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One of the most famous models in physics is the Ising model, invented by Wilhelm Lenz as a PhD problem to his student Ernst Ising. The one-dimensional version of it was solved in Ising's thesis in 1924; later, in 1944, Lars Onsager solved the two-dimensional case in the absense of external magnetic field and in a square lattice.

Although primarily a physical model, it is quite fair to say that the model became part of the mathematics literature, since its descriptions and formulations involve many interesting tools from graph theory, combinatorics, measure theory, convex analysis and so on.

Physically, the Ising model can me thought as a system of many little magnets in which case the spins $\pm 1$ represent a magnetic moment. It can also represent a lattice gas, where the spins now represent whether a site is occupied by a particle or not.

I've heard before that the Ising model has a vast number of applications, some of them really interesting and curious. But after coming across the paper Social applications of two-dimensional Ising models, in which the authors use the Ising model to study socio-economic opinions, urban segregation and language change, I got really curious on what else can be done using it. So, the title says it all: what are other interesting and/or surprising applications of the Ising model?

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    $\begingroup$ The Wikipedia article you link to has a section on applications: en.wikipedia.org/wiki/Ising_model#Applications $\endgroup$ Jan 16 at 22:55
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    $\begingroup$ The linked paper on social applications can also be found at arxiv.org/0706.3983 $\endgroup$ Jan 16 at 23:40
  • $\begingroup$ Probably not surprising, but Geoff Hinton's inspiration for the restricted Boltzmann-machine (a stochastic neural network) is the Ising spin-glass, which is a version of the Ising model with random couplings. $\endgroup$
    – PeaBrane
    Jan 19 at 20:49
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An application of the Ising model in social sciences is to voter models: The dynamics of the Ising model tries to align neighbouring spins, similarly, perhaps, to humans deciding on their political, religious, or consumer preferences [1].

An application to genetics is described in [2], where it is shown that the Ising model with only nearest-neighbor interactions between genetic markers can detect susceptibility loci for type-1 diabetes not previously found by other methods of analysis.

In computer science, the Ising model describes the complexity class of a topological quantum computer based on Majorana fermions [3].

In physics "Ising superconductivity" describes the pairing of electrons in two-dimensional superconducting layered compounds [4].

  1. Phase transition and power-law coarsening in Ising-doped voter model, Adam Lipowski, Dorota Lipowska, Antonio L. Ferreira.

  2. The Ising model in physics and statistical genetics, J. Majewski, H. Li, J. Ott.

  3. A short introduction to topological quantum computation, Ville T. Lahtinen and Jiannis K. Pachos.

  4. Recent progresses in two-dimensional Ising superconductivity, W. Li et al.

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The Ising model defines a universality class, meaning lots of systems simplify to something that looks basically like a magnet. Renormalisation tells us that lots of systems share universal asymptotic dynamics, which is a more formal way of saying they simplify to the same thing. So, anything lying in the Ising model's universality class answers your question. This includes lots of systems that lie on a network or have some dynamical description emphasising interactions, as well as lots of systems that have a second-order phase transition or exhibit some anomalous breaking of a symmetry group under certain conditions. Between these two examples, that's quite a lot of applied mathematics. An interesting meta-commentary on any mathematical model of correlated variables itself being an Ising model can be found in this paper. It also describes non-magnetic physical systems, like string theories and conformal field theories. The point of the model is that, for something so simple, it is incredibly rich -- this is probably why it's stuck about for so long -- and naturally, that makes it difficult to enumerate all the ways in which it has been useful.

I give a few remarks about universality itself here, where I use it to explain why the outputs of neural networks look the way they do.

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The Ising model can be used to estimate the maximum tolerable noise level in a scalable quantum computer [1].

  1. Topological quantum memory, Eric Dennis, Alexei Kitaev, Andrew Landahl, John Preskill (2001)
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The study of the Ising model (and Potts model in general) has led to the construction of a particular dependent percolation model, known as the random cluster model. Formally speaking, this is not quite an "application" of the Ising model but rather a representation/generalization of it (among others like cluster expansion, dimer, currents, etc).

But nowadays, the random cluster model is studied extensively as a percolation model by itself, and offered some deep insights into duality and criticality of dependent percolation models.

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