596 research outputs found
Monte Carlo Tests of SLE Predictions for the 2D Self-Avoiding Walk
The conjecture that the scaling limit of the two-dimensional self-avoiding
walk (SAW) in a half plane is given by the stochastic Loewner evolution (SLE)
with leads to explicit predictions about the SAW. A remarkable
feature of these predictions is that they yield not just critical exponents,
but probability distributions for certain random variables associated with the
self-avoiding walk. We test two of these predictions with Monte Carlo
simulations and find excellent agreement, thus providing numerical support to
the conjecture that the scaling limit of the SAW is SLE.Comment: TeX file using APS REVTeX 4.0. 10 pages, 5 figures (encapsulated
postscript
Elephants can always remember: Exact long-range memory effects in a non-Markovian random walk
We consider a discrete-time random walk where the random increment at time
step depends on the full history of the process. We calculate exactly the
mean and variance of the position and discuss its dependence on the initial
condition and on the memory parameter . At a critical value
where memory effects vanish there is a transition from a weakly localized
regime (where the walker returns to its starting point) to an escape regime.
Inside the escape regime there is a second critical value where the random walk
becomes superdiffusive. The probability distribution is shown to be governed by
a non-Markovian Fokker-Planck equation with hopping rates that depend both on
time and on the starting position of the walk. On large scales the memory
organizes itself into an effective harmonic oscillator potential for the random
walker with a time-dependent spring constant . The solution of
this problem is a Gaussian distribution with time-dependent mean and variance
which both depend on the initiation of the process.Comment: 10 page
Rigorous Non-Perturbative Ornstein-Zernike Theory for Ising Ferromagnets
We rigorously derive the Ornstein-Zernike asymptotics of the pair-correlation
functions for finite-range Ising ferromagnets in any dimensions and at any
temperature above critical
Amortized Causal Discovery: Learning to Infer Causal Graphs from Time-Series Data
Standard causal discovery methods must fit a new model whenever they encounter samples from a new underlying causal graph. However, these samples often share relevant information - for instance, the dynamics describing the effects of causal relations - which is lost when following this approach. We propose Amortized Causal Discovery, a novel framework that leverages such shared dynamics to learn to infer causal relations from time-series data. This enables us to train a single, amortized model that infers causal relations across samples with different underlying causal graphs, and thus makes use of the information that is shared. We demonstrate experimentally that this approach, implemented as a variational model, leads to significant improvements in causal discovery performance, and show how it can be extended to perform well under hidden confounding
Monte Carlo Methods for the Self-Avoiding Walk
This article is a pedagogical review of Monte Carlo methods for the
self-avoiding walk, with emphasis on the extraordinarily efficient algorithms
developed over the past decade.Comment: 81 pages including lots of figures, 700138 bytes Postscript
(NYU-TH-94/05/02) [To appear in Monte Carlo and Molecular Dynamics
Simulations in Polymer Science, edited by Kurt Binder, Oxford University
Press, expected late 1994
Identifying influential spreaders and efficiently estimating infection numbers in epidemic models: a walk counting approach
We introduce a new method to efficiently approximate the number of infections
resulting from a given initially-infected node in a network of susceptible
individuals. Our approach is based on counting the number of possible infection
walks of various lengths to each other node in the network. We analytically
study the properties of our method, in particular demonstrating different forms
for SIS and SIR disease spreading (e.g. under the SIR model our method counts
self-avoiding walks). In comparison to existing methods to infer the spreading
efficiency of different nodes in the network (based on degree, k-shell
decomposition analysis and different centrality measures), our method directly
considers the spreading process and, as such, is unique in providing estimation
of actual numbers of infections. Crucially, in simulating infections on various
real-world networks with the SIR model, we show that our walks-based method
improves the inference of effectiveness of nodes over a wide range of infection
rates compared to existing methods. We also analyse the trade-off between
estimate accuracy and computational cost, showing that the better accuracy here
can still be obtained at a comparable computational cost to other methods.Comment: 6 page
Self-avoiding walks crossing a square
We study a restricted class of self-avoiding walks (SAW) which start at the
origin (0, 0), end at , and are entirely contained in the square on the square lattice . The number of distinct
walks is known to grow as . We estimate as well as obtaining strict upper and lower bounds,
We give exact results for the number of SAW of
length for and asymptotic results for .
We also consider the model in which a weight or {\em fugacity} is
associated with each step of the walk. This gives rise to a canonical model of
a phase transition. For the average length of a SAW grows as ,
while for it grows as
. Here is the growth constant of unconstrained SAW in . For we provide numerical evidence, but no proof, that the
average walk length grows as .
We also consider Hamiltonian walks under the same restriction. They are known
to grow as on the same lattice. We give
precise estimates for as well as upper and lower bounds, and prove that
Comment: 27 pages, 9 figures. Paper updated and reorganised following
refereein
Efficiency of the Incomplete Enumeration algorithm for Monte-Carlo simulation of linear and branched polymers
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
sites for large varies as for linear polymers, but as for branched (undirected and directed) polymers, where
. On the binary tree, our numerical studies for of order
gives . We argue that exactly in this
case.Comment: replaced with published versio
Uncovering the topology of configuration space networks
The configuration space network (CSN) of a dynamical system is an effective
approach to represent the ensemble of configurations sampled during a
simulation and their dynamic connectivity. To elucidate the connection between
the CSN topology and the underlying free-energy landscape governing the system
dynamics and thermodynamics, an analytical soluti on is provided to explain the
heavy tail of the degree distribution, neighbor co nnectivity and clustering
coefficient. This derivation allows to understand the universal CSN network
topology observed in systems ranging from a simple quadratic well to the native
state of the beta3s peptide and a 2D lattice heteropolymer. Moreover CSN are
shown to fall in the general class of complex networks describe d by the
fitness model.Comment: 6 figure
Equilibrium size of large ring molecules
The equilibrium properties of isolated ring molecules were investigated using
an off-lattice model with no excluded volume but with dynamics that preserve
the topological class. Using an efficient set of long range moves, chains of
more than 2000 monomers were studied. Despite the lack of any excluded volume
interaction, the radius of gyration scaled like that of a self avoiding walk,
as had been previously conjectured. However this scaling was only seen for
chains greater than 500 monomers.Comment: 11 pages, 3 eps figures, latex, psfi
- âŠ