity bound. When this assumption is not met, the numerical behaviour of Algorithm 1 is highly disappointing, even when the problem is primal-dual feasible. This is due to the presence of implicit equalities, inequality constraints that hold as equalities at all the feasible points. Controlled perturbations of the inequality constraints of the primal-dual problems are introduced - geometrically equivalent to enlarging the primal-dual feasible region and then systematically contracting it back to its initial shape - in order for the perturbed problems to satisfy the assumption. Thus Algorithm 1 can successfully be employed to solve each of the perturbed problems.\ud We show that, when there exist primal-dual strictly feasible points of the original problems, the resulting method, Algorithm 2, finds such a point in a finite number of changes to the perturbation parameters. When implicit equalities are present, but the original problem and its dual are feasible, Algorithm 2 asymptotically detects all the primal-dual implicit equalities and generates a point in the relative interior of the primal-dual feasible set. Algorithm 2 can also asymptotically detect primal-dual infeasibility. Successful numerical experience with Algorithm 2 on linear programs from NETLIB and CUTEr, both with and without any significant preprocessing of the problems, indicates that Algorithm 2 may be used as an algorithmic preprocessor for removing implicit equalities, with theoretical guarantees of convergence

The Casimir Energy in Curved Space and its Supersymmetric Counterpart

We study $d$-dimensional Conformal Field Theories (CFTs) on the cylinder, $S^{d-1}\times \mathbb{R}$, and its deformations. In $d=2$ the Casimir energy (i.e. the vacuum energy) is universal and is related to the central charge $c$. In $d=4$ the vacuum energy depends on the regularization scheme and has no intrinsic value. We show that this property extends to infinitesimally deformed cylinders and support this conclusion with a holographic check. However, for $\mathcal{N}=1$ supersymmetric CFTs, a natural analog of the Casimir energy turns out to be scheme independent and thus intrinsic. We give two proofs of this result. We compute the Casimir energy for such theories by reducing to a problem in supersymmetric quantum mechanics. For the round cylinder the vacuum energy is proportional to $a+3c$. We also compute the dependence of the Casimir energy on the squashing parameter of the cylinder. Finally, we revisit the problem of supersymmetric regularization of the path integral on Hopf surfaces.Comment: 53 pages; v2: minor changes, references added, version published in JHE

Interior point method in tensor optimal transport

We study a tensor optimal transport (TOT) problem for $d\ge 2$ discrete measures. This is a linear programming problem on $d$-tensors. We introduces an interior point method (ipm) for $d$-TOT with a corresponding barrier function. Using a "short-step" ipm following central path within $\varepsilon$ precision we estimate the number of iterations.Comment: Corrected typos and added a short additional subsection, 11 page

Lower Bounds for the Average and Smoothed Number of Pareto Optima

Smoothed analysis of multiobjective 0-1 linear optimization has drawn considerable attention recently. The number of Pareto-optimal solutions (i.e., solutions with the property that no other solution is at least as good in all the coordinates and better in at least one) for multiobjective optimization problems is the central object of study. In this paper, we prove several lower bounds for the expected number of Pareto optima. Our basic result is a lower bound of \Omega_d(n^(d-1)) for optimization problems with d objectives and n variables under fairly general conditions on the distributions of the linear objectives. Our proof relates the problem of lower bounding the number of Pareto optima to results in geometry connected to arrangements of hyperplanes. We use our basic result to derive (1) To our knowledge, the first lower bound for natural multiobjective optimization problems. We illustrate this for the maximum spanning tree problem with randomly chosen edge weights. Our technique is sufficiently flexible to yield such lower bounds for other standard objective functions studied in this setting (such as, multiobjective shortest path, TSP tour, matching). (2) Smoothed lower bound of min {\Omega_d(n^(d-1.5) \phi^{(d-log d) (1-\Theta(1/\phi))}), 2^{\Theta(n)}}$ for the 0-1 knapsack problem with d profits for phi-semirandom distributions for a version of the knapsack problem. This improves the recent lower bound of Brunsch and Roeglin