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 can be proven to display mean-field behavior and make the required analysis and computation as accessible as possible.

On this webpage you find an overview of the NoBLE articles by Remco van der Hofstad and Robert Fitzner. Further, we provide the implementation of the computer-assisted proof described in these article. The technique was developed and implemented by Robert Fitzner under the supervision of Remco van der Hofstad.

In case you are more interested in the technique than the results obtain, we advise you to read the thesis (2013) of Robert Fitzner as it contains more details and explanations than can be found in the articles. Click here to go to the website dedicated to the thesis..

We developed the NoBLE in the following articles:

Generalized approach to the non-backtracking lace expansion

Nearest-neighbor percolation function is continuous for d>10

NoBLE for lattice trees and lattice animals

In the first paper we derive the analysis used to show mean-field behavior in a general setting. In the second and third paper we derive the NoBLE for percolation, lattice animals and lattice trees and provide all details required for the general analysis to be applied. As described in the thesis the analysis of the first paper also applies to the self-avoiding walk.
Remco van der Hofstad and Robert Fitzner applied NoBLE to self-avoiding walk (SAW), lattice tree (LT), lattice animals (LA) and percolation. These models were known to show mean-field behavior in sufficiently high dimension. In the following table we review the known results and state in which dimensions we have proven mean-field behavior.

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 computation 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. The implementation of the computer-assisted proof consists of three files. The first file is used to compute simple random walks integrals. The second part the bound of the general analysis are implemented. The third part is the model dependent implementation of the bounds on the NoBLE-coefficient. At this moment we have only prepared the implementation for percolation for publication.
Mathematica notebook PDF Version description
SRW-Integrals Numerical bounds on the required SRW integral
General Analysis implementation of the model-independent bounds
Percolation model-dependent bounds, successful in d≥11
Lattice trees model-dependent bounds, successful in d≥16
Lattice animals model-dependent bounds, successful in d≥17
Lattice animals, uniform bounds model-dependent bounds, uniform bound for all d≥30
Using the SRW file, given above, we compute for a given dimension the number of SRW ending at given positions and several SRW-integrals. This computation can take hours. For this reason we provided precomputed values for d=7,8,9,...,20. When put into the correct directory/folder these files will be loaded automatically, which reduces the necessary computational time to seconds. When you do not download the files they will be automatically created on your computer, when you compile/evaluate the SRW file. If you run the computation in the same dimension once more, then the (by your computer) pre-computed values will be used.

Where to put the files? In the starting directory of Mathematica. Where to find that? Open any Mathematica notebook, put $InitialDirectory into an input cell and evaluate it. The directory will be shown. Usually it is your user directory.
Number of SRWs , SRW integrals
The SRW file uses some involved function to automatically compute which SRW-Integrals are required and how to compute a bound on the involved SRW-Integral Ln. If you would like to understand the SRW file we recommend you to read the basic version of the SRW-Integrals file(PDF), which we used at an early stage of the project. In this basic version everything is done manually. Later we changed to the more flexibility solution, which also guarantees the desired numerical precision.