Scaling of loop-erased walks in 2 to 4 dimensions

We simulate loop-erased random walks on simple (hyper-)cubic lattices of dimensions 2,3, and 4. These simulations were mainly motivated to test recent two loop renormalization group predictions for logarithmic corrections in $d=4$, simulations in lower dimensions were done for completeness and in order to test the algorithm. In $d=2$, we verify with high precision the prediction $D=5/4$, where the number of steps $n$ after erasure scales with the number $N$ of steps before erasure as $n\sim N^{D/2}$. In $d=3$ we again find a power law, but with an exponent different from the one found in the most precise previous simulations: $D = 1.6236\pm 0.0004$. Finally, we see clear deviations from the naive scaling $n\sim N$ in $d=4$. While they agree only qualitatively with the leading logarithmic corrections predicted by several authors, their agreement with the two-loop prediction is nearly perfect.Comment: 3 pages, including 3 figure

Field theory conjecture for loop-erased random walks

We give evidence that the functional renormalization group (FRG), developed to study disordered systems, may provide a field theoretic description for the loop-erased random walk (LERW), allowing to compute its fractal dimension in a systematic expansion in epsilon=4-d. Up to two loop, the FRG agrees with rigorous bounds, correctly reproduces the leading logarithmic corrections at the upper critical dimension d=4, and compares well with numerical studies. We obtain the universal subleading logarithmic correction in d=4, which can be used as a further test of the conjecture.Comment: 5 page

Probability distribution of the sizes of largest erased-loops in loop-erased random walks

We have studied the probability distribution of the perimeter and the area of the k-th largest erased-loop in loop-erased random walks in two-dimensions for k = 1 to 3. For a random walk of N steps, for large N, the average value of the k-th largest perimeter and area scales as N^{5/8} and N respectively. The behavior of the scaled distribution functions is determined for very large and very small arguments. We have used exact enumeration for N <= 20 to determine the probability that no loop of size greater than l (ell) is erased. We show that correlations between loops have to be taken into account to describe the average size of the k-th largest erased-loops. We propose a one-dimensional Levy walk model which takes care of these correlations. The simulations of this simpler model compare very well with the simulations of the original problem.Comment: 11 pages, 1 table, 10 included figures, revte

Spanning Forests on Random Planar Lattices

The generating function for spanning forests on a lattice is related to the q-state Potts model in a certain q -> 0 limit, and extends the analogous notion for spanning trees, or dense self-avoiding branched polymers. Recent works have found a combinatorial perturbative equivalence also with the (quadratic action) O(n) model in the limit n -> -1, the expansion parameter t counting the number of components in the forest. We give a random-matrix formulation of this model on the ensemble of degree-k random planar lattices. For k = 3, a correspondence is found with the Kostov solution of the loop-gas problem, which arise as a reformulation of the (logarithmic action) O(n) model, at n = -2. Then, we show how to perform an expansion around the t = 0 theory. In the thermodynamic limit, at any order in t we have a finite sum of finite-dimensional Cauchy integrals. The leading contribution comes from a peculiar class of terms, for which a resummation can be performed exactly.Comment: 43 pages, Dedicated to Edouard Brezin and Giorgio Parisi, on the occasion of their special birthda

A review of Monte Carlo simulations of polymers with PERM

In this review, we describe applications of the pruned-enriched Rosenbluth method (PERM), a sequential Monte Carlo algorithm with resampling, to various problems in polymer physics. PERM produces samples according to any given prescribed weight distribution, by growing configurations step by step with controlled bias, and correcting "bad" configurations by "population control". The latter is implemented, in contrast to other population based algorithms like e.g. genetic algorithms, by depth-first recursion which avoids storing all members of the population at the same time in computer memory. The problems we discuss all concern single polymers (with one exception), but under various conditions: Homopolymers in good solvents and at the $\Theta$ point, semi-stiff polymers, polymers in confining geometries, stretched polymers undergoing a forced globule-linear transition, star polymers, bottle brushes, lattice animals as a model for randomly branched polymers, DNA melting, and finally -- as the only system at low temperatures, lattice heteropolymers as simple models for protein folding. PERM is for some of these problems the method of choice, but it can also fail. We discuss how to recognize when a result is reliable, and we discuss also some types of bias that can be crucial in guiding the growth into the right directions.Comment: 29 pages, 26 figures, to be published in J. Stat. Phys. (2011

Spanning forests and the q-state Potts model in the limit q \to 0

We study the q-state Potts model with nearest-neighbor coupling v=e^{\beta J}-1 in the limit q,v \to 0 with the ratio w = v/q held fixed. Combinatorially, this limit gives rise to the generating polynomial of spanning forests; physically, it provides information about the Potts-model phase diagram in the neighborhood of (q,v) = (0,0). We have studied this model on the square and triangular lattices, using a transfer-matrix approach at both real and complex values of w. For both lattices, we have computed the symbolic transfer matrices for cylindrical strips of widths 2 \le L \le 10, as well as the limiting curves of partition-function zeros in the complex w-plane. For real w, we find two distinct phases separated by a transition point w=w_0, where w_0 = -1/4 (resp. w_0 = -0.1753 \pm 0.0002) for the square (resp. triangular) lattice. For w > w_0 we find a non-critical disordered phase, while for w < w_0 our results are compatible with a massless Berker-Kadanoff phase with conformal charge c = -2 and leading thermal scaling dimension x_{T,1} = 2 (marginal operator). At w = w_0 we find a "first-order critical point": the first derivative of the free energy is discontinuous at w_0, while the correlation length diverges as w \downarrow w_0 (and is infinite at w = w_0). The critical behavior at w = w_0 seems to be the same for both lattices and it differs from that of the Berker-Kadanoff phase: our results suggest that the conformal charge is c = -1, the leading thermal scaling dimension is x_{T,1} = 0, and the critical exponents are \nu = 1/d = 1/2 and \alpha = 1.Comment: 131 pages (LaTeX2e). Includes tex file, three sty files, and 65 Postscript figures. Also included are Mathematica files forests_sq_2-9P.m and forests_tri_2-9P.m. Final journal versio

Supplementary data for article: Selaković, Ž.; Tran, J. P.; Kota, K. P.; Lazić, M.; Retterer, C.; Besh, R.; Panchal, R. G.; Soloveva, V.; Sean, V. A.; Jay, W. B.; et al. Second Generation of Diazachrysenes: Protection of Ebola Virus Infected Mice and Mechanism of Action. European Journal of Medicinal Chemistry 2019, 162, 32–50. https://doi.org/10.1016/j.ejmech.2018.10.061

Supplementary material for: [https://www.sciencedirect.com/science/article/pii/S0223523418309383?via%3Dihub]Related to published version: [http://cherry.chem.bg.ac.rs/handle/123456789/360]Related to accepted version: [http://cherry.chem.bg.ac.rs/handle/123456789/2966

Second generation of diazachrysenes: Protection of Ebola virus infected mice and mechanism of action

Ebola virus (EBOV) causes a deadly hemorrhagic fever in humans and non-human primates. There is currently no FDA-approved vaccine or medication to counter this disease. Here, we report on the design, synthesis and anti-viral activities of two classes of compounds which show high potency against EBOV in both in vitro cell culture assays and in vivo mouse models Ebola viral disease. These compounds incorporate the structural features of cationic amphiphilic drugs (CAD), i.e they possess both a hydrophobic domain and a hydrophilic domain consisting of an ionizable amine functional group. These structural features enable easily diffusion into cells but once inside an acidic compartment their amine groups became protonated, ionized and remain trapped inside the acidic compartments such as late endosomes and lysosomes. These compounds, by virtue of their lysomotrophic functions, blocked EBOV entry. However, unlike other drugs containing a CAD moiety including chloroquine and amodiaquine, compounds reported in this study display faster kinetics of accumulation in the lysosomes, robust expansion of late endosome/lysosomes, relatively more potent suppression of lysosome fusion with other vesicular compartments and inhibition of cathepsins activities, all of which play a vital role in anti-EBOV activity. Furthermore, the diazachrysene 2 (ZSML08) that showed most potent activity against EBOV in in vitro cell culture assays also showed significant survival benefit with 100% protection in mouse models of Ebola virus disease, at a low dose of 10 mg/kg/day. Lastly, toxicity studies in vivo using zebrafish models suggest no developmental defects or toxicity associated with these compounds. Overall, these studies describe two new pharmacophores that by virtue of being potent lysosomotrophs, display potent anti-EBOV activities both in vitro and in vivo animal models of EBOV disease. © 2018 Elsevier Masson SASSupplementary material: [http://cherry.chem.bg.ac.rs/handle/123456789/2967]Peer-reviewed manuscript: [http://cherry.chem.bg.ac.rs/handle/123456789/2966