arXiv subjects
Publications and source records attributed to Sumedha.
In this paper we solve the Blume-Capel model on a complete graph in the presence of random crystal field with a distribution, $P(Δ_i) =pδ(Δ_i-Δ)+(1-p) δ(Δ_i+Δ)$, using large deviation techniques. We find that the first order transition of the pure system is destroyed for $0.046 0.954$) even at zero temperature.
We study the random field $p$-spin model with Ising spins on a fully connected graph using the theory of large deviations in this paper. This is a good model to study the effect of quenched random field on systems which have a sharp first order transition in the pure state. For $p=2$, the phase-diagram of the model, for bimodal distribution of the random field, has been well studied and is known to undergo a continuous transition for lower values of the random field ($h$) and a first order transition beyond a threshold, $h_{tp}(\approx 0.439)$. We find the phase diagram of the model, for all $p \ge 2$, with bimodal random field distribution, using large deviation techniques. We also look at the fluctuations in the system by calculating the magnetic susceptibility. For $p=2$, beyond the tri-critical point in the regime of first order transition, we find that for $h_{tp} h_o=1/p!$), the system does not show ferromagnetic order even at zero temperature. We find that the magnetic susceptibility for $p \ge 3$ is discontinuous at the transition point for $h<h_o$.
The satisfiability threshold for constraint satisfaction problems is that value of the ratio of constraints (or clauses) to variables, above which the probability that a random instance of the problem has a solution is zero in the large system limit. Two different approaches to obtaining this threshold have been discussed in the literature - using first or second-moment methods which give rigorous bounds or using the non-rigorous but powerful replica-symmetry breaking (RSB) approach, which gives very accurate predictions on random graphs. In this paper, we lay out a different route to obtaining this threshold on a Bethe lattice. We need make no assumptions about the solution-space structure, a key assumption in the RSB approach. Despite this, our expressions and threshold values exactly match the best predictions of the cavity method under the 1-RSB assumption. Our method hence provides alternate interpretations as well as motivations for the key equations in the RSB approach.
We study and solve some variations of the random K-satisfiability problem - balanced K-SAT and biased random K-SAT - on a regular tree, using techniques we have developed earlier(arXiv:1110.2065). In both these problems, as well as variations of these that we have looked at, we find that the SAT-UNSAT transition obtained on the Bethe lattice matches the exact threshold for the same model on a random graph for K=2 and is very close to the numerical value obtained for K=3. For higher K it deviates from the numerical estimates of the solvability threshold on random graphs, but is very close to the dynamical 1-RSB threshold as obtained from the first non-trivial fixed point of the survey propagation algorithm.
We consider the K-satisfiability problem on a regular d-ary rooted tree. For this model, we demonstrate how we can calculate in closed form, the moments of the total number of solutions as a function of d and K, where the average is over all realizations, for a fixed assignment of the surface variables. We find that different moments pick out different 'critical' values of d, below which they diverge as the total number of variables on the tree goes to infinity and above which they decay. We show that K-SAT on the random graph also behaves similarly. We also calculate exactly the fraction of instances that have solutions for all K. On the tree, this quantity decays to 0 (as the number of variables increases) for any d>1. However the recursion relations for this quantity have a non-trivial fixed-point solution which indicates the existence of a different transition in the interior of an infinite rooted tree.
We study a one-dimensional model of microtubule assembly/disassembly in which GTP bound to tubulins within the microtubule undergoes stochastic hydrolysis. In contrast to models that only consider a cap of GTP-bound tubulin, stochastic hydrolysis allows GTP-bound tubulin remnants to exist within the microtubule. We find that these buried GTP remnants enable an alternative mechanism of recovery from shrinkage, and enhances fluctuations of filament lengths. Under conditions for which this alternative mechanism dominates, an increasing depolymerization rate leads to a decrease in dissociation rate and thus a net increase in assembly.
In this paper, we propose a thermodynamic mechanism for the formation of transcriptional foci via the joint agglomeration of DNA-looping proteins and protein-binding domains on DNA: The competition between the gain in protein-DNA binding free energy and the entropy loss due to DNA looping is argued to result in an effective attraction between loops. A mean-field approximation can be described analytically via a mapping to a restricted random-graph ensemble having local degree constraints and global constraints on the number of connected components. It shows the emergence of protein clusters containing a finite fraction of all looping proteins. If the entropy loss due to a single DNA loop is high enough, this transition is found to be of first order.
Soft-constraint affinity propagation (SCAP) is a new statistical-physics based clustering technique. First we give the derivation of a simplified version of the algorithm and discuss possibilities of time- and memory-efficient implementations. Later we give a detailed analysis of the performance of SCAP on artificial data, showing that the algorithm efficiently unveils clustered and hierarchical data structures. We generalize the algorithm to the problem of semi-supervised clustering, where data are already partially labeled, and clustering assigns labels to previously unlabeled points. SCAP uses both the geometrical organization of the data and the available labels assigned to few points in a computationally efficient way, as is shown on artificial and biological benchmark data.
Motivation: Similarity-measure based clustering is a crucial problem appearing throughout scientific data analysis. Recently, a powerful new algorithm called Affinity Propagation (AP) based on message-passing techniques was proposed by Frey and Dueck \cite{Frey07}. In AP, each cluster is identified by a common exemplar all other data points of the same cluster refer to, and exemplars have to refer to themselves. Albeit its proved power, AP in its present form suffers from a number of drawbacks. The hard constraint of having exactly one exemplar per cluster restricts AP to classes of regularly shaped clusters, and leads to suboptimal performance, {\it e.g.}, in analyzing gene expression data. Results: This limitation can be overcome by relaxing the AP hard constraints. A new parameter controls the importance of the constraints compared to the aim of maximizing the overall similarity, and allows to interpolate between the simple case where each data point selects its closest neighbor as an exemplar and the original AP. The resulting soft-constraint affinity propagation (SCAP) becomes more informative, accurate and leads to more stable clustering. Even though a new {\it a priori} free-parameter is introduced, the overall dependence of the algorithm on external tuning is reduced, as robustness is increased and an optimal strategy for parameter selection emerges more naturally. SCAP is tested on biological benchmark data, including in particular microarray data related to various cancer types. We show that the algorithm efficiently unveils the hierarchical cluster structure present in the data sets. Further on, it allows to extract sparse gene expression signatures for each cluster.
We study the dynamics of a population subject to selective pressures, evolving either on RNA neutral networks or in toy fitness landscapes. We discuss the spread and the neutrality of the population in the steady state. Different limits arise depending on whether selection or random drift are dominant. In the presence of strong drift we show that observables depend mainly on $M μ$, $M$ being the population size and $μ$ the mutation rate, while corrections to this scaling go as 1/M: such corrections can be quite large in the presence of selection if there are barriers in the fitness landscape. Also we find that the convergence to the large $M μ$ limit is linear in $1/M μ$. Finally we introduce a protocol that minimizes drift; then observables scale like 1/M rather than $1/(Mμ)$, allowing one to determine the large $M$ limit faster when $μ$ is small; furthermore the genotypic diversity increases from $O(\ln M)$ to $O(M)$.
RNA secondary structure is an important computational model to understand how genetic variation maps into phenotypic (structural) variation. Evolutionary innovation in RNA structures is facilitated by neutral networks, large connected sets of RNA sequences that fold into the same structure. Our work extends and deepens previous studies on neutral networks. First, we show that even the 1-mutant neighborhood of a given sequence (genotype) G0 with structure (phenotype) P contains many structural variants that are not close to P. This holds for biological and generic RNA sequences alike. Second, we analyze the relation between new structures in the 1-neighborhoods of genotypes Gk that are only a moderate Hamming distance k away from G0, and the structure of G0 itself, both for biological and for generic RNA structures. Third, we analyze the relation between mutational robustness of a sequence and the distances of structural variants near this sequence. Our findings underscore the role of neutral networks in evolutionary innovation, and the role that high robustness can play in diminishing the potential for such innovation.
We study the efficiency of the incomplete enumeration algorithm for linear and branched polymers. There is a qualitative difference in the efficiency in these two cases. The average time to generate an independent sample of $n$ sites for large $n$ varies as $n^2$ for linear polymers, but as $exp(c n^α)$ for branched (undirected and directed) polymers, where $0<α<1$. On the binary tree, our numerical studies for $n$ of order $10^4$ gives $α= 0.333 \pm 0.005$. We argue that $α=1/3$ exactly in this case.
We study rooted spiral trees in 2,3 and 4 dimensions on a hyper cubical lattice using exact enumeration and Monte-Carlo techniques. On the square lattice, we also obtain exact lower bound of 1.93565 on the growth constant $λ$. Series expansions give $θ=-1.3667\pm 0.001$ and $ν= 1.3148\pm0.001$ on a square lattice. With Monte-Carlo simulations we get the estimates as $θ=-1.364\pm0.01$, and $ν= 1.312\pm0.01$. These results are numerical evidence against earlier proposed dimensional reduction by four in this problem. In dimensions higher than two, the spiral constraint can be implemented in two ways. In either case, our series expansion results do not support the proposed dimensional reduction.
We study rooted self avoiding polygons and self avoiding walks on deterministic fractal lattices of finite ramification index. Different sites on such lattices are not equivalent, and the number of rooted open walks W_n(S), and rooted self-avoiding polygons P_n(S) of n steps depend on the root S. We use exact recursion equations on the fractal to determine the generating functions for P_n(S), and W_n(S) for an arbitrary point S on the lattice. These are used to compute the averages $< P_n(S)>, , $ and $ $ over different positions of S. We find that the connectivity constant $μ$, and the radius of gyration exponent $ν$ are the same for the annealed and quenched averages. However, $ ~ n log μ+ (α_q -2) log n$, and $ ~ n log μ+ (γ_q -1)log n$, where the exponents $α_q$ and $γ_q$ take values different from the annealed case. These are expressed as the Lyapunov exponents of random product of finite-dimensional matrices. For the 3-simplex lattice, our numerical estimation gives $ α_q \simeq 0.72837 \pm 0.00001$; and $γ_q \simeq 1.37501 \pm 0.00003$, to be compared with the annealed values $α_a = 0.73421$ and $γ_a = 1.37522$.
We study the adsorption-desorption phase transition of directed branched polymer in $d+1$ dimensions in contact with a line by mapping it to a $d$ dimensional hard core lattice gas at negative activity. We solve the model exactly in 1+1 dimensions, and calculate the crossover exponent related to fraction of monomers adsorbed at the critical point of surface transition, and we also determine the density profile of the polymer in different phases. We also obtain the value of crossover exponent in 2+1 dimensions and give the scaling function of the sticking fraction for 1+1 and 2+1 dimensional directed branched polymer.
We relate $ϕ(\bf{x},s)$, the average number of sites at a transverse distance $\bf{x}$ in the directed animals with $s$ sites in $d$ transverse dimensions, to the two-point correlation function of a lattice gas with nearest neighbor exclusion in $d$ dimensions. For large $s$, $ϕ(\bf{x},s)$ has the scaling form $\frac{s}{R_s^d} f(|\bf{x}|/R_s)$, where $R_s$ is the root mean square radius of gyration of animals of $s$ sites. We determine the exact scaling function for $d =1$ to be $f(r) = \frac{\sqrtπ}{2 \sqrt{3}}erfc(r/\sqrt{3})$. We also show that $ϕ(\bf{x}=0,s)$ can be determined in terms of the animals number generating function of the directed animals.