The Razumov-Stroganov conjecture: Stochastic processes, loops and combinatorics

A fascinating conjectural connection between statistical mechanics and combinatorics has in the past five years led to the publication of a number of papers in various areas, including stochastic processes, solvable lattice models and supersymmetry. This connection, known as the Razumov-Stroganov conjecture, expresses eigenstates of physical systems in terms of objects known from combinatorics, which is the mathematical theory of counting. This note intends to explain this connection in light of the recent papers by Zinn-Justin and Di Francesco.Comment: 6 pages, 4 figures, JSTAT News & Perspective

Why haven't loose globular clusters collapsed yet?

We report on the discovery of a surprising observed correlation between the slope of the low-mass stellar global mass function (GMF) of globular clusters (GCs) and their central concentration parameter c=log(r_t/r_c), i.e. the logarithmic ratio of tidal and core radii. This result is based on the analysis of a sample of twenty Galactic GCs with solid GMF measurements from deep HST or VLT data. All the high-concentration clusters in the sample have a steep GMF, most likely reflecting their initial mass function. Conversely, low-concentration clusters tend to have a flatter GMF implying that they have lost many stars via evaporation or tidal stripping. No GCs are found with a flat GMF and high central concentration. This finding appears counter-intuitive, since the same two-body relaxation mechanism that causes stars to evaporate and the cluster to eventually dissolve should also lead to higher central density and possibly core-collapse. Therefore, more concentrated clusters should have lost proportionately more stars and have a shallower GMF than low concentration clusters, contrary to what is observed. It is possible that severely depleted GCs have also undergone core collapse and have already recovered a normal radial density profile. It is, however, more likely that GCs with a flat GMF have a much denser and smaller core than suggested by their surface brightness profile and may well be undergoing collapse at present. In either case, we may have so far seriously underestimated the number of post core-collapse clusters and many may be lurking in the Milky Way.Comment: Four pages, one figure, accepted for publication in ApJ Letter

Inhomogeneous loop models with open boundaries

We consider the crossing and non-crossing O(1) dense loop models on a semi-infinite strip, with inhomogeneities (spectral parameters) that preserve the integrability. We compute the components of the ground state vector and obtain a closed expression for their sum, in the form of Pfaffian and determinantal formulas.Comment: 42 pages, 31 figures, minor corrections, references correcte

Open boundary Quantum Knizhnik-Zamolodchikov equation and the weighted enumeration of Plane Partitions with symmetries

We propose new conjectures relating sum rules for the polynomial solution of the qKZ equation with open (reflecting) boundaries as a function of the quantum parameter $q$ and the $\tau$-enumeration of Plane Partitions with specific symmetries, with $\tau=-(q+q^{-1})$. We also find a conjectural relation \`a la Razumov-Stroganov between the $\tau\to 0$ limit of the qKZ solution and refined numbers of Totally Symmetric Self Complementary Plane Partitions.Comment: 27 pages, uses lanlmac, epsf and hyperbasics, minor revision

Normal metal tunnel junction-based superconducting quantum interference proximity transistor: the N-SQUIPT

We report the fabrication and characterization of an alternative design for a superconducting quantum interference proximity transistor (SQUIPT) based on a normal metal (N) probe. The absence of direct Josephson coupling between the proximized metal nanowire and the N probe allows us to observe the full modulation of the wire density of states around zero voltage and current \textit{via} the application of an external magnetic field. This results into a drastic suppression of power dissipation which can be as low as a few $\sim 10^{-17}$ W. In this context the interferometer allows an improvement of up to four orders of magnitude with respect to earlier SQUIPT designs, and makes it ideal for extra-low power cryogenic applications. In addition, the N-SQUIPT has been recently predicted to be the enabling candidate for the implementation of coherent caloritronic devices based on proximity effect.Comment: 5 pages, 4 color figure

Quantum Knizhnik-Zamolodchikov equation: reflecting boundary conditions and combinatorics

We consider the level 1 solution of quantum Knizhnik-Zamolodchikov equation with reflecting boundary conditions which is relevant to the Temperley--Lieb model of loops on a strip. By use of integral formulae we prove conjectures relating it to the weighted enumeration of Cyclically Symmetric Transpose Complement Plane Partitions and related combinatorial objects

Sum rules for the ground states of the O(1) loop model on a cylinder and the XXZ spin chain

The sums of components of the ground states of the O(1) loop model on a cylinder or of the XXZ quantum spin chain at Delta=-1/2 (of size L) are expressed in terms of combinatorial numbers. The methods include the introduction of spectral parameters and the use of integrability, a mapping from size L to L+1, and knot-theoretic skein relations.Comment: final version to be publishe

Dynamical and photometric imprints of feedback processes on the early evolution of E/S0 galaxies

We show that the observed Velocity Dispersion Function of E/S0 galaxies matches strikingly well the distribution function of virial velocities of massive halos virializing at z > 1.5, as predicted by the standard hierarchical clustering scenario in a \LambdaCDM cosmology, for a constant ratio sigma/V_vir = 0.55 \pm 0.05, close to the value expected at virialization if it typically occurred at z > 3. This strongly suggests that dissipative processes and later merging events had little impact on the matter density profile. Adopting the above sigma/V_vir ratio, the observed relationships between photometric and dynamical properties which define the fundamental plane of elliptical galaxies, such as the luminosity-sigma (Faber-Jackson) and the luminosity-effective radius relations, as well as the M_BH-sigma relation, are nicely reproduced. Their shapes turn out to be determined by the mutual feedback of star-formation (and supernova explosions)and nuclear activity, along the lines discussed by Granato et al. (2004). To our knowledge, this is the first semi-analytic model for which simultaneous fits of the fundamental plane relations and of the epoch-dependent luminosity function of spheroidal galaxies have been presented.Comment: 14 pages, 6 figures, submitted to Ap