# A general method for lower bounds on fluctuations of random variables

## TL;DR

This paper introduces a universal method to establish lower bounds on the fluctuations of random variables, addressing a gap in the systematic analysis of their variability across various complex problems.

## Contribution

The paper presents a novel, general approach for deriving lower bounds on fluctuations, applicable to multiple problems in probability and statistical physics.

## Key findings

- New lower bounds for stochastic traveling salesman problem
- Lower bounds for random minimal matching and assignment problems
- Results for spin glasses, first-passage percolation, and random matrices

## Abstract

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.