Condensation of degrees emerging through a first-order phase transition in classical random graphs

Due to their conceptual and mathematical simplicity, Erd\"os-R\'enyi or classical random graphs remain as a fundamental paradigm to model complex interacting systems in several areas. Although condensation phenomena have been widely considered in complex network theory, the condensation of degrees has hitherto eluded a careful study. Here we show that the degree statistics of the classical random graph model undergoes a first-order phase transition between a Poisson-like distribution and a condensed phase, the latter characterized by a large fraction of nodes having degrees in a limited sector of their configuration space. The mechanism underlying the first-order transition is discussed in light of standard concepts in statistical physics. We uncover the phase diagram characterizing the ensemble space of the model and we evaluate the rate function governing the probability to observe a condensed state, which shows that condensation of degrees is a rare statistical event akin to similar condensation phenomena recently observed in several other systems. Monte Carlo simulations confirm the exactness of our theoretical results.Comment: 8 pages, 6 figure

Level compressibility for the Anderson model on regular random graphs and the eigenvalue statistics in the extended phase

We calculate the level compressibility $\chi(W,L)$ of the energy levels inside $[-L/2,L/2]$ for the Anderson model on infinitely large random regular graphs with on-site potentials distributed uniformly in $[-W/2,W/2]$. We show that $\chi(W,L)$ approaches the limit $\lim_{L \rightarrow 0^+} \chi(W,L) = 0$ for a broad interval of the disorder strength $W$ within the extended phase, including the region of $W$ close to the critical point for the Anderson transition. These results strongly suggest that the energy levels follow the Wigner-Dyson statistics in the extended phase, consistent with earlier analytical predictions for the Anderson model on an Erd\"os-R\'enyi random graph. Our results are obtained from the accurate numerical solution of an exact set of equations valid for infinitely large regular random graphs.Comment: 7 pages, 3 figure

Tame and wild theorem for the category of filtered by standard modules for a quasi-hereditary algebra

We introduce the notion of interlaced weak ditalgebras and apply reduction procedures to their module categories to prove the tame-wild dichotomy for the category ${\cal F}(\Delta)$ of filtered by standard modules for a quasi-hereditary algebra. Moreover, in the tame case, we show that given a fixed dimension $d$, for every $d$-dimensional indecomposable module $M \in {\cal F}(\Delta)$, with the only possible exception of those lying in a finite number of isomorphism classes, the module $M$ coincides with its Auslander-Reiten translate in ${\cal F}(\Delta)$. Our results are based on a theorem by Koenig, K\"ulshammer, and Ovsienko relating ${\cal F}(\Delta)$ with the module category of some special type of ditalgebra.Comment: 51 page

Integral field observations of the blue compact galaxy Haro14. Star formation and feedback in dwarf galaxies

(Abridged) Low-luminosity, gas-rich blue compact galaxies (BCG) are ideal laboratories to investigate many aspects of the star formation in galaxies. We study the morphology, stellar content, kinematics, and the nebular excitation and ionization mechanism in the BCG Haro 14 by means of integral field observations with VIMOS in the VLT. From these data we build maps in continuum and in the brighter emission lines, produce line-ratio maps, and obtain the velocity and velocity dispersion fields. We also generate the integrated spectrum of the major HII regions and young stellar clusters identified in the maps to determine reliable physical parameters and oxygen abundances. We find as follows: i) the current star formation in Haro 14 is spatially extended with the major HII regions placed along a linear structure, elongated in the north-south direction, and in a horseshoe-like curvilinear feature that extends about 760 pc eastward; the continuum emission is more concentrated and peaks close to the galaxy center; ii) two different episodes of star formation are present: the recent starburst, with ages $\leq$ 6 Myrs and the intermediate-age clusters, with ages between 10 and 30 Myrs; these stellar components rest on a several Gyr old underlying host galaxy; iii) the H$\alpha$/H$\beta$ pattern is inhomogeneous, with excess color values varying from E(B-V)=0.04 up to E(B-V)=1.09; iv) shocks play a significant role in the galaxy; and v) the velocity field displays a complicated pattern with regions of material moving toward us in the east and north galaxy areas. The morphology of Haro 14, its irregular velocity field, and the presence of shocks speak in favor of a scenario of triggered star formation. Ages of the knots are consistent with the ongoing burst being triggered by the collective action of stellar winds and supernovae originated in the central clusters.Comment: 18 pages, 17 figures. Accepted for publication in A&

Probing star formation and feedback in dwarf galaxies. Integral field view of the blue compact galaxy Tololo 1937-423

(Abridged) Blue compact galaxies (BCG) are gas-rich, low-mass, small systems that form stars at unusually high rates. This makes them excellent laboratories for investigating the process of star-formation (SF) at galactic scales and the effects of massive stellar feedback on the interstellar (and intergalactic) medium. We analyzed the BCG Tololo 1937-423 using optical integral field spectroscopy with VIMOS at the Very Large Telescope to probe its morphology, stellar content, nebular excitation and ionization properties, and the kinematics of its warm ionized gas. We found that Tololo 1937-423 is currently undergoing an extended starburst, with nine major $H\alpha$ clumps. The galaxy presents a single continuum peak that is not cospatial with any knot in emission lines, indicating at least two relatively recent episodes of SF. The inhomogeneous dust distribution reaches its maximum roughly at the position of the continuum peak. We found shocked regions in the galaxy outer regions and at the edges of the SF knots. The oxygen abundance is similar in all the SF regions, suggesting a chemically homogeneous ionized interstellar medium over spatial scales of several kpc. The ionized gas kinematics displays an overall regular rotation around a northwest-southeast axis. The morphology of the galaxy and the two different episodes of SF suggest a scenario of triggered (induced by supernova shock waves) SF acting in Tololo 1937-423. The inferred ages for the different SF episodes (~13-80 Myr for the central post-starburst and 5-7 Myr for the ongoing SF) are consistent with triggered SF, with the most recent SF episode caused by the collective effect of stellar winds and supernova explosions from the central post-starburst. The velocity dispersion pattern, with higher velocity dispersions found at the edges of the SF regions, and shocked regions in the galaxy, also favor this scenario.Comment: 16 pages, 18 figures. Accepted for publication in A&