ترغب بنشر مسار تعليمي؟ اضغط هنا

Asymptotic expansions of some Toeplitz determinants via the topological recursion

86   0   0.0 ( 0 )
 نشر من قبل Olivier Marchal
 تاريخ النشر 2016
  مجال البحث فيزياء
والبحث باللغة English
 تأليف Olivier Marchal




اسأل ChatGPT حول البحث

In this article, we study the large $n$ asymptotic expansions of $ntimes n$ Toeplitz determinants whose symbols are indicator functions of unions of arc-intervals of the unit circle. In particular, we use an Hermitian matrix model reformulation of the problem to provide a rigorous derivation of the general form of the large $n$ expansion when the symbol is an indicator function of either a single arc-interval or several arc-intervals with a discrete rotational symmetry. Moreover, we prove that the coefficients in the expansions can be reconstructed, up to some constants, from the Eynard-Orantin topological recursion applied to some explicit spectral curves. In addition, when the symbol is an indicator function of a single arc-interval, we provide the corresponding normalizing constants using a Selberg integral and illustrate the theoretical results with numeric simulations up to order $oleft(frac{1}{n^4}right)$. We also briefly discuss the situation when the number of arc-intervals increases with $n$, as well as more general Toeplitz determinants to which we may apply the present strategy.



قيم البحث

اقرأ أيضاً

We find the asymptotic behaviors of Toeplitz determinants with symbols which are a sum of two contributions: one analytical and non-zero function in an annulus around the unit circle, and the other proportional to a Dirac delta function. The formulas are found by using the Wiener-Hopf procedure. The determinants of this type are found in computing the spin-correlation functions in low-lying excited states of some integrable models, where the delta function represents a peak at the momentum of the excitation. As a concrete example of applications of our results, using the derived asymptotic formulas we compute the spin-correlation functions in the lowest energy band of the frustrated quantum XY chain in zero field, and the ground state magnetization.
169 - I. Krasovsky 2010
We review the asymptotic behavior of a class of Toeplitz (as well as related Hankel and Toeplitz + Hankel) determinants which arise in integrable models and other contexts. We discuss Szego, Fisher-Hartwig asymptotics, and how a transition between th em is related to the Painleve V equation. Certain Toeplitz and Hankel determinants reduce, in certain double-scaling limits, to Fredholm determinants which appear in the theory of group representations, in random matrices, random permutations and partitions. The connection to Toeplitz determinants helps to evaluate the asymptotics of related Fredholm determinants in situations of interest, and we review the corresponding results.
We prove the analogue of the strong Szeg{H o} limit theorem for a large class of bordered Toeplitz determinants. In particular, by applying our results to the formula of Au-Yang and Perk cite{YP} for the next-to-diagonal correlations $langle sigma_{0 ,0}sigma_{N-1,N} rangle$ in the anisotropic square lattice Ising model, we rigorously justify that the next-to-diagonal long-range order is the same as the diagonal and horizontal ones in the low temperature regime. The anisotropy-dependence of the subleading term in the asymptotics of the next-to-diagonal correlations is also established. We use Riemann-Hilbert and operator theory techniques, independently and in parallel, to prove these results.
361 - Martin Lohmann 2015
We reconsider the Gram-Hadamard bound as it is used in constructive quantum field theory and many body physics to prove convergence of Fermionic perturbative expansions. Our approach uses a recursion for the amplitudes of the expansion, discovered or iginally by Djokic arXiv:1312.1185. It explains the standard way to bound the expansion from a new point of view, and for some of the amplitudes provides new bounds, which avoid the use of Fourier transform, and are therefore superior to the standard bounds for models like the cold interacting Fermi gas.
161 - B Eynard 2019
Topological recursion associates to a spectral curve, a sequence of meromorphic differential forms. A tangent space to the moduli space of spectral curves (its space of deformations) is locally described by meromorphic 1-forms, and we use form-cycle duality to re-express it in terms of cycles (generalized cycles). This formulation allows to express the ABCD tensors of Quantum Airy Structures acting on the vector space of cycles, in an intrinsic spectral-curve geometric way.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا