The Annals of Probability
- Ann. Probab.
- Volume 47, Number 4 (2019), 2140-2171.
A general method for lower bounds on fluctuations of random variables
There are many ways of establishing upper bounds on fluctuations of random variables, but there is no systematic approach for lower bounds. As a result, lower bounds are unknown in many important problems. This paper introduces a general method for lower bounds on fluctuations. The method is used to obtain new results for the stochastic traveling salesman problem, the stochastic minimal matching problem, the random assignment problem, the Sherrington–Kirkpatrick model of spin glasses, first-passage percolation and random matrices. A long list of open problems is provided at the end.
Ann. Probab., Volume 47, Number 4 (2019), 2140-2171.
Received: August 2017
Revised: July 2018
First available in Project Euclid: 4 July 2019
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Primary: 60E15: Inequalities; stochastic orderings 60C05: Combinatorial probability 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)
Variance lower bound first-passage percolation random assignment problem stochastic minimal matching problem stochastic traveling salesman problem spin glass Sherrington–Kirkpatrick model random matrix determinant
Chatterjee, Sourav. A general method for lower bounds on fluctuations of random variables. Ann. Probab. 47 (2019), no. 4, 2140--2171. doi:10.1214/18-AOP1304. https://projecteuclid.org/euclid.aop/1562205705