Circular law as a legal basis for a circular economy

The article is devoted to the study of the legal basis for the development of the circular economy, clarifying the essence of the concept of circular law, and analysis of international and national legal acts in this area. Research has shown humanity’s awareness of modern global threats and its gradual movement from the traditional to circular economic model. The legal basis for such a transition lies in circular law, which is a set of legal norms that define and regulate social relations in the field of circular economy. The article explores legal acts in the field of circular economy at the universal, regional and national levels across the globe. The main purpose of the article is to analyze the legal basis for the development of the circular economy. International law is currently being enriched with a number of new legal acts devoted to this subject matter. This may indicate the emergence of a new branch of international law – a circular law. An increased attention of international financial institutions towards financing of circular projects stipulates the further development and spread of the circular concept.

The strong circular law: A combinatorial view

Let [Formula: see text] be an [Formula: see text] complex random matrix, each of whose entries is an independent copy of a centered complex random variable [Formula: see text] with finite nonzero variance [Formula: see text]. The strong circular law, proved by Tao and Vu, states that almost surely, as [Formula: see text], the empirical spectral distribution of [Formula: see text] converges to the uniform distribution on the unit disc in [Formula: see text]. A crucial ingredient in the proof of Tao and Vu, which uses deep ideas from additive combinatorics, is controlling the lower tail of the least singular value of the random matrix [Formula: see text] (where [Formula: see text] is fixed) with failure probability that is inverse polynomial. In this paper, using a simple and novel approach (in particular, not using machinery from additive combinatorics or any net arguments), we show that for any fixed complex matrix [Formula: see text] with operator norm at most [Formula: see text] and for all [Formula: see text], [Formula: see text] where [Formula: see text] is the least singular value of [Formula: see text] and [Formula: see text] are positive absolute constants. Our result is optimal up to the constants [Formula: see text] and the inverse exponential-type error rate improves upon the inverse polynomial error rate due to Tao and Vu. Our proof relies on the solution to the so-called counting problem in inverse Littlewood–Offord theory, developed by Ferber, Luh, Samotij, and the author, a novel complex anti-concentration inequality, and a “rounding trick” based on controlling the [Formula: see text] operator norm of heavy-tailed random matrices.

Rate of convergence to the Circular Law via smoothing inequalities for log-potentials

The aim of this paper is to investigate the Kolmogorov distance of the Circular Law to the empirical spectral distribution of non-Hermitian random matrices with independent entries. The optimal rate of convergence is determined by the Ginibre ensemble and is given by [Formula: see text]. A smoothing inequality for complex measures that quantitatively relates the uniform Kolmogorov-like distance to the concentration of logarithmic potentials is shown. Combining it with results from Local Circular Laws, we apply it to prove nearly optimal rate of convergence to the Circular Law in Kolmogorov distance. Furthermore, we show that the same rate of convergence holds for the empirical measure of the roots of Weyl random polynomials.