Delocalization transition of the selective interface model: distribution of pseudo-critical temperatures

According to recent progress in the finite size scaling theory of critical disordered systems, the nature of the phase transition is reflected in the distribution of pseudo-critical temperatures $T_c(i,L)$ over the ensemble of samples $(i)$ of size $L$. In this paper, we apply this analysis to the delocalization transition of an heteropolymeric chain at a selective fluid-fluid interface. The width $\Delta T_c(L)$ and the shift $[T_c(\infty)-T_c^{av}(L)]$ are found to decay with the same exponent $L^{-1/\nu_{R}}$, where $1/\nu_{R} \sim 0.26$. The distribution of pseudo-critical temperatures $T_c(i,L)$ is clearly asymmetric, and is well fitted by a generalized Gumbel distribution of parameter $m \sim 3$. We also consider the free energy distribution, which can also be fitted by a generalized Gumbel distribution with a temperature dependent parameter, of order $m \sim 0.7$ in the critical region. Finally, the disorder averaged number of contacts with the interface scales at $T_c$ like $L^{\rho}$ with $\rho \sim 0.26 \sim 1/\nu_R$.Comment: 9 pages,6 figure

Probing the tails of the ground state energy distribution for the directed polymer in a random medium of dimension d=1,2,3d=1,2,3 via a Monte-Carlo procedure in the disorder

In order to probe with high precision the tails of the ground-state energy distribution of disordered spin systems, K\"orner, Katzgraber and Hartmann \cite{Ko_Ka_Ha} have recently proposed an importance-sampling Monte-Carlo Markov chain in the disorder. In this paper, we combine their Monte-Carlo procedure in the disorder with exact transfer matrix calculations in each sample to measure the negative tail of ground state energy distribution $P_d(E_0)$ for the directed polymer in a random medium of dimension $d=1,2,3$. In $d=1$, we check the validity of the algorithm by a direct comparison with the exact result, namely the Tracy-Widom distribution. In dimensions $d=2$ and $d=3$, we measure the negative tail up to ten standard deviations, which correspond to probabilities of order $P_d(E_0) \sim 10^{-22}$. Our results are in agreement with Zhang's argument, stating that the negative tail exponent $\eta(d)$ of the asymptotic behavior $\ln P_d (E_0) \sim - | E_0 |^{\eta(d)}$ as $E_0 \to -\infty$ is directly related to the fluctuation exponent $\theta(d)$ (which governs the fluctuations $\Delta E_0(L) \sim L^{\theta(d)}$ of the ground state energy $E_0$ for polymers of length $L$) via the simple formula $\eta(d)=1/(1-\theta(d))$. Along the paper, we comment on the similarities and differences with spin-glasses.Comment: 13 pages, 16 figure

Statistics of low energy excitations for the directed polymer in a 1+d1+d random medium (d=1,2,3d=1,2,3)

We consider a directed polymer of length $L$ in a random medium of space dimension $d=1,2,3$. The statistics of low energy excitations as a function of their size $l$ is numerically evaluated. These excitations can be divided into bulk and boundary excitations, with respective densities $\rho^{bulk}_L(E=0,l)$ and $\rho^{boundary}_L(E=0,l)$. We find that both densities follow the scaling behavior $\rho^{bulk,boundary}_L(E=0,l) = L^{-1-\theta_d} R^{bulk,boundary}(x=l/L)$, where $\theta_d$ is the exponent governing the energy fluctuations at zero temperature (with the well-known exact value $\theta_1=1/3$ in one dimension). In the limit $x=l/L \to 0$, both scaling functions $R^{bulk}(x)$ and $R^{boundary}(x)$ behave as $R^{bulk,boundary}(x) \sim x^{-1-\theta_d}$, leading to the droplet power law $\rho^{bulk,boundary}_L(E=0,l)\sim l^{-1-\theta_d}$ in the regime $1 \ll l \ll L$. Beyond their common singularity near $x \to 0$, the two scaling functions $R^{bulk,boundary}(x)$ are very different : whereas $R^{bulk}(x)$ decays monotonically for $0<x<1$, the function $R^{boundary}(x)$ first decays for $0<x<x_{min}$, then grows for $x_{min}<x<1$, and finally presents a power law singularity $R^{boundary}(x)\sim (1-x)^{-\sigma_d}$ near $x \to 1$. The density of excitations of length $l=L$ accordingly decays as $\rho^{boundary}_L(E=0,l=L)\sim L^{- \lambda_d}$ where $\lambda_d=1+\theta_d-\sigma_d$. We obtain $\lambda_1 \simeq 0.67$, $\lambda_2 \simeq 0.53$ and $\lambda_3 \simeq 0.39$, suggesting the possible relation $\lambda_d= 2 \theta_d$.Comment: 15 pages, 25 figure

A Solvable Model of Secondary Structure Formation in Random Hetero-Polymers

We propose and solve a simple model describing secondary structure formation in random hetero-polymers. It describes monomers with a combination of one-dimensional short-range interactions (representing steric forces and hydrogen bonds) and infinite range interactions (representing polarity forces). We solve our model using a combination of mean field and random field techniques, leading to phase diagrams exhibiting second-order transitions between folded, partially folded and unfolded states, including regions where folding depends on initial conditions. Our theoretical results, which are in excellent agreement with numerical simulations, lead to an appealing physical picture of the folding process: the polarity forces drive the transition to a collapsed state, the steric forces introduce monomer specificity, and the hydrogen bonds stabilise the conformation by damping the frustration-induced multiplicity of states.Comment: 24 pages, 14 figure

Random wetting transition on the Cayley tree : a disordered first-order transition with two correlation length exponents

We consider the random wetting transition on the Cayley tree, i.e. the problem of a directed polymer on the Cayley tree in the presence of random energies along the left-most bonds. In the pure case, there exists a first-order transition between a localized phase and a delocalized phase, with a correlation length exponent $\nu_{pure}=1$. In the disordered case, we find that the transition remains first-order, but that there exists two diverging length scales in the critical region : the typical correlation length diverges with the exponent $\nu_{typ}=1$, whereas the averaged correlation length diverges with the bigger exponent $\nu_{av}=2$ and governs the finite-size scaling properties. We describe the relations with previously studied models that are governed by the same "Infinite Disorder Fixed Point". For the present model, where the order parameter is the contact density $\theta_L=l_a/L$ (defined as the ratio of the number $l_a$ of contacts over the total length $L$), the notion of "infinite disorder fixed point" means that the thermal fluctuations of $\theta_L$ within a given sample, become negligeable at large scale with respect to sample-to-sample fluctuations. We characterize the statistics over the samples of the free-energy and of the contact density. In particular, exactly at criticality, we obtain that the contact density is not self-averaging but remains distributed over the samples in the thermodynamic limit, with the distribution ${\cal P}_{T_c}(\theta) = 1/(\pi \sqrt{\theta (1-\theta)})$.Comment: 15 pages, 1 figur

Modelling high redshift Lyman α emitters

We present a new model for high redshift Lyman α emitters (LAEs) in the cosmological context which takes into account the resonant scattering of Lyα photons through expanding gas. The GALICS semi-analytic model provides us with the physical properties of a large sample of high redshift galaxies. We implement, in post-processing, a gas outflow model for each galaxy based on simple scaling arguments. The coupling with a library of numerical experiments of Lyα transfer through expanding (or static) dusty shells of gas allows us to derive the Lyα escape fraction and profile of each galaxy. Results obtained with this new approach are compared with simpler models often used in the literature. The predicted distribution of Lyα photons escape fraction shows that galaxies with a low star formation rate (SFR) have a fesc of the order of unity, suggesting that, for those objects, Lyα may be used to trace the SFR assuming a given conversion law. In galaxies forming stars intensely, the escape fraction spans the whole range from 0 to 1. The model is able to get a good match to the ultraviolet (UV) and Lyα luminosity function data at 3 < z < 5. We find that we are in good agreement with both the bright Lyα data and the faint LAE population observed by Rauch et al. at z= 3 whereas a simpler constant Lyαescape fraction model fails to do so. Most of the Lyα profiles of our LAEs are redshifted by the diffusion in the expanding gas which suppresses intergalactic medium absorption and scattering. The bulk of the observed Lyα equivalent width (EW) distribution is recovered by our model, but we fail to obtain the very large values sometimes detected. Our predictions for stellar masses and UV luminosity functions of LAEs show a satisfactory agreement with observational estimates. The UV-brightest galaxies are found to show only low Lyα EWs in our model, as it is reported by many observations of high redshift LAEs. We interpret this effect as the joint consequence of old stellar populations hosted by UV-bright galaxies, and high H i column densities that we predict for these objects, which quench preferentially resonant Lyα photons via dust extinctio

The Phase Diagram of Random Heteropolymers

We propose a new analytic approach to study the phase diagram of random heteropolymers, based on the cavity method. For copolymers we analyze the nature and phenomenology of the glass transition as a function of sequence correlations. Depending on these correlations, we find that two different scenarios for the glass transition can occur. We show that, beside the much studied possibility of an abrupt freezing transition at low temperature, the system can exhibit, upon cooling, a first transition to a soft glass phase with fully broken replica symmetry and a continuously growing degree of freezing as the temperature is lowered.Comment: 4 pages, 3 figures; published versio

Semi-Analytic Galaxy Evolution (SAGE): Model Calibration and Basic Results

This paper describes a new publicly available codebase for modelling galaxy formation in a cosmological context, the "Semi-Analytic Galaxy Evolution" model, or SAGE for short. SAGE is a significant update to that used in Croton et al. (2006) and has been rebuilt to be modular and customisable. The model will run on any N-body simulation whose trees are organised in a supported format and contain a minimum set of basic halo properties. In this work we present the baryonic prescriptions implemented in SAGE to describe the formation and evolution of galaxies, and their calibration for three N-body simulations: Millennium, Bolshoi, and GiggleZ. Updated physics include: gas accretion, ejection due to feedback, and reincorporation via the galactic fountain; a new gas cooling--radio mode active galactic nucleus (AGN) heating cycle; AGN feedback in the quasar mode; a new treatment of gas in satellite galaxies; and galaxy mergers, disruption, and the build-up of intra-cluster stars. Throughout, we show the results of a common default parameterization on each simulation, with a focus on the local galaxy population.Comment: 15 pages, 9 figures, accepted for publication in ApJS. SAGE is a publicly available codebase for modelling galaxy formation in a cosmological context, available at https://github.com/darrencroton/sage Questions and comments can be sent to Darren Croton: [email protected]

Random RNA under tension

The Laessig-Wiese (LW) field theory for the freezing transition of random RNA secondary structures is generalized to the situation of an external force. We find a second-order phase transition at a critical applied force f = f_c. For f f_c, the extension L as a function of pulling force f scales as (f-f_c)^(1/gamma-1). The exponent gamma is calculated in an epsilon-expansion: At 1-loop order gamma = epsilon/2 = 1/2, equivalent to the disorder-free case. 2-loop results yielding gamma = 0.6 are briefly mentioned. Using a locking argument, we speculate that this result extends to the strong-disorder phase.Comment: 6 pages, 10 figures. v2: corrected typos, discussion on locking argument improve

Anderson transitions : multifractal or non-multifractal statistics of the transmission as a function of the scattering geometry

The scaling theory of Anderson localization is based on a global conductance $g_L$ that remains a random variable of order O(1) at criticality. One realization of such a conductance is the Landauer transmission for many transverse channels. On the other hand, the statistics of the one-channel Landauer transmission between two local probes is described by a multifractal spectrum that can be related to the singularity spectrum of individual eigenstates. To better understand the relations between these two types of results, we consider various scattering geometries that interpolate between these two cases and analyse the statistics of the corresponding transmissions. We present detailed numerical results for the power-law random banded matrices (PRBM model). Our conclusions are : (i) in the presence of one isolated incoming wire and many outgoing wires, the transmission has the same multifractal statistics as the local density of states of the site where the incoming wire arrives; (ii) in the presence of backward scattering channels with respect to the case (i), the statistics of the transmission is not multifractal anymore, but becomes monofractal. Finally, we also describe how these scattering geometries influence the statistics of the transmission off criticality.Comment: 12 pages, 9 figure