# On concentration properties of disordered Hamiltonians

## TL;DR

This paper introduces a straightforward method to demonstrate how disordered Hamiltonians concentrate around specific energy levels, assuming the free energy is differentiable with respect to inverse temperature.

## Contribution

It provides an elementary approach to concentration results for disordered Hamiltonians based on the differentiability of the limiting free energy.

## Key findings

- Hamiltonians concentrate around the energy level F'(β)
- Concentration holds under free energy and Gibbs average
- Approach simplifies previous methods

## Abstract

We present an elementary approach to concentration of disordered Hamiltonians. Assuming differentiability of the limiting free energy $F$ with respect to the inverse temperature $\beta$, we show that the Hamiltonian concentrates around the energy level $F'(\beta)$ under the free energy and Gibbs average.