Combinatorial formulation of Ising model revisited

横向伊辛链在临界磁场下自相关函数的长时渐近行为再探

Long-time asymptotics of the autocorrelation function of the transverse Ising chain at the critical magnetic field Revisited

通过Riemann-Hilbert问题，重新分析横向伊辛链在临界磁场下自相关函数的长时渐近行为，改进了Deift-Zhou的结果，确定了渐近展开中的次主导增长项。

The six-vertex model on random planar maps revisited

We address the six vertex model on a random lattice, which in combinatorial terms corresponds to the enumeration of weighted 4-valent planar maps equipped with an Eulerian orientation. This problem was exactly, albeit non-rigorously solved by Ivan Kostov in 2000 using matrix integral techniques. We convert Kostov's work to a combinatorial argument involving functional equations coming from recursive decompositions of the maps, which we solve rigorously using complex analysis. We then investigate modular properties of the solution, which lead to simplifications in certain special cases. In part…

Exact bosonization of the Ising model

We present exact combinatorial versions of bosonization identities, which equate the product of two Ising correlators with a free field (bosonic) correlator. The role of the discrete free field is played by the height function of an associated bipartite dimer model. Some applications to the asymptotic analysis of Ising correlators are discussed.

Revisiting the combinatorics of the 2D Ising model

We provide a concise exposition with original proofs of combinatorial formulas for the 2D Ising model partition function, multi-point fermionic observables, spin and energy density correlations, for general graphs and interaction constants, using the language of Kac-Ward matrices. We also give a brief account of the relations between various alternative formalisms which have been used in the combinatorial study of the planar Ising model: dimers and Grassmann variables, spin and disorder operators, and, more recently, s-holomorphic observables. In addition, we point out that these formulas can …

Limit theorems for dependent combinatorial data, with applications in statistical inference

The Ising model is a celebrated example of a Markov random field, introduced in statistical physics to model ferromagnetism. This is a discrete exponential family with binary outcomes, where the sufficient statistic involves a quadratic term designed to capture correlations arising from pairwise interactions. However, in many situations the dependencies in a network arise not just from pairs, but from peer-group effects. A convenient mathematical framework for capturing higher-order dependencies, is the $p$-tensor Ising model, where the sufficient statistic consists of a multilinear polynomial…

Exact low-temperature series expansion for the partition function of the two-dimensional zero-field s=1/2 Ising model on the infinite square lattice

In this paper, we provide the exact expression for the coefficients in the low-temperature series expansion of the partition function of the two-dimensional Ising model on the infinite square lattice. This is equivalent to exact determination of the number of spin configurations at a given energy. With these coefficients, we show that the ferromagnetic--to--paramagnetic phase transition in the square lattice Ising model can be explained through equivalence between the model and the perfect gas of energy clusters model, in which the passage through the critical point is related to the complete …