At the top of a hill south east of Florence, at the end of a long boulevard climbing up through pine, cypress and oak trees, is the Galileo Galilei Institute for Theoretical Physics. Cats doze by the steps leading up to the grand terracotta–coloured building. This is the neighbourhood where Galileo was kept under house arrest in the last years of his life, and there are worse locations to be exiled in quiet contemplation of the physical world. A light breeze brings jasmine and birdsong through the open windows of the lecture theatre where around fifty academics have gathered for a conference on lattice models: exact methods and combinatorics, part of a wider school on statistical mechanics, integrability and combinatorics.
This focus week brings together mathematicians interested in combinatorics, the study of countable phenomena such as colouring problems or bell ringing, with physicists studying statistical mechanics. Lattice models, where systems are studied by being broken up into a regular grid, have been very successful in statistical mechanics, providing previously unattainable exact solutions. They are most obviously popular in studying condensed matter systems which are already lattice-like, but have also found use in traffic systems, DNA, percolation and field theories.
The mathematics here has a strong visual element, so I can follow talks where much of the manipulation is shown through different ways of crossing or twisting lines without getting too lost. I asked Frank Göhmann of Wuppertal where he thought the field was headed. ‘I think the most interesting problems are in non-equilibrium physics and integrability,’ he said. ‘Integrable and non-integrable systems show very different behaviour in non-equilibrium situations.’
Some recent relevant papers:
Open two-species exclusion processes with integrable boundaries by N Crampe, K Mallick, E Ragoucy and M Vanicat.
High-precision phase diagram of spin glasses from duality analysis with real-space renormalization and graph polynomials by Masayuki Ohzeki and Jesper Lykke Jacobsen.
Scaling functions in the square Ising model by S Hassani and J-M Maillard.
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