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Register a new userAsymptotic expansion of a partition function related to the sinh-model
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Added by
Karol Kozlowski Kajetan
Publication date
2014
fields
Physics
and research's language is
English
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This paper develops a method to carry out the large-$N$ asymptotic analysis of a class of $N$-dimensional integrals arising in the context of the so-called quantum separation of variables method. We push further ideas developed in the context of random matrices of size $N$, but in the present problem, two scales $1/N^{alpha}$ and $1/N$ naturally occur. In our case, the equilibrium measure is $N^{alpha}$-dependent and characterised by means of the solution to a $2times 2$ Riemann--Hilbert problem, whose large-$N$ behavior is analysed in detail. Combining these results with techniques of concentration of measures and an asymptotic analysis of the Schwinger-Dyson equations at the distributional level, we obtain the large-$N$ behavior of the free energy explicitly up to $o(1)$. The use of distributional Schwinger-Dyson is a novelty that allows us treating sufficiently differentiable interactions and the mixing of scales $1/N^{alpha}$ and $1/N$, thus waiving the analyticity assumptions often used in random matrix theory.
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Using the Griffiths-Simon construction of the $varphi^4$ model and the lace expansion for the Ising model, we prove that, if the strength $lambdage0$ of nonlinearity is sufficiently small for a large class of short-range models in dimensions $d>4$, then the critical $varphi^4$ two-point function $langlevarphi_ovarphi_xrangle_{mu_c}$ is asymptotically $|x|^{2-d}$ times a model-dependent constant, and the critical point is estimated as $mu_c=mathscr{hat J}-fraclambda2langlevarphi_o^2rangle_{mu_c}+O(lambda^2)$, where $mathscr{hat J}$ is the massless point for the Gaussian model.
We use techniques in the shuffle algebra to present a formula for the partition function of a one-dimensional log-gas comprised of particles of (possibly) different integer charges at certain inverse temperature $beta$ in terms of the Berezin integral of an associated non-homogeneous alternating tensor. This generalizes previously known results by removing the restriction on the number of species of odd charge. Our methods provide a unified framework extending the de Bruijn integral identities from classical $beta$-ensembles ($beta$ = 1, 2, 4) to multicomponent ensembles, as well as to iterated integrals of more general determinantal integrands.
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.
The lace expansion for the Ising two-point function was successfully derived in Sakai (Commun. Math. Phys., 272 (2007): 283--344). It is an identity that involves an alternating series of the lace-expansion coefficients. In the same paper, we claimed that the expansion coefficients obey certain diagrammatic bounds which imply faster $x$-space decay (as the two-point function cubed) above the critical dimension $d_c$ ($=4$ for finite-variance models), if the spin-spin coupling is ferromagnetic, translation-invariant, summable and symmetric with respect to the underlying lattice symmetries. However, we recently found a flaw in the proof of Lemma 4.2 in Sakai (2007), a key lemma to the aforementioned diagrammatic bounds. In this paper, we no longer use the problematic Lemma 4.2 of Sakai (2007), and prove new diagrammatic bounds on the expansion coefficients that are slightly more complicated than those in Proposition 4.1 of Sakai (2007) but nonetheless obey the same fast decay above the critical dimension $d_c$. Consequently, the lace-expansion results for the Ising and $varphi^4$ models so far are all saved. The proof is based on the random-current representation and its source-switching technique of Griffiths, Hurst and Sherman, combined with a double expansion: a lace expansion for the lace-expansion coefficients.
We present a new dynamical proof of the Thouless-Anderson-Palmer (TAP) equations for the classical Sherrington-Kirkpatrick spin glass at sufficiently high temperature. In our derivation, the TAP equations are a simple consequence of the decay of the two point correlation functions. The methods can also be used to establish the decay of higher order correlation functions. We illustrate this by proving a suitable decay bound on the three point functions from which we derive an analogue of the TAP equations for the two point functions.
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