## Correlations of the Planar Ising Model

In the physical sciences the two-dimensional Ising model provides the fundamental paradigm of a statistical mechanical system exhibiting a
phase transition and for critical phenomena occurring at this transition. Lars Onsager, celebrated for his contributions to Chemistry amongst other
things, attained a pinnacle of 20th Century theoretical physics - the "crown of theoretical physics" in the opinion of Lev Landau - with his
solution for the thermodynamics of the square lattice Ising model.
The standard reference works on this model are the older work **MW_1973** and the more recent works **McC_2010** and **Pal_2007**.

He revealed **Ons_1944** in a characteristically singular way that this model also possesses a very rich mathematical content - this is now known to include
the combinatorics of dimer coverings on regular planar lattices **Ken_2004**;
the Onsager algebra arising from the transfer operator theory ;
the appearance of the {$ E_8 $} root system in the spectrum of the quantum version of the model in a transverse magnetic field **Zam_1989**;
and the Szegö theorems on the asymptotics of Toeplitz determinants as their rank grows **DIK_2013**.
Yet another example is the pivotal role of integrable systems, particularly the Painlevé transcendents, in the description of the planar Ising model correlation functions.

Another famous result was the evaluation by Jimbo and Miwa (1980) **JM_1980_XVII** of the diagonal two-point correlation function of the square lattice Ising model in terms of a
particular solution of the sixth Painlevé equation. This equation is the master case of a family of six non-linear, second-order differential equations discovered
at the beginning of the 20th Century. Recently this result has been extended in a number of directions by the author.
The next-to-diagonal correlations of this Ising model were evaluated **Wit_2007** as the Cauchy-Hilbert transform of orthogonal polynomials with respect to a certain
weight defined on the unit circle. A particular coefficient of the orthogonal polynomials themselves is the {$\tau$}-function found by Jimbo
and Miwa. Furthermore the diagonal correlations of the Ising model on a triangular lattice with general couplings in the three directions has been evaluated in terms of
a three-variable extension of the sixth Painlevé system (called Garnier systems) **Wit_2016** and specialisations of this recover the row and diagonal correlations
of the rectangular lattice Ising model. Numerous consequences flow from this, for instance recurrence relations for the correlations that are analogues of the discrete Painlevé equations.

However there are a number of gaps in our understanding of the correlations. It is proposed in this project to address this problem and make further identifications either with known
generalisations of the Painlevé equations, or where there is a new integrable system to deduce the elements of its underlying theory.
In addition we seek to exploit such identifications in order to solve technical, analytic issues arising in the applications such as deriving systems of coupled,
non-linear recurrence relations for the correlations. The techniques that we will utilise here include but are not limited to the theory of isomonodromic deformations of linear,
meromorphic differential equations and the theory of orthogonal/bi-orthogonal polynomial systems with regular semi-classical weights.
This would suit a student with a background in classical analysis and combinatorics, and who is interested in applications to statistical mechanics.

**DIK_2013**
P. Deift, A. Its, and I. Krasovsky. *Toeplitz matrices and Toeplitz determinants under the impetus of the Ising model: some history and some recent results*. Comm. Pure Appl. Math., 66(9):1360--1438, 2013.

**JM_1980_XVII**
M. Jimbo and T. Miwa. *Studies on holonomic quantum fields. XVII*. Proc. Japan Acad. Ser. A Math. Sci., 56(9):405--410, 1980.

**Ken_2004**
R. Kenyon. *An introduction to the dimer model*. In School and Conference on Probability Theory, ICTP Lect. Notes, XVII, pg. 267--304, 2004.

**MW_1973**
B.M. McCoy and T.T. Wu. *The Two-Dimensional Ising Model*. Harvard University Press, Harvard, 1973.

**McC_2010**
B.M. McCoy. *Advanced Statistical Mechanics*, Vol. 146 of International Series of Monographs on Physics. Oxford University Press, Oxford, 2010.

**Ons_1944**
L. Onsager. *Crystal statistics. I. A two-dimensional model with an order-disorder transition*. Phys. Rev. (2), 65:117--149, 1944.

**Pal_2007**
J. Palmer. *Planar Ising Correlations and the Deformation Analysis of Scaling*, Vol. 49 of Progress in Mathematical Physics. Birkhäuser, Boston, 2007.

**Wit_2007**
N.S. Witte. *Isomonodromic deformation theory and the next-to-diagonal correlations of the anisotropic square lattice Ising model*. J. Phys. A: Math. Theor., 40(24):F491--F501, 2007.

**Wit_2016**
N.S. Witte. *The diagonal two-point correlations of the Ising model on the anisotropic triangular lattice and Garnier systems*. Nonlinearity, 29(1):131, 2016.

**Zam_1989**
A.B. Zamolodchikov. *Integrals of motion and {$S$}-matrix of the (scaled) {$T=T_c$} Ising model with magnetic field*. Internat. J. Modern Phys. A, 4(16):4235--4248, 1989.