Thesis: Non-backtracking lace expansion

Remco van der Hofstad and Robert Fitzner developed the non-backtracking lace expansion (NoBLE) to prove mean-field behavior for several nearest-neighbor models in statistical physics. The main aim of NoBLE is to explicity compute for which dimensions nearest-neighbor systems display mean-field behavior and make the required analysis and computation as accessible as possible.

On this webpage you can find the results of my thesis and the implementation of the computer-assisted proof of the NoBLE as described in my thesis. We recommend readers, that would like to understand the details of the technique lace expansion to read this thesis before any of the other articles, as it contains more details and explanation then can be found in the condensed/polished/published articles. The original version can be found here

As it is a 300+ page document written by me it includes many typos, for which I apologize. While creating the articles for publication we also found a small number of mistakes within equations. Mostly mistakes were simply wrong indices and numerations and none of these mistakes turn out to be conceptional and the results all remain true as stated in 2013.
In the following table we review the known results and state in which dimensions mean-field behavior is proven in the thesis.

mean-field behavior

self-avoiding walk

lattice trees

lattice animals


expected for





proved before 2013


sufficiently high

sufficiently high


proved by us





Computer-assisted proof:

The results were obtained using a computer-assisted proof. The author implemented the computations using Mathematica notebooks. In the following tables these notebooks can be downloaded. Next to the Mathematica format .nb we provide the file as PDF file, in which also the parameters used for the lowest applicable dimension can be retrieved.

Disclaimer: These Mathematica notebooks have been created by Robert Fitzner and reviewed by Remco van der Hofstad. Due to the length of each individual file (up to 40 pages) we can not guaranty that the files do no contain minor errors. However, we are convinced that any such mistake, will be so minor that it does not influence the final result that mean-field behaviour hold in the dimension given here.

Using Fourier space bounds on weighted diagrams (see Section 3.3.)

Model Mathematica notebook PDF Version works in
SAW (NoBLE expansion) d≥8 (6.5.2013)
SAW (Exp. of Chapter 6) d≥7 (14.5.2013)
Lattice tree d≥29 (9.5.2013)
Lattice animal d≥49 (8.5.2013)
Percolation d≥38 (23.4.2013)

Using x space bounds on weighted diagrams (see Section 3.5.)

We split the implementation of the analysis of Section 3.5-3.6 into two parts. In the first file we compute SRW-integrals and implement the computations of the analysis given in Section 3.6. This file is used for all models. The second file depend for the given model a bound on the perturbation in form of the coefficient and then compute whether the analysis of Section 3.5 succeeds to prove mean-field behavior for the given model in a given dimension.
Model Mathematica notebook PDF Version works in
SRW-Integrals a tool(24.1.2014)
Lattice tree d≥20 (14.5.2013)
Lattice animal d≥21 (9.5.2013)
Percolation d≥15 (11.5.2013)