Global behavior of a fourth order difference equation

We determine the forbidden set, introduce an explicit formula for the solutions, and discuss the global behavior of solutions of a fourth order difference equation

The Martin-Benito-Mena Marugan-Olmedo prescription for the Dapor-Liegener model of Loop Quantum Cosmology

Recently, an alternative Hamiltonian constraint for Loop Quantum Cosmology has been put forward by Dapor and Liegener, inspired by previous work on regularization due to Thiemann. Here, we quantize this Hamiltonian following a prescription for cosmology proposed by Mart\'{\i}n-Benito, Mena Marug\'an, and Olmedo. To this effect, we first regularize the Euclidean and Lorentzian parts of the Hamiltonian constraint separately in the case of a Bianchi I cosmology. This allows us to identify a natural symmetrization of the Hamiltonian which is apparent in anisotropic scenarios. Preserving this symmetrization in isotropic regimes, we then determine the Hamiltonian constraint corresponding to a Friedmann-Lema\^itre-Robertson-Walker cosmology, which we proceed to quantize. We compute the action of this Hamiltonian operator in the volume eigenbasis and show that it takes the form of a fourth-order difference equation, unlike in standard Loop Quantum Cosmology, where it is known to be of second order. We investigate the superselection sectors of our constraint operator, proving that they are semilattices supported only on either the positive or the negative semiaxis, depending on the triad orientation. Remarkably, the decoupling between semiaxes allows us to write a closed expression for the generalized eigenfunctions of the geometric part of the constraint. This expression is totally determined by the values at the two points of the semilattice that are closest to the origin, namely the two contributions with smallest eigenvolume. This is in clear contrast with the situation found for the standard Hamiltonian of Loop Quantum Cosmology, where only the smallest value is free. This result indicates that the degeneracy of the new geometric Hamiltonian operator is equal to two, doubling the possible number of solutions with respect to the conventional quantization considered until now.Comment: 15 pages, published in Physical Review

On the accumulation points of non-periodic orbits of a difference equation of fourth order

PreprintIn this paper, we are interested in analyzing the dynamics of the fourth-order difference equation x_{n+4}=max{x_{n+3},x_{n+2},x_{n+1},0}-x_n, with arbitrary real initial conditions. We fully determine the accumulation point sets of the non-periodic solutions that, in fact, are configured as proper compact intervals of the real line. This study complements the previous knowledge of the dynamics of the difference equation already achieved in [M. Csörnyei, M. Laczkovich, Monatsh. Math. 132 (2001), 215-236] and [A. Linero Bas, D. Nieves Roldán, J. Difference Equ. Appl. 27 (2021), no. 11, 1608-1645]This work has been supported by the grant MTM2017-84079-P funded by MCIN/AEI/10.13039/501100011033 and by ERDF “A way of making Europe”, by the European Union. The second autor acknowledges the group research recognition 2021 SGR 01039 from Agència de Gestió d’Ajuts Universitaris i de RecercaPreprin

Application of the τ\tau-Function Theory of Painlev\'e Equations to Random Matrices: PIV, PII and the GUE

Tracy and Widom have evaluated the cumulative distribution of the largest eigenvalue for the finite and scaled infinite GUE in terms of a PIV and PII transcendent respectively. We generalise these results to the evaluation of $\tilde{E}_N(\lambda;a) := \Big < \prod_{l=1}^N \chi_{(-\infty, \lambda]}^{(l)} (\lambda - \lambda_l)^a \Big>$, where $\chi_{(-\infty, \lambda]}^{(l)} = 1$ for $\lambda_l \in (-\infty, \lambda]$ and $\chi_{(-\infty, \lambda]}^{(l)} = 0$ otherwise, and the average is with respect to the joint eigenvalue distribution of the GUE, as well as to the evaluation of F_N(\lambda;a) := \Big . Of particular interest are $\tilde{E}_N(\lambda;2)$ and $F_N(\lambda;2)$, and their scaled limits, which give the distribution of the largest eigenvalue and the density respectively. Our results are obtained by applying the Okamoto $\tau$-function theory of PIV and PII, for which we give a self contained presentation based on the recent work of Noumi and Yamada. We point out that the same approach can be used to study the quantities $\tilde{E}_N(\lambda;a)$ and $F_N(\lambda;a)$ for the other classical matrix ensembles.Comment: 40 pages, Latex2e plus AMS and XY packages. to appear Commun. Math. Phy

High-precision e-expansions of massive four-loop vacuum bubbles

In this paper we calculate at high-precision the expansions in e=(4-D)/2 of the master integrals of 4-loop vacuum bubble diagrams with equal masses, using a method based on the solution of systems of difference equations. We also show that the analytical expression of a related on-shell 3-loop self-mass master integral contains new transcendental constants made up of complete elliptic integrals of first and second kind.Comment: 7 pages, 2 figures, LaTex, to be published in Physics Letters