Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userContinuum and thermodynamic limits for a simple random-exchange model
82
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We discuss various limits of a simple random exchange model that can be used for the distribution of wealth. We start from a discrete state space - discrete time version of this model and, under suitable scaling, we show its functional convergence to a continuous space - discrete time model. Then, we show a thermodynamic limit of the empirical distribution to the solution of a kinetic equation of Boltzmann type. We solve this equation and we show that the solutions coincide with the appropriate limits of the invariant measure for the Markov chain. In this way we complete Boltzmanns program of deriving kinetic equations from random dynamics for this simple model. Three families of invariant measures for the mean field limit are discovered and we show that only two of those families can be obtained as limits of the discrete system and the third is extraneous. Finally, we cast our results in the framework of integer partitions and strengthen some results already available in the literature.
rate research
Read More
We show that at any location away from the spectral edge, the eigenvalues of the Gaussian unitary ensemble and its general beta siblings converge to Sine_beta, a translation invariant point process. This process has a geometric description in term of the Brownian carousel, a deterministic function of Brownian motion in the hyperbolic plane. The Brownian carousel, a description of the a continuum limit of random matrices, provides a convenient way to analyze the limiting point processes. We show that the gap probability of Sine_beta is continuous in the gap size and $beta$, and compute its asymptotics for large gaps. Moreover, the stochastic differential equation version of the Brownian carousel exhibits a phase transition at beta=2.
Motivated by limits of critical inhomogeneous random graphs, we construct a family of sequences of measured metric spaces that we call continuous multiplicative graphs, that are expected to be the universal limit of graphs related to the multiplicative coalescent (the ErdH{o}s--Renyi random graph, more generally the so-called rank-one inhomogeneous random graphs of various types, and the configuration model). At the discrete level, the construction relies on a new point of view on (discrete) inhomogeneous random graphs that involves an embedding into a Galton--Watson forest. The new representation allows us to demonstrate that a processus that was already present in the pionnering work of Aldous [Ann. Probab., vol.~25, pp.~812--854, 1997] and Aldous and Limic [Electron. J. Probab., vol.~3, pp.~1--59, 1998] about the multiplicative coalescent actually also (essentially) encodes the limiting metric: The discrete embedding of random graphs into a Galton--Watson forest is paralleled by an embedding of the encoding process into a Levy process which is crucial in proving the very existence of the local time functionals on which the metric is based; it also yields a transparent approach to compactness and fractal dimensions of the continuous objects. In a companion paper, we show that the continuous Levy graphs are indeed the scaling limit of inhomogeneous random graphs.
We propose an observation-driven time-varying SVAR model where, in agreement with the Lucas Critique, structural shocks drive both the evolution of the macro variables and the dynamics of the VAR parameters. Contrary to existing approaches where parameters follow a stochastic process with random and exogenous shocks, our observation-driven specification allows the evolution of the parameters to be driven by realized past structural shocks, thus opening the possibility to gauge the impact of observed shocks and hypothetical policy interventions on the future evolution of the economic system.
In this paper, we propose a spatially constrained clustering problem belonging to the family of p-regions problems. Our formulation is motivated by the recent developments of economic complexity on the evolution of the economic output through key interactions among industries within economic regions. The objective of this model consists in aggregating a set of geographic areas into a prescribed number of regions (so-called innovation ecosystems) such that the resulting regions preserve the most relevant interactions among industries. We formulate the p-Innovation Ecosystems model as a mixed-integer programming (MIP) problem and propose a heuristic solution approach. We explore a case involving the municipalities of Colombia to illustrate how such a model can be applied and used for policy and regional development.
Based on some analytic structural properties of the Gini and Kolkata indices for social inequality, as obtained from a generic form of the Lorenz function, we make a conjecture that the limiting (effective saturation) value of the above-mentioned indices is about 0.865. This, together with some more new observations on the citation statistics of individual authors (including Nobel laureates), suggests that about $14%$ of people or papers or social conflicts tend to earn or attract or cause about $86%$ of wealth or citations or deaths respectively in very competitive situations in markets, universities or wars. This is a modified form of the (more than a) century old $80-20$ law of Pareto in economy (not visible today because of various welfare and other strategies) and gives an universal value ($0.86$) of social (inequality) constant or number.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
