Real income growth and revealed preference inconsistency

If a smooth demand function violates the strong axiom of revealed preference, the income and prices can follow a cycle and returm to their starting values even though real income is always rising. We show how real income growth along the "worst" revealed preference cycle depends on the range of price variation and on violations of the Slutsky conditions. We relate this result to proposed reforms of the consumer price index and use it to justify a new index of local demand inconsistency. We also use the Slutsky matrix to determine an upper bound on the number of observations required to detect revealed preference inconsistency

Real income growth and revealed preference inconsistency.

If a smooth demand function violates the strong axiom of revealed preference, the income and prices can follow a cycle and returm to their starting values even though real income is always rising. We show how real income growth along the "worst" revealed preference cycle depends on the range of price variation and on violations of the Slutsky conditions. We relate this result to proposed reforms of the consumer price index and use it to justify a new index of local demand inconsistency. We also use the Slutsky matrix to determine an upper bound on the number of observations required to detect revealed preference inconsistency.

How to recognize convexity of a set from its marginals

We investigate the regularity of the marginals onto hyperplanes for sets of finite perimeter. We prove, in particular, that if a set of finite perimeter has log-concave marginals onto a.e. hyperplane then the set is convex

Logarithmic fluctuations for internal DLA

Let each of n particles starting at the origin in Z^2 perform simple random walk until reaching a site with no other particles. Lawler, Bramson, and Griffeath proved that the resulting random set A(n) of n occupied sites is (with high probability) close to a disk B_r of radius r=\sqrt{n/\pi}. We show that the discrepancy between A(n) and the disk is at most logarithmic in the radius: i.e., there is an absolute constant C such that the following holds with probability one: B_{r - C \log r} \subset A(\pi r^2) \subset B_{r+ C \log r} for all sufficiently large r.Comment: 38 pages, 5 figures, v2 addresses referee comments. To appear in Journal of the AM

A free boundary problem for the localization of eigenfunctions

We study a variant of the Alt, Caffarelli, and Friedman free boundary problem with many phases and a slightly different volume term, which we originally designed to guess the localization of eigenfunctions of a Schr\"odinger operator in a domain. We prove Lipschitz bounds for the functions and some nondegeneracy and regularity properties for the domains.Comment: 174 page

Internal DLA and the Gaussian free field

In previous works, we showed that the internal DLA cluster on \Z^d with t particles is a.s. spherical up to a maximal error of O(\log t) if d=2 and O(\sqrt{\log t}) if d > 2. This paper addresses "average error": in a certain sense, the average deviation of internal DLA from its mean shape is of constant order when d=2 and of order r^{1-d/2} (for a radius r cluster) in general. Appropriately normalized, the fluctuations (taken over time and space) scale to a variant of the Gaussian free field.Comment: 29 pages, minor revisio

Quantitative stability for sumsets in RnR^n

Given a measurable set $A\subset \R^n$ of positive measure, it is not difficult to show that $|A+A|=|2A|$ if and only if $A$ is equal to its convex hull minus a set of measure zero. We investigate the stability of this statement: If $(|A+A|-|2A|)/|A|$ is small, is $A$ close to its convex hull? Our main result is an explicit control, in arbitrary dimension, on the measure of the difference between $A$ and its convex hull in terms of $(|A+A|-|2A|)/|A|$