On the range of a constrained random walk

1988 ◽  
Vol 25 (3) ◽  
pp. 451-463 ◽  
Author(s):  
W. Th. F. Den Hollander ◽  
G. H. Weiss
Keyword(s):  
Random Walk ◽  
Discrete Time ◽  
Large Time ◽  
Statistical Properties ◽  
Time Limit ◽  
The Mean ◽  
Large Time Limit ◽  
Lattice Random Walk

We study statistical properties of the range (= number of distinct sites visited) of a lattice random walk in discrete time constrained to visit a given site at a given time. In particular, we calculate the mean and obtain a bound on the variance of the range in the large time limit. The results are applied to a problem involving an unconstrained random walk in the presence of randomly distributed traps. A key role is played by the associated random walk that is obtained from the original random walk via a Cramer transform.

On the range of a constrained random walk

1988 ◽  
Vol 25 (03) ◽  
pp. 451-463
Author(s):  
W. Th. F. Den Hollander ◽  
G. H. Weiss
Keyword(s):  
Random Walk ◽  
Discrete Time ◽  
Large Time ◽  
Statistical Properties ◽  
Time Limit ◽  
The Mean ◽  
Large Time Limit ◽  
Lattice Random Walk

We study statistical properties of the range (= number of distinct sites visited) of a lattice random walk in discrete time constrained to visit a given site at a given time. In particular, we calculate the mean and obtain a bound on the variance of the range in the large time limit. The results are applied to a problem involving an unconstrained random walk in the presence of randomly distributed traps. A key role is played by the associated random walk that is obtained from the original random walk via a Cramer transform.

scholarly journals ON THE STATISTICAL PROPERTIES OF THE LARGE TIME ZERO TEMPERATURE DYNAMICS OF THE SK MODEL

Fractals ◽  
2003 ◽  
Vol 11 (supp01) ◽  
pp. 161-171 ◽  
Author(s):  
GIORGIO PARISI
Keyword(s):  
Large Time ◽  
Weak Sense ◽  
Statistical Properties ◽  
Time Limit ◽  
Zero Temperature ◽  
Replica Symmetry ◽  
Equilibrium Dynamics ◽  
Temperature Dynamics ◽  
Kirkpatrick Model ◽  
Large Time Limit

Here we study the zero temperature dynamics of the Sherrington Kirkpatrick model and we investigate the statistical properties of the configurations that are obtained in the large time limit. We find that the replica symmetry is broken (in a weak sense). We also present some general considerations on the synchronic approach to the off-equilibrium dynamics, that has motivated the present study.

On the absorption probabilities and mean time for absorption for discrete Markov chains

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nikolaos Halidias
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Markov Chains ◽  
Generating Function ◽  
Discrete Time ◽  
Probability Generating Function ◽  
The Mean ◽  
Mean Time ◽  
Absorption Probabilities

Abstract In this note we study the probability and the mean time for absorption for discrete time Markov chains. In particular, we are interested in estimating the mean time for absorption when absorption is not certain and connect it with some other known results. Computing a suitable probability generating function, we are able to estimate the mean time for absorption when absorption is not certain giving some applications concerning the random walk. Furthermore, we investigate the probability for a Markov chain to reach a set A before reach B generalizing this result for a sequence of sets A 1 , A 2 , … , A k {A_{1},A_{2},\dots,A_{k}} .

Amplitudes of Composition Fluctuations in the Simplest Chemical Reaction Equilibrium

2004 ◽  
Vol 218 (9) ◽  
pp. 1033-1040 ◽  
Author(s):  
M. Šolc ◽  
J. Hostomský
Keyword(s):  
Random Walk ◽  
Discrete Time ◽  
Continuous Time ◽  
Time Scale ◽  
Numerical Study ◽  
Mean Amplitude ◽  
Time Interval ◽  
The Mean ◽  
Composition Fluctuations ◽  
Number Of Particles

AbstractWe present a numerical study of equilibrium composition fluctuations in a system where the reaction X1 ⇔ X2 having the equilibrium constant equal to 1 takes place. The total number of reacting particles is N. On a discrete time scale, the amplitude of a fluctuation having the lifetime 2r reaction events is defined as the difference between the number of particles X1 in the microstate most distant from the microstate N/2 visited at least once during the fluctuation lifetime, and the equilibrium number of particles X1, N/2. On the discrete time scale, the mean value of this amplitude, m̅(r̅), is calculated in the random walk approximation. On a continuous time scale, the average amplitude of fluctuations chosen randomly and regardless of their lifetime from an ensemble of fluctuations occurring within the time interval (0,z), z → ∞, tends with increasing N to ~1.243 N0.25. Introducing a fraction of fluctuation lifetime during which the composition of the system spends below the mean amplitude m̅(r̅), we obtain a value of the mean amplitude of equilibrium fluctuations on the continuous time scale equal to ~1.19√N. The results suggest that using the random walk value m̅(r̅) and taking into account a) the exponential density of fluctuations lifetimes and b) the fact that the time sequence of reaction events represents the Poisson process, we obtain values of fluctuations amplitudes which differ only slightly from those derived for the Ehrenfest model.

scholarly journals Large-time limit of the quantum Zeno effect

2017 ◽  
Vol 58 (3) ◽  
pp. 032103 ◽  
Author(s):  
Paolo Facchi ◽  
Marilena Ligabò
Keyword(s):  
Large Time ◽  
Time Limit ◽  
Quantum Zeno Effect ◽  
Zeno Effect ◽  
Large Time Limit

scholarly journals THE SCHRÖDINGER REPRESENTATION FOR FERMIONS AND A LOCAL EXPANSION OF THE SCHWINGER MODEL

2000 ◽  
Vol 15 (03) ◽  
pp. 429-447 ◽  
Author(s):  
DAVID NOLLAND ◽  
PAUL MANSFIELD
Keyword(s):  
Large Time ◽  
Higher Dimensions ◽  
Numerical Approach ◽  
Time Limit ◽  
Algebraic Equations ◽  
Schwinger Model ◽  
Local Expansion ◽  
Schrödinger Representation ◽  
Functional Representation ◽  
Large Time Limit

We discuss the functional representation of fermions, and obtain exact expressions for wave-functionals of the Schwinger model. Known features of the model such as bosonization and the vacuum angle arise naturally. Contrary to expectations, the vacuum wave-functional does not simplify at large distances, but it may be reconstructed as a large time limit of the Schrödinger functional, which has an expansion in local terms. The functional Schrödinger equation reduces to a set of algebraic equations for the coefficients of these terms. These ideas generalize to a numerical approach to QCD in higher dimensions.

scholarly journals The Large Time Limit of the Evolution Kernel

1990 ◽  
Vol 83 (2) ◽  
pp. 305-317 ◽  
Author(s):  
H. Higurashi ◽  
R. Fukuda
Keyword(s):  
Large Time ◽  
Time Limit ◽  
Evolution Kernel ◽  
Large Time Limit

scholarly journals Large time limit and local $L^2$-index theorems for families

2015 ◽  
Vol 9 (2) ◽  
pp. 621-664 ◽  
Author(s):  
Sara Azzali ◽  
Sebastian Goette ◽  
Thomas Schick
Keyword(s):  
Large Time ◽  
Time Limit ◽  
Index Theorems ◽  
Large Time Limit

scholarly journals The effect of a non-zero Lagrangian time scale on bounded shear dispersion

2011 ◽  
Vol 691 ◽  
pp. 69-94 ◽  
Author(s):  
Matthew S. Spydell ◽  
Falk Feddersen
Keyword(s):  
Velocity Field ◽  
Time Dependence ◽  
Time Scale ◽  
Large Time ◽  
Time Limit ◽  
Flow Shear ◽  
Shear Dispersion ◽  
Lagrangian Time Scale ◽  
Large Time Limit ◽  
Diffusive Time

AbstractPrevious studies of shear dispersion in bounded velocity fields have assumed random velocities with zero Lagrangian time scale (i.e. velocities are$\delta $-function correlated in time). However, many turbulent (geophysical and engineering) flows with mean velocity shear exist where the Lagrangian time scale is non-zero. Here, the longitudinal (along-flow) shear-induced diffusivity in a two-dimensional bounded velocity field is derived for random velocities with non-zero Lagrangian time scale${\tau }_{L} $. A non-zero${\tau }_{L} $results in two-time transverse (across-flow) displacements that are correlated even for large (relative to the diffusive time scale${\tau }_{D} $) times. The longitudinal (along-flow) shear-induced diffusivity${D}_{S} $is derived, accurate for all${\tau }_{L} $, using a Lagrangian method where the velocity field is periodically extended to infinity so that unbounded transverse particle spreading statistics can be used to determine${D}_{S} $. The non-dimensionalized${D}_{S} $depends on time and two parameters: the ratio of Lagrangian to diffusive time scales${\tau }_{L} / {\tau }_{D} $and the release location. Using a parabolic velocity profile, these dependencies are explored numerically and through asymptotic analysis. The large-time${D}_{S} $is enhanced relative to the classic Taylor diffusivity, and this enhancement increases with$ \sqrt{{\tau }_{L} } $. At moderate${\tau }_{L} / {\tau }_{D} = 0. 1$this enhancement is approximately a factor of 3. For classic shear dispersion with${\tau }_{L} = 0$, the diffusive time scale${\tau }_{D} $determines the time dependence and large-time limit of the shear-induced diffusivity. In contrast, for sufficiently large${\tau }_{L} $, a shear time scale${\tau }_{S} = \mathop{ ({\tau }_{L} {\tau }_{D} )}\nolimits ^{1/ 2} $, anticipated by a simple analysis of the particle’s domain-crossing time, determines both the${D}_{S} $time dependence and the large-time limit. In addition, the scalings for turbulent shear dispersion are recovered from the large-time${D}_{S} $using properties of wall-bounded turbulence.

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