Higher Order Analogues of Tracy-Widom Distributions via the Lax Method

We study the distribution of the largest eigenvalue in formal Hermitian one-matrix models at multicriticality, where the spectral density acquires an extra number of k-1 zeros at the edge. The distributions are directly expressed through the norms of orthogonal polynomials on a semi-infinite interval, as an alternative to using Fredholm determinants. They satisfy non-linear recurrence relations which we show form a Lax pair, making contact to the string literature in the early 1990's. The technique of pseudo-differential operators allows us to give compact expressions for the logarithm of the gap probability in terms of the Painleve XXXIV hierarchy. These are the higher order analogues of the Tracy-Widom distribution which has k=1. Using known Backlund transformations we show how to simplify earlier equivalent results that are derived from Fredholm determinant theory, valid for even k in terms of the Painleve II hierarchy.Comment: 24 pages. Improved discussion of Backlund transformations, in addition to other minor improvements in text. Typos corrected. Matches published versio

Low-temperature properties of the heavy-fermion system U Cd

We present electrical-resistivity, magnetic-susceptibility, specific-heat, and thermal-expansion data for UCd11. The low-temperature specific heat indicates that the electronic subsystem has a highly enhanced specific heat which is partially removed by a phase transition at 5.0 K. © 1984 The American Physical Society

Hopping or Jumping on the Cliffs: The Unusual Phylogeographical and Demographic Structure of an Extremely Narrow Endemic Mediterranean Plant

Several past and recent climatic and geological events have greatly influenced the current distribution of coastal species around the Mediterranean Basin. As a consequence, the reconstruction of the distributional history of these species is challenging. In this study, we used both chloroplast and nuclear SNPs to assess the levels of genetic differentiation, contemporary/historical levels of gene flow, and demographic history for the three only known (one mainland and two insular) populations of Eokochia saxicola, a rare Mediterranean coastal rocky halophyte. Plastid genome analysis revealed very low intraspecific haplotype variation and partial admixture among Capri and Palinuro populations with at least two independent colonization events for the Strombolicchio islet. Nuclear SNPs variation consistently identified three distinct genetic clusters corresponding to our sampling localities. Furthermore, strong genetic isolation was confirmed by both historical and contemporary levels of migration among the three populations. The DIYABC analysis identified two introductions temporally separated from Palinuro to Capri (ca.25 Mya) and subsequently to Strombolicchio (ca.09 Mya) as the most likely hypothesis for the current distribution of E. saxicola. Regardless of their small population sizes, all study sites supported high-genetic diversity maintained by outcrossing and random mating between individuals owing largely to wind pollination, an exclusive trait among Mediterranean narrow endemics. In conclusion, the patterns observed confirm that some Mediterranean endemics are not necessarily “evolutionary dead-ends” but rather represent species that have extensive demographic stability and a strong evolutionary legacy

The influence of a relict distribution on genetic structure and variation in the Mediterranean tree, Platanus orientalis

The distribution of plant species around the Mediterranean basin is a product of the influence of both geographical barriers and of climatic changes experienced during the Tertiary, with the transition from a warm to cool periods. Several species, once largely distributed across the Northern Hemisphere, retracted to refugial areas in southern Europe where they are described as Tertiary relicts. Platanus orientalis is a typical representative of Tertiary flora in southwest Eurasia; its distribution spreads from the Caucasus to the Mediterranean, with its western border in Southern Italy and Sicily. We analysed genetic diversity and differentiation in the centre and western part of its distribution range using nuclear microsatellites and compared genetic parameters between core and western populations. We found an overall decrease in genetic diversity estimates (heterozygosity, private allelic richness) from central towards western populations, with those from Southern Italy and Sicily showing the lowest values. The low level of genetic diversity probably results from historic range fragmentation experienced by P. orientalis in its westernmost distribution as confirmed by high level genetic isolation of these populations. Ornamental hybrids were genetically distinguished from P. orientalis as contained private alleles, indicating that gene flow with natural populations is rare. Population assignment and neighbour-joining (NJ) analysis of populations identified four groups belonging to two main phyletic groups (the Southern Italian-Sicilian-Balkan and Cretan-Bulgarian-Turkish lineages) that seem to have different biogeographic origin and also excluded an artificial origin for southern Italian and Sicilian populations as previously suggested. These results show that quantifying the genetic variation of a Tertiary relict in a geographical context, and the potential effect of hybridization with introduced ornamental hybrids, can provide useful insights on factors influencing population genetic structure. Such information is crucial to predict how such taxa will respond to increasing anthropogenic influence on the environment