We generalize a previous inequality related to a sharp version of the Littlewood conjecture on the minimal L1-norm of N -term exponential sums f on the unit circle. The new result concerns replacing the expression log(1+ t|f |) with log (∑K k=1 tk|fk| 2 ) . The proof occurs on the level of finite Toeplitz matrices, where it reduces to an inequality between their polarized determinants (or “mixe...

Following the recent proposal made by [J. Bouttier, Phys. Rev. E 76, 041140 (2007)], we study analytically the mobility properties of a single vacancy in the close-packed dimer model on the square lattice. Using the spanning web representation, we find determinantal expressions for various observable quantities. In the limiting case of large lattices, they can be reduced to the calculation of T...

We prove that Fredholm determinants of the form det(1 − Ks), where Ks is the restriction of either the discrete Bessel kernel or the discrete 2F1 kernel to {s, s + 1, . . . }, can be expressed through solutions of discrete Painlevé II and V equations, respectively. These Fredholm determinants can also be viewed as distribution functions of the first part of the random partitions distributed acc...

We prove that Fredholm determinants of the form det(1−Ks), where Ks is the restriction of either the discrete Bessel kernel or the discrete 2 F1-kernel to {s, s + 1, . . . }, can be expressed, respectively, through solutions of discrete Painlevé II (dPII) and Painlevé V (dPV) equations. These Fredholm determinants can also be viewed as distribution functions of the first part of the random part...