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Spontaneous breaking of symmetry in liquid crystal (LC) films often reveals itself as a microscopic pattern of molecular alignment. In a smectic-A LC, the emergence of positional order at the transition from the nematic phase leads to periodic textures that can be used as optical microarrays, templates for soft lithography, and ordering matrices for the organization and manipulation of functional nanoparticles. While both 1d and 2d patterns have been obtained as a function of the LC film thickness and applied fields, the connection has not been made between pattern formation and the peculiar critical behavior of LCs at the nematic–smectic transition, still eluding a comprehensive theoretical explanation. In this article, we demonstrate that an intense bend distortion applied to the LC molecular director while cooling from the nematic phase produces a frustrated smectic phase with depressed transition temperature, and the characteristic 1d periodic texture previously observed in thin films and under applied electric fields. In light of De Gennes’ analogy with the normal-superconductor transition of a metal, we identify the 1d texture as the equivalent of the intermediate state in type I superconductors. The bend distortion is analog to the magnetic field in metals and penetrates in the frustrated phase as an array of undercooled nematic domains, periodically intermixed with bend-free smectic-A domains. Our findings provide fundamental evidence for theories of the nematic–smectic transition, highlighting the deep connection between phase frustration and pattern formation, and perspectives on the design of functional smectic microarrays.

International audience; The dynamics of a multimode quantum cascade laser, are studied in a model based on effective semiconductor Maxwell-Bloch equations, encompassing key features for the radiation-medium interaction such as an asymmetric frequency dependent gain and refractive index as well as the phase-amplitude coupling provided by the linewidth enhancement factor. By considering its role and that of the free spectral range, we find the conditions in which the traveling wave emitted by the laser at the threshold can be destabilized by adjacent modes, thus leading the laser emission towards chaotic or regular multimode dynamics. In the latter case our simulations show that the field oscillations are associated to self-confined structures which travel along the laser cavity, bridging mode-locking and solitary wave propagation. In addition, we show how a RF modulation of the bias current leads to active mode-locking yielding high-contrast, picosecond pulses. Our results compare well with recent experiments on broad-band THz-QCLs and may help in the understanding of the conditions for the generation of ultrashort pulses and comb operation in mid-IR and THz spectral regions.

For every physical model defined on a generic graph or factor graph, the Bethe M -layer construction allows building a different model for which the Bethe approximation is exact in the large M limit, and coincides with the original model for ##IMG## [http://ej.iop.org/images/1742-5468/2017/11/113303/jstataa8c3cieqn001.gif] $M=1$ . The ##IMG## [http://ej.iop.org/images/1742-5468/2017/11/113303/jstataa8c3cieqn002.gif] $1/M$ perturbative series is then expressed by a diagrammatic loop expansion in terms of so-called fat diagrams. Our motivation is to study some important second-order phase transitions that do exist on the Bethe lattice, but are either qualitatively different or absent in the corresponding fully connected case. In this case, the standard approach based on a perturbative expansion around the naive mean field theory (essentially a fully connected model) fails. On physical grounds, we expect that when the construction is applied to a lattice in finite dimension there is a small region of the external parameters, close to the Bethe critical point, where strong deviations from mean-field behavior will be observed. In this region, the ##IMG## [http://ej.iop.org/images/1742-5468/2017/11/113303/jstataa8c3cieqn003.gif] $1/M$ expansion for the corrections diverges, and can be the starting point for determining the correct non-mean-field critical exponents using renormalization group arguments. In the end, we will show that the critical series for the generic observable can be expressed as a sum of Feynman diagrams with the same numerical prefactors of field theories. However, the contribution of a given diagram is not evaluated by associating Gaussian propagators to its lines, as in field theories: one has to consider the graph as a portion of the original lattice, replacing the internal lines with appropriate one-dimensional chains, and attaching to the internal points the appropriate number of infinite-size Bethe trees to restore the correct local connectivity of the original model. The actual contribution of each (fat) diagram is the so-called line-connected observable, which also includes contributions from sub-diagrams with appropriate prefactors. In order to compute the corrections near to the critical point, Feynman diagrams (with their symmetry factors) can be read directly from the appropriate field-theoretical literature; the computation of momentum integrals is also quite similar; the extra work consists of computing the line-connected observable of the associated fat diagram in the limit of all lines becoming infinitely long.

The typical complexity of Constraint Satisfaction Problems (CSPs) can be investigated by means of random ensembles of instances. The latter exhibit many threshold phenomena besides their satisfiability phase transition, in particular a clustering or dynamic phase transition (related to the tree reconstruction problem) at which their typical solutions shatter into disconnected components. In this paper we study the evolution of this phenomenon under a bias that breaks the uniformity among solutions of one CSP instance, concentrating on the bicoloring of k-uniform random hypergraphs. We show that for small k the clustering transition can be delayed in this way to higher density of constraints, and that this strategy has a positive impact on the performances of Simulated Annealing algorithms. We characterize the modest gain that can be expected in the large k limit from the simple implementation of the biasing idea studied here. This paper contains also a contribution of a more methodological nature, made of a review and extension of the methods to determine numerically the discontinuous dynamic transition threshold., Comment: 32 pages, 16 figures

We present a complete characterization of the fluctuations and correlations of the squared overlap in the Edwards-Anderson Spin-Glass model in zero field. The analysis reveals that the energy-energy correlations (and thus the specific heat) have a different critical behavior than the fluctuations of the link overlap in spite of the fact that the average energy and average link overlap have the same critical properties. More precisely the link-overlap fluctuations are larger than the specific heat according to a computation at first order in the $6-\epsilon$ expansion. An unexpected outcome is that the link-overlap fluctuations have a subdominant power-law contribution characterized by an anomalous logarithmic prefactor which is missing in the specific heat. In order to compute the $\epsilon$ expansion we consider the problem of the renormalization of quadratic composite operators in a generic multicomponent cubic field theory: the results obtained have a range of applicability beyond spin-glass theory., Comment: 9 pages

International audience; We study a well known machine learning model -the perceptron- as a simple model of jamming of hard objects. We exhibit two regimes: 1) a convex optimisation regime where jamming is hypostatic and non-critical. 2) a non convex optimisation regime where jamming is isostatic and critical. We characterise the critical jamming phase through exponents describing the distributions law of forces and gaps. Surprisingly we find that these exponents coincide with the corresponding ones recently computed in high dimensional hard spheres. In addition, modifying the perceptron to a random linear programming problem, we show that isostaticity is not a sufficient condition for singular force and gap distributions. For that, fragmentation of the space of solutions (replica symmetry breaking) appears to be a crucial ingredient. We hypothesise universality for a large class of non-convex constrained satisfaction problems with continuous variables.

We report an analytical study of the vibrational spectrum of the simplest model of jamming, the soft perceptron. We identify two distinct classes of soft modes. The first kind of modes are related to isostaticity and appear only in the close vicinity of the jamming transition. The second kind of modes instead are present everywhere in the glass phase and are related to the hierarchical structure of the potential energy landscape. Our results highlight the universality of the spectrum of normal modes in disordered systems, and open the way towards a detailed analytical understanding of the vibrational spectrum of low-temperature glasses., Comment: 6 pages, 3 figures, submitted to PNAS

It is well know that systems with an interaction decaying as a power of the distance may have critical exponents that are different from those of short-range systems. The boundary between long-range and short-range is known, however the behavior in the crossover region is not well understood. In this paper we propose a general form for the crossover function and we compute it in a particular limit. We compare our predictions with the results of numerical simulations for two-dimensional long-range percolation., 17 pages, 6 figures

The sound attenuation in the THz region is studied down to T 16 K in glassy glycerol by inelastic x-ray scattering. At striking variance with the decrease found below 100 K in the GHz data, the attenuation in the THz range does not show any T dependence. This result (i) indicates the presence of two different attenuation mechanisms, active, respectively, in the high- and low-frequency limits, (ii) demonstrates the nondynamic origin of the attenuation of THz sound waves, and confirms a similar conclusion obtained in SiO2 glass by molecular dynamics, and (iii) supports the low-frequency attenuation mechanism proposed by Fabian and Allen [ Phys. Rev. Lett. 82, 1478 (1999)].

We consider the theory of the glass phase and jamming of hard spheres in the large space dimension limit. Building upon the exact expression for the free-energy functional obtained previously, we find that the Random First Order Transition (RFOT) scenario is realized here with two thermodynamic transitions: the usual Kauzmann point associated with entropy crisis, and a further transition at higher pressures in which a glassy structure of micro-states is developed within each amorphous state. This kind of glass-glass transition into a phase dominating the higher densities was described years ago by Elisabeth Gardner, and may well be a generic feature of RFOT. Micro states that are small excitations of an amorphous matrix -- separated by low entropic or energetic barriers -- thus emerge naturally, and modify the high pressure (or low temperature) limit of the thermodynamic functions., Comment: 26 pages, 7 figures -- to be published in a Special Issue of The Journal of Physical Chemistry B in honor of Peter G. Wolynes -- paper I is arXiv:1208.0421

We consider the critical properties of points of continuous glass transition as one can find in liquids in presence of constraints or in liquids in porous media. Through a one loop analysis we show that the critical Replica Field Theory describing these points can be mapped in the $\phi^4$-Random Field Ising Model. We confirm our analysis studying the finite size scaling of the $p$-spin model defined on sparse random graph, where a fraction of variables is frozen such that the phase transition is of a continuous kind., Comment: The paper has been completely revised. A completely new part with simulations of a p-spin glass model on random graph has been included. An appendix with the Mathematica worksheet used in the calculation of the diagrams has also been added

We study a chain of identical glassy systems in a constrained equilibrium where each bond of the chain is forced to remain at a preassigned distance to the previous one. We apply this description to Mean Field Glassy systems in the limit of long chain where each bond is close to the previous one. We show that in specific conditions this pseudo-dynamic process can formally describe real relaxational dynamics the long time. In particular, in mean field spin glass models we can recover in this way the equations of Langevin dynamics in the long time limit at the dynamical transition temperature and below. We interpret the formal identity as an evidence that in these situations the configuration space is explored in a quasi-equilibrium fashion. Our general formalism, that relates dynamics to equilibrium puts slow dynamics in a new perspective and opens the way to the computation of new dynamical quantities in glassy systems., A few typos corrected in the new version

International audience; We analyze Mode Coupling discontinuous transition in the limit of vanishing discontinuity, approaching the so called "$A_3$" point. In these conditions structural relaxation and fluctuations appear to have universal form independent from the details of the system. The analysis of this limiting case suggests new ways for looking at the Mode Coupling equations in the general case.

We analyze critical points that can be induced in glassy systems by the presence of constraints. These critical points are predicted by the Mean Field Thermodynamic approach and they are precursors of the standard glass transition in absence of constraints. Through a deep analysis of the soft modes appearing in the replica field theory we can establish the universality class of these points. In the case of the "annealed potential" of a symmetric coupling between two copies of the system, the critical point is in the Ising universality class. More interestingly, is the case of the "quenched potential" where the a single copy is coupled with an equilibrium reference configuration, or the "pinned particle" case where a fraction of particles is frozen in fixed positions. In these cases we find the Random Field Ising Model (RFIM) universality class. The effective random field is a "self-generated" disorder that reflects the random choice of the reference configuration. The RFIM representation of the critical theory predicts non-trivial relations governing the leading singular behavior of relevant correlation functions, that can be tested in numerical simulations., Comment: 27 pages, 1 figure. New version, accepted for publication in JSTAT. Added a detailed discussion of the annealed construction and the particle pinning one

International audience; We study the probability distribution of the pseudo-critical temperature in a mean-field and in a short-range spin-glass model: the Sherrington-Kirkpatrick (SK) and the Edwards-Anderson (EA) model. In both cases, we put in evidence the underlying connection between the fluctuations of the pseudo-critical point and and the Extreme Value Statistics of random variables. For the SK model, both with Gaussian and binary couplings, the distribution of the pseudo-critical temperature is found to be the Tracy-Widom distribution. For the EA model, the distribution is found to be the Gumbel distribution. Being the EA model representative of uniaxial magnetic materials with quenched disorder like $Fe_{0.5} Mn_{0.5} Ti O_3$ or $Eu_{0.5} Ba_{0.5} Mn O_3$, its pseudo-critical point distribution should be a priori experimentally accessible.

International audience; We develop a field-theoretical description of dynamical heterogeneities and fluctuations in supercooled liquids close to the (avoided) MCT singularity. Using quasi-equilibrium arguments we eliminate time from the description and we completely characterize fluctuations in the beta regime. We identify different sources of fluctuations and show that the most relevant ones are associated to variations of "self-induced disorder" in the initial condition of the dynamics. It follows that heterogeneites can be describes through a cubic field theory with an effective random field term. The phenomenon of perturbative dimensional reduction ensues, well known in random field problems, which implies an upper critical dimension of the theory equal to 8. We apply our theory to finite size scaling for mean-field systems and we test its prediction against numerical simulations.

International audience; Liquid and amorphous metals are an outstanding example of systems combining great relevance in both industrial applications and basic science. They find broad technological application, ranging from the production of industrial coatings (walls of refinery cokers, drill pipes for oil drilling) to medical equipment (reconstructive devices, surgical blades) or high-performance sporting goods. Most metallic materials, indeed, need to be refined in the molten state before being manufactured. Liquid metals, in particular the monoatomic ones, have long been recognized as the prototype of simple liquids, in the sense that they encompass most of the physical properties of real fluids without the complications which may be present in a particular system. Until the sixties the understanding of the physical properties of liquid metals proceeded rather slowly. It was John Ziman, indeed, who first developed a theory for the electron transport in liquid metals in 1961, and subsequently several experimental and theoretical studies were presented at the first Liquid Metals International Conference held in Brookhaven in 1966 (see, for example, refs. [1-3]). From the experimental point of view, the 1950s saw major efforts related to the development of neutron scattering facilities, which constitute a unique probe in accessing the microscopic structure and dynamics in condensed matter and, in particular, in the liquid and amorphous state. A further experimental breakthrough, however, has happened in the last twenty years when X-rays generated by synchrotron radiation facilities could be used as a probe for structure and dynamics in frequency and wavelength regions similar to those explored by neutrons. Concerning the nature of the dynamical structure in liquid and amorphous metals, in the last two decades many theoretical and experimental groups have switched their studies to researching effects due to non-hydrodynamic processes in liquids, i.e. collective processes which cannot be predicted by ordinary hydrodynamics. Examples of such non-hydrodynamic propagating excitations are shear waves, the famous "fast sound" [4] in binary liquids with disparate masses, or charge waves in ionic melts [5], while the most obvious non-hydrodynamic relaxation process is structural

International audience; We investigate the finite-size fluctuations due to quenched disorder of the critical temperature of the Sherrington-Kirkpatrick spin glass. In order to accomplish this task, we perform a finite-size analysis of the spectrum of the susceptibility matrix obtained via the Plefka expansion. By exploiting results from random matrix theory, we obtain that the fluctuations of the critical temperature are described by the Tracy-Widom distribution with a non-trivial scaling exponent 2/3.

International audience; A real space Renormalization Group approach is presented for a non-mean field spin-glass. This approach has been conceived in the effort to develop an alternative method to the Renormalization Group approaches based on the replica method. Indeed, non-perturbative effects in the latter are quite generally out of control, in such a way that these approaches are non-predictive. On the contrary, we show that the real space method developed in this work yields precise predictions for the critical behavior and exponents of the model.

We study the adversarial satisfiability problem, where the adversary can choose whether the variables are negated in clauses or not, in order to make the resulting formula unsatisfiable. This problem belongs to a general class of adversarial optimization problems that often arise in practice and are algorithmically much harder than the standard optimization problems. We use the cavity method to compute large deviations of the entropy in the random satisfiability problem with respect to the configurations of negations. We conclude that in the thermodynamic limit the best strategy the adversary can adopt is to simply balance the number of times every variable is negated and the number of times it is not negated. We also conduct a numerical study of the problem, and find that there are very strong pre-asymptotic effects that may be due to the fact that for small sizes exponential and factorial growth is hardly distinguishable. As a side result we compute the satisfiability threshold for balanced configurations of negations, and also the random regular satisfiability, i.e. when all variables belong to the same number of clauses.

The large scale behavior of the simplest non-mean-field spin-glass system is analyzed, and the critical exponent related to the divergence of the correlation length is computed at two loops within the $ϵ$-expansion technique with two independent methods. The techniques presented show how the underlying ideas of the renormalization group apply also in this disordered model, in such a way that an $ϵ$-expansion can be consistently set up. By pushing such calculation to high orders in $ϵ$, a consistent non-mean-field theory for such disordered system could be established, giving a substantial contribution the development of a predictive theory for real spin glasses.

We introduce a version of the cavity method for diluted mean-field spin models that allows the computation of thermodynamic quantities similar to the Franz-Parisi quenched potential in sparse random graph models. This method is developed in the particular case of partially decimated random constraint satisfaction problems. This allows to develop a theoretical understanding of a class of algorithms for solving constraint satisfaction problems, in which elementary degrees of freedom are sequentially assigned according to the results of a message passing procedure (belief-propagation). We confront this theoretical analysis to the results of extensive numerical simulations., 32 pages, 24 figures

We extend our theory of amorphous packings of hard spheres to binary mixtures and more generally to multicomponent systems. The theory is based on the assumption that amorphous packings produced by typical experimental or numerical protocols can be identified with the infinite pressure limit of long lived metastable glassy states. We test this assumption against numerical and experimental data and show that the theory correctly reproduces the variation with mixture composition of structural observables, such as the total packing fraction and the partial coordination numbers., Comment: 10 pages, 3 figures

We study link-diluted $\pm J$ Ising spin glass models on the hierarchical lattice and on a three-dimensional lattice close to the percolation threshold. We show that previously computed zero temperature fixed points are unstable with respect to temperature perturbations and do not belong to any critical line in the dilution-temperature plane. We discuss implications of the presence of such spurious unstable fixed points on the use of optimization algorithms, and we show how entropic effects should be taken into account to obtain the right physical behavior and critical points., Comment: 4 pages, 4 figures. A major typo error in formula (8) has been corrected

Hard spheres interacting through a square well potential were simulated by using two different methods: Brownian cluster dynamics (BCD) and event driven Brownian dynamics (EDBD). The structure of the equilibrium states obtained by both methods was compared and found to be almost identical. Self-diffusion coefficients (D) were determined as a function of the interaction strength. The same values were found by using BCD or EDBD. Contrary to EDBD, BCD allows one to study the effect of bond rigidity and hydrodynamic interaction within the clusters. When the bonds are flexible, the effect of attraction on D is relatively weak compared to systems with rigid bonds. D increases first with increasing attraction strength, and then decreases for stronger interaction. Introducing intracluster hydrodynamic interaction weakly increases D for a given interaction strength. Introducing bond rigidity causes a strong decrease in D which no longer shows a maximum as function of the attraction strength. (C) 2008 American Institute of Physics.

In this paper we study finite interaction range corrections to the mosaic picture of the glass transition as emerges from the study of the Kac limit of large interaction range for disordered models. To this aim we consider point to set correlation functions, or overlaps, in a one dimensional random energy model as a function of the range of interaction. In the Kac limit, the mosaic length defines a sharp first order transition separating a high overlap phase from a low overlap one. Correspondingly we find that overlap curves as a function of the window size and different finite interaction ranges cross roughly at the mosaic lenght. Nonetheless we find very slow convergence to the Kac limit and we discuss why this could be a problem for measuring the mosaic lenght in realistic models., Comment: 18 pages, 7 figures, contribution for the special issue "Viewing the World through Spin Glasses" in honour of Professor David Sherrington

In this paper we review a recent proposal to understand the long time limit of glassy dynamics in terms of an appropriate Markov Chain. [1]. The advantages of the resulting construction are many. The first one is that it gives a quasi equilibrium description on how glassy systems explore the phase space in the slow relaxation part of their dynamics. The second one is that it gives an alternative way to obtain dynamical equations starting from a dynamical rule that is static in spirit. This provides a way to overcome the difficulties encountered in the short time part of the dynamics where current conservation must be enforced. We study this approach in detail in a prototypical mean field disordered spin system, namely the p-spin spherical model, showing how we can obtain the well known equations that describes its dynamics. Then we apply the same approach to structural glasses. We first derive a set of dynamical Ornstein-Zernike equations which are very general in nature. Finally we consider two possible closure schemes for them, namely the Hypernetted Chain approximation of liquid theory and a closure of the BBGKY hierarchy that has been recently introduced by G. Szamel. From both approaches we finally find a set of dynamical Mode-Coupling like equations that are supposed to describe the system in the long time/slow dynamics regime., Comment: 13 pages, 3 figures

Cavity solitons (CSs) are localized structures appearing as intensity peaks in the homogeneous background of the field emitted by a nonlinear microresonator. We experimentally and theoretically study the switch-on process of cavity solitons in semiconductor amplifiers. The switching time has two contributions: a lethargic lapse following the application of the switching pulse and a characteristic buildup time. While the latter is not significantly affected by the control parameters, the former crucially depends on them. Optimization of the parameters leads to a switch-on time of less than one nanosecond, assessing the CS competitiveness for all-optical applications.

The low temperature phase of discontinuous mean-field spin glasses is characterized by the appearance of an exponential number of metastable states. Which ones among these states dominate the out-of-equilibrium dynamics of these systems? In order to answer this question, we compare high-precision numerical simulations of a diluted $p$-spin model with a cavity computation of the threshold energy. Our main conclusion is that the aging dynamics is dominated by different layers of metastable states depending on the cooling schedule. In order to perform our analysis, we present a method for computing the marginality condition of diluted spin glasses at non-zero temperature., Comment: 19 pages, 6 eps figures

We reconsider the one-step replica-symmetry-breaking (1RSB) solutions of two random combinatorial problems: k-XORSAT and k-SAT. We present a general method for establishing the stability of these solutions with respect to further steps of replica-symmetry breaking. Our approach extends the ideas of [A.Montanari and F. Ricci-Tersenghi, Eur.Phys.J. B 33, 339 (2003)] to more general combinatorial problems. It turns out that 1RSB is always unstable at sufficiently small clauses density alpha or high energy. In particular, the recent 1RSB solution to 3-SAT is unstable at zero energy for alpha< alpha_m, with alpha_m\approx 4.153. On the other hand, the SAT-UNSAT phase transition seems to be correctly described within 1RSB., 26 pages, 7 eps figures

We derive analytical solutions for p-spin models with finite connectivity at zero temperature. These models are the statistical mechanics equivalent of p-XORSAT problems in theoretical computer science. We give a full characterization of the phase diagram: location of the phase transitions (static and dynamic), together with a description of the clustering phenomenon taking place in configurational space. We use two alternative methods: the cavity approach and a rigorous derivation., Comment: 14 pages, 14 figures. v3: small errors corrected, simpler notation used

A laser–plasma interaction experiment was performed in order to match effective conditions for wake-field electron acceleration. A 30-fs pulse was made to interact with a preformed plasma generated via the exploding foil technique from a 1-μm-thick plastic film. The irradiance of the femtosecond pulse in the plasma was 1020 Wcm−2. The interaction conditions were investigated via imaging and spectroscopy at the fundamental and the second harmonic of the laser frequency, both forward and backward. Our data clearly show that conditions suitable for electron acceleration are achieved close to the propagation axis and can be easily reproduced from shot to shot. In contrast, significant growth of instabilities occurs at the boundaries of the interaction region. These observations are consistent with a preliminary evidence of forward acceleration of high-energy electrons. Optical, X-ray, and γ-ray data obtained for different positions of the foil target with respect to the laser focal plane further support this promising scenario.

We discuss the slow relaxation phenomenon in glassy systems by means of replicas by constructing a static field theory approach to the problem. At the mean field level we study how criticality in the four point correlation functions arises because of the presence of soft modes and we derive an effective replica field theory for these critical fluctuations. By using this at the Gaussian level we obtain many physical quantities: the correlation length, the exponent parameter that controls the Mode-Coupling dynamical exponents for the two-point correlation functions, and the prefactor of the critical part of the four point correlation functions. Moreover we perform a one-loop computation in order to identify the region in which the mean field Gaussian approximation is valid. The result is a Ginzburg criterion for the glass transition. We define and compute in this way a proper Ginzburg number. Finally, we present numerical values of all these quantities obtained from the Hypernetted Chain approximation for the replicated liquid theory., 34 pages, 1 figure - to be published on J.Chem.Phys. for a special issue on the Glass Transition

We study the set of solutions of random k-satisfiability formulae through the cavity method. It is known that, for an interval of the clause-to-variables ratio, this decomposes into an exponential number of pure states (clusters). We refine substantially this picture by: (i) determining the precise location of the clustering transition; (ii) uncovering a second `condensation' phase transition in the structure of the solution set for k larger or equal than 4. These results both follow from computing the large deviation rate of the internal entropy of pure states. From a technical point of view our main contributions are a simplified version of the cavity formalism for special values of the Parisi replica symmetry breaking parameter m (in particular for m=1 via a correspondence with the tree reconstruction problem) and new large-k expansions., Comment: 30 pages, 14 figures, typos corrected, discussion of appendix C expanded with a new figure