Density Matrix Renormalization Group and Reaction-Diffusion Processes

The density matrix renormalization group (DMRG) is applied to some one-dimensional reaction-diffusion models in the vicinity of and at their critical point. The stochastic time evolution for these models is given in terms of a non-symmetric ``quantum Hamiltonian'', which is diagonalized using the DMRG method for open chains of moderate lengths (up to about 60 sites). The numerical diagonalization methods for non-symmetric matrices are reviewed. Different choices for an appropriate density matrix in the non-symmetric DMRG are discussed. Accurate estimates of the steady-state critical points and exponents can then be found from finite-size scaling through standard finite-lattice extrapolation methods. This is exemplified by studying the leading relaxation time and the density profiles of diffusion-annihilation and of a branching-fusing model in the directed percolation universality class.Comment: 16 pages, latex, 5 PostScript figures include

A lattice polymer study of DNA renaturation dynamics

DNA renaturation is the recombination of two complementary single strands to form a double helix. It is experimentally known that renaturation proceeds through the formation of a double stranded nucleus of several base pairs (the rate limiting step) followed by a much faster zippering. We consider a lattice polymer model undergoing Rouse dynamics and focus on the nucleation of two diffusing strands. We study numerically the dependence of various nucleation rates on the strand lengths and on an additional local nucleation barrier. When the local barrier is sufficiently high, all renaturation rates considered scale with the length as predicted by Kramers' rate theory and are also in agreement with experiments: their scaling behavior is governed by exponents describing equilibrium properties of polymers. When the local barrier is lowered renaturation occurs in a regime of genuine non-equilibrium behavior and the scaling deviates from the rate theory prediction.Comment: 13 pages, 6 figures. To appear in Journal of Statistical Mechanic

Critical point shift in a fluid confined between opposing walls

The properties of a fluid, or Ising magnet, confined in a $L \times \infty$ geometry with opposing surface fields at the walls are studied by density matrix renormalization techniques. In particular we focus on the effect of gravity on the system, which is modeled by a bulk field whose strength varies linearly with the distance from the walls. It is well known that in the absence of gravity phase coexistence is restricted to temperatures below the wetting temperature. We find that gravity restores phase coexistence up to the bulk critical temperature, in agreement with previous mean field results. A detailed study of the scaling to the critical point, as $L \to \infty$, is performed. The temperature shift scales as $1/L^{y_T}$, while the gravitational constant scales as $1/L^{1+y_H}$, with $y_T$ and $y_H$ the bulk thermal and magnetic exponents respectively. For weak surface fields and $L$ not too large, we also observe a regime where the gravitational constant scales as $1/L^{1+y_H - \Delta_1 y_T}$ ($\Delta_1$ is the surface gap exponent) with a crossover, for sufficiently large $L$, to a scaling of type $1/L^{1+y_H}$.Comment: 9 pages, RevTeX, 11 PostScript figures included. Minor corrections. Final version as publishe

A Transfer Matrix study of the staggered BCSOS model

The phase diagram of the staggered six vertex, or body centered solid on solid model, is investigated by transfer matrix and finite size scaling techniques. The phase diagram contains a critical region, bounded by a Kosterlitz-Thouless line, and a second order line describing a deconstruction transition. In part of the phase diagram the deconstruction line and the Kosterlitz-Thouless line approach each other without merging, while the deconstruction changes its critical behaviour from Ising-like to a different universality class. Our model has the same type of symmetries as some other two-dimensional models, such as the fully frustrated XY model, and may be important for understanding their phase behaviour. The thermal behaviour for weak staggering is intricate. It may be relevant for the description of surfaces of ionic crystals of CsCl structure.Comment: 13 pages, RevTex, 1 Postscript file with all figures, to be published in Phys. Rev.

Models of DNA denaturation dynamics: universal properties

We briefly review some of the models used to describe DNA denaturation dynamics, focusing on the value of the dynamical exponent $z$, which governs the scaling of the characteristic time $\tau\sim L^z$ as a function of the sequence length $L$. The models contain different degrees of simplifications, in particular sometimes they do not include a description for helical entanglement: we discuss how this aspect influences the value of $z$, which ranges from $z=0$ to $z \approx 3.3$. Connections with experiments are also mentioned

Linear model for fast background subtraction in oligonucleotide microarrays

One important preprocessing step in the analysis of microarray data is background subtraction. In high-density oligonucleotide arrays this is recognized as a crucial step for the global performance of the data analysis from raw intensities to expression values. We propose here an algorithm for background estimation based on a model in which the cost function is quadratic in a set of fitting parameters such that minimization can be performed through linear algebra. The model incorporates two effects: 1) Correlated intensities between neighboring features in the chip and 2) sequence-dependent affinities for non-specific hybridization fitted by an extended nearest-neighbor model. The algorithm has been tested on 360 GeneChips from publicly available data of recent expression experiments. The algorithm is fast and accurate. Strong correlations between the fitted values for different experiments as well as between the free-energy parameters and their counterparts in aqueous solution indicate that the model captures a significant part of the underlying physical chemistry.Comment: 21 pages, 5 figure