Permutation Glass

A permutation glass is a statistical physics system whose state space consists of permutations of an ordered list and whose energy parameters are drawn from a quenched distribution of values. The simplest permutation glass has a Hamiltonian of the form

where I_A = 1 if A is true and I_A = 0 otherwise, and (theta_1, theta_2, ..., theta_N) in the set perm(omega_1, omega_2, ..., omega_N). Namely the equation defines a state space consisting of permutations of the initially ordered list (omega_1, omega_2, ..., omega_N) where there is an energy cost lambda_k for each omega_k "incorrect" placement of omega_k, that is a placement where omega_k is not in its original position in the initially ordered list.

We can use two figures to depict the permutation glass schematically:

The left figure is a "permutation graph" depiction of four microstates in a permutation system with N=15. In each graph, j is equivalent to the number of diagonal lines in the permutation graph. The number of "correct" connections are shown as vertical lines. The right figure is a "matching problem" depiction of a j=10 microstate for a 2N = 30 permutation system. The spatial location of each pair is not important in determining the energy of the state. For this state, the matching pairs are 3, 6, 11, 14, and 15.

This respository contains the notebooks that reproduce the results of the associated paper.

Reproducing figures and tables

The notebooks that reproduce the figures and tables in the paper are as follows

• temp_vs_signorm.ipynb: Reproduces Figure 3; Runs in < 1 sec
• perm_glass_simulation.ipynb: Reproduces Figure 4; Runs in < 5 minutes
• prob_neglambda_vs_N.ipynb: Reproduces Figure 5; Runs in < 1 sec

References

[1] Williams, Mobolaji. "Permutation glass." Physical Review E 97.1 (2018): 012139.

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@article{williams2018permutation,
  title={Permutation glass},
  author={Williams, Mobolaji},
  journal={Physical Review E},
  volume={97},
  number={1},
  pages={012139},
  year={2018},
  publisher={APS}
}