A Simple Re-Derivation of Onsager's Solution of the 2D Ising Model using Experimental Mathematics

By Manuel Kauers and Doron Zeilberger

.pdf     .tex  

First Written: May 22, 2018   ;   This version: Aug. 5, 2018.

[Appeared in Mathematical Intelligencer v. 41 issue 1 (March 2019), pp. 1-6.]

One of the crowning achievements of human ingenuity is Lars Onsager's 1944 exact solution of the celebrated two-dimensional Ising model in zero magnetic field. This proof is notoriously complicated and ad-hoc, as are all the subsequent `simpler' proofs. Here we present a new derivation that is embarrassingly simple, using (very basic) symbolic computation, and `experimental mathematics' (aka guessing). While non-rigorous it is absolutely certain, and should be totally acceptable to physicists who do not share mathematicians' fanatical (and often misplaced) insistence on rigor.

Added Aug. 30, 2018: Read a fascinating message from Tony Guttmann.

Maple Package

  • Lars.txt (version of May 2, 2018), Doing Onsager by pure guessing

    [One of the procedures, vSeriesCh(K), requires the data file: Ons48PC.txt ]

    C program for fast computation of the trace of a power of a matrix

    Sample Input and Output for Maple package Lars.txt

    • If you want to see an ab initio derivation of Onsager's formula, using experimental mathematics,
      the input leads to the output.

    • If you want to see the v-Series up to 11 terms, Directly in terms of the definition (but by "cheating", using the pre-computed weight-enumerators for the n by n torodial rectangle for n=24)
      input leads to the output.

    The Ising Polnomials

    In the following list of length 16, the i-th entry is the number of so-called lattice polygons with 2i sides, that can be placed in a torodial rectangle with N lattice points side-lenghts are both larger than 2i+2.

    L:=[ 0,N,2*N,(N*(9 + N))/2,2*N*(6 + N),(N*(7 + N)*(32 + N))/6,N*(130 + 21*N + N^2),N*(11766 + 1715*N + 102*N^2 + N^3)/24, N*(5876 + 776*N + 49*N^2 + N^3)/3,(N*(980904 + 118830*N + 7415*N^2 + 210*N^3 + N^4))/120, N*(423624 + 47666*N + 2855*N^2 + 94*N^3 + N^4)/12,N*(112852800 + 11919274*N + 678945*N^2 + 23725*N^3 + 375*N^4 + N^5)/720, N*(42723120 + 4272044*N + 231260*N^2 + 8175*N^3 + 160*N^4 + N^5)/60, N*(16620978240 + 1584498216*N + 81728374*N^2 + 2851695*N^3 + 62545*N^4+ 609*N^5 + N^6)/5040, N*(5589930384 + 510961484*N + 25204804*N^2 + 857825*N^3 + 19891*N^4 +251*N^5 + N^6)/360, N*(2990184306480 + 263323487916*N + 12469823436*N^2 + 411847009*N^3 +9754920*N^4 + 143794*N^5 + 924*N^6 + N^7)/40320 ]:

    Acknowledgement: We were greatly helped by