# Convergence of Stochastic Approximation Monte Carlo and modified Wang-Landau algorithms: Tests for the Ising model

## TL;DR

This paper compares the convergence behaviors of Wang-Landau, 1/t-modified Wang-Landau, and SAMC algorithms in estimating the density of states for the 2D Ising model, highlighting how refinement schemes affect accuracy.

## Contribution

It introduces a 1/t-modification to Wang-Landau and analyzes the convergence sensitivity of SAMC in the context of the Ising model.

## Key findings

- Wang-Landau deviation saturates without modification
- 1/t-modification improves convergence of Wang-Landau
- SAMC convergence is highly sensitive to refinement onset

## Abstract

We investigate the behavior of the deviation of the estimator for the density of states (DOS) with respect to the exact solution in the course of Wang-Landau and Stochastic Approximation Monte Carlo (SAMC) simulations of the two-dimensional Ising model. We find that the deviation saturates in the Wang-Landau case. This can be cured by adjusting the refinement scheme. To this end, the 1/t-modification of the Wang-Landau algorithm has been suggested. A similar choice of refinement scheme is employed in the SAMC algorithm. The convergence behavior of all three algorithms is examined. It turns out that the convergence of the SAMC algorithm is very sensitive to the onset of the refinement. Finally, the internal energy and specific heat of the Ising model are calculated from the SAMC DOS and compared to exact values.