Replica Field Theory for Deterministic Models: Binary Sequences with Low Autocorrelation

TL;DR

This paper applies replica symmetry theory and advanced mathematical techniques to analyze deterministic models with complex landscapes, specifically focusing on binary sequences with low autocorrelation, providing new insights into their high-temperature behavior.

Contribution

It introduces a replica-based analytical framework for deterministic models with complex landscapes, including the low autocorrelation model, and develops a novel approach using one-link integral techniques.

Findings

Replica theory successfully describes the high-temperature phase.

The model's properties are reconstructed using replica calculations.

A new method employing Fourier and unitary transformations is proposed.

Abstract

We study systems without quenched disorder with a complex landscape, and we use replica symmetry theory to describe them. We discuss the Golay-Bernasconi-Derrida approximation of the low autocorrelation model, and we reconstruct it by using replica calculations. Then we consider the full model, its low $T$ properties (with the help of number theory) and a Hartree-Fock resummation of the high-temperature series. We show that replica theory allows to solve the model in the high $T$ phase. Our solution is based on one-link integral techniques, and is based on substituting a Fourier transform with a generic unitary transformation. We discuss this approach as a powerful tool to describe systems with a complex landscape in the absence of quenched disorder.