N. Zenine — Page 2 | Searcharxiv

arXiv2005

Differential Galois groups of high order Fuchsian ODE's

We present a simple, but efficient, way to calculate connection matrices between sets of independent local solutions, defined at two neighboring singular points, of Fuchsian differential equations of quite large orders, such as those found for the third and fourth contribution ($χ^{(3)}$ and $χ^{(4)}$) to the magnetic susceptibility of square lattice Ising model. We use the previous connection matrices to get the exact explicit expressions of all the monodromy matrices of the Fuchsian differential equation for $χ^{(3)}$ (and $χ^{(4)}$) expressed in the same basis of solutions. These monodromy matrices are the generators of the differential Galois group of the Fuchsian differential equations for $χ^{(3)}$ (and $χ^{(4)}$), whose analysis is just sketched here.

math-ph↗

arXiv2005

Beyond series expansions: mathematical structures for the susceptibility of the square lattice Ising model

We first study the properties of the Fuchsian ordinary differential equations for the three and four-particle contributions $ χ^{(3)}$ and $ χ^{(4)}$ of the square lattice Ising model susceptibility. An analysis of some mathematical properties of these Fuchsian differential equations is sketched. For instance, we study the factorization properties of the corresponding linear differential operators, and consider the singularities of the three and four-particle contributions $ χ^{(3)}$ and $ χ^{(4)}$, versus the singularities of the associated Fuchsian ordinary differential equations, which actually exhibit new ``Landau-like'' singularities. We sketch the analysis of the corresponding differential Galois groups. In particular we provide a simple, but efficient, method to calculate the so-called ``connection matrices'' (between two neighboring singularities) and deduce the singular behaviors of $ χ^{(3)}$ and $ χ^{(4)}$. We provide a set of comments and speculations on the Fuchsian ordinary differential equations associated with the $ n$-particle contributions $ χ^{(n)}$ and address the problem of the apparent discrepancy between such a holonomic approach and some scaling results deduced from a Painlevé oriented approach.

hep-th↗

arXiv2005

Square lattice Ising model susceptibility: connection matrices and singular behavior of $χ^{(3)}$ and $χ^{(4)}$

We present a simple, but efficient, way to calculate connection matrices between sets of independent local solutions, defined at two neighboring singular points, of Fuchsian differential equations of quite large orders, such as those found for the third and fourth contribution ($χ^{(3)}$ and $χ^{(4)}$) to the magnetic susceptibility of square lattice Ising model. We deduce all the critical behaviors of the solutions $χ^{(3)}$ and $χ^{(4)}$, as well as the asymptotic behavior of the coefficients in the corresponding series expansions. We confirm that the newly found quadratic number singularities of the Fuchsian ODE associated to $χ^{(3)}$ are not singularities of the particular solution $χ^{(3)}$ itself. We use the previous connection matrices to get the exact expressions of all the monodromy matrices of the Fuchsian differential equation for $χ^{(3)}$ (and $χ^{(4)}$) expressed in the same basis of solutions. These monodromy matrices are the generators of the differential Galois group of the Fuchsian differential equations for $χ^{(3)}$ (and $χ^{(4)}$), whose analysis is just sketched here. As far as the physics implications of the solutions are concerned, we find challenging qualitative differences when comparing the corrections to scaling for the full susceptibillity $χ$ at high temperature (resp. low temperature) and the first two terms $χ^{(1)}$ and $χ^{(3)}$ (resp. $χ^{(2)}$ and $χ^{(4)}$) .

math-ph↗

arXiv2005

Ising model susceptibility: Fuchsian differential equation for $χ^{(4)}$ and its factorization properties

We give the Fuchsian linear differential equation satisfied by $χ^{(4)}$, the ``four-particle'' contribution to the susceptibility of the isotropic square lattice Ising model. This Fuchsian differential equation is deduced from a series expansion method introduced in two previous papers and is applied with some symmetries and tricks specific to $χ^{(4)}$. The corresponding order ten linear differential operator exhibits a large set of factorization properties. Among these factorizations one is highly remarkable: it corresponds to the fact that the two-particle contribution $χ^{(2)}$ is actually a solution of this order ten linear differential operator. This result, together with a similar one for the order seven differential operator corresponding to the three-particle contribution, $χ^{(3)}$, leads us to a conjecture on the structure of all the $ n$-particle contributions $ χ^{(n)}$.

cond-mat.stat-mech↗

arXiv2004

Square lattice Ising model susceptibility: Series expansion method and differential equation for $χ^{(3)}$

In a previous paper (J. Phys. A {\bf 37} (2004) 9651-9668) we have given the Fuchsian linear differential equation satisfied by $χ^{(3)}$, the ``three-particle'' contribution to the susceptibility of the isotropic square lattice Ising model. This paper gives the details of the calculations (with some useful tricks and tools) allowing one to obtain long series in polynomial time. The method is based on series expansion in the variables that appear in the $(n-1)$-dimensional integrals representing the $n$-particle contribution to the isotropic square lattice Ising model susceptibility $χ$. The integration rules are straightforward due to remarkable formulas we derived for these variables. We obtain without any numerical approximation $χ^{(3)}$ as a fully integrated series in the variable $w=s/2/(1+s^{2})$, where $ s =sh (2K)$, with $K=J/kT$ the conventional Ising model coupling constant. We also give some perspectives and comments on these results.

hep-th↗

arXiv2004

The Fuchsian differential equation of the square lattice Ising model $χ(3)$ susceptibility

Using an expansion method in the variables $ x_i$ that appear in the $(n-1)$-dimensional integrals representing the $n$-particle contribution to the Ising square lattice model susceptibility $χ$, we generate a long series of coefficients for the 3-particle contribution $χ^{(3)}$, using a $ N^4$ polynomial time algorithm. We give the Fuchsian differential equation of order seven for $χ^{(3)}$ that reproduces all the terms of our long series. An analysis of the properties of this Fuchsian differential equation is performed.

math-ph↗