In this paper we present a new and flexible method to show that, in one dimension, various self-repellent random walks converge
to self-repellent Brownian motion in the limit of weak interaction after appropriate space-time scaling. Our method is based
on cutting the path into pieces of an appropriately scaled length, controlling the interaction between the different pieces,
and applying an invariance principle to the single pieces. In this way, we show that the self-repellent random walk large
deviation rate function for the empirical drift of the path converges to the self-repellent Brownian motion large deviation
rate function after appropriate scaling with the interaction parameters. The method is considerably simpler than the approach
followed in our earlier work, which was based on functional analytic arguments applied to variational representations and
only worked in a very limited number of situations.
We consider two examples of a weak interaction limit: (1) vanishing self-repellence, (2) diverging step variance. In example
(1), we recover our earlier scaling results for simple random walk with vanishing self-repellence and show how these can be
extended to random walk with steps that have zero mean and a finite exponential moment. Moreover, we show that these scaling
results are stable against adding self-attraction, provided the self-repellence dominates. In example (2), we prove a conjecture
by Aldous for the scaling of self-avoiding walk with diverging step variance. Moreover, we consider self-avoiding walk on
a two-dimensional horizontal strip such that the steps in the vertical direction are uniform over the width of the strip and
find the scaling as the width tends to infinity.
Received: 6 March 2002 / Revised version: 11 October 2002 / Published online: 21 February 2003
Mathematics Subject Classification (2000): 60F05, 60F10, 60J55, 82D60
Key words or phrases: Self-repellent random walk and Brownian motion – Invariance principles – Large deviations – Scaling limits – Universality 相似文献
We investigate the scaling of the largest critical percolation cluster on a large d-dimensional torus, for nearest-neighbor percolation in sufficiently high dimensions, or when d > 6 for sufficiently spread-out percolation. We use a relatively simple coupling argument to show that this largest critical
cluster is, with high probability, bounded above by a large constant times V2/3 and below by a small constant times , where V is the volume of the torus. We also give a simple criterion in terms of the subcritical percolation two-point function on
under which the lower bound can be improved to small constant times , i.e. we prove random graph asymptotics for the largest critical cluster on the high-dimensional torus. This establishes a conjecture by [1], apart from logarithmic
corrections. We discuss implications of these results on the dependence on boundary conditions for high-dimensional percolation.
Our method is crucially based on the results in [11, 12], where the scaling was proved subject to the assumption that a suitably defined critical window contains the percolation threshold on . We also strongly rely on mean-field results for percolation on proved in [17–20]. 相似文献
Self-assembly of polystyrene sulfonate and modified cowpea chlorotic mottle virus protein yields monodisperse icosahedral nanoparticles of 16 nm size. 相似文献
In the present paper a simple and efficient alternate friction model is presented to simulate stick-slip vibrations. The alternate friction model consists of a set of ordinary non-stiff differential equations and has the advantage that the system can be integrated with any standard ODE-solver. Comparison with a smoothing method reveals that the alternate friction model is more efficient from a computational point of view. A shooting method for calculating limit cycles, based on the alternate friction model, is presented. Time-dependent static friction is studied as well as application in a system with 2-DOF. 相似文献
In this paper, a random graph process {G(t)}t≥1 is studied and its degree sequence is analyzed. Let {Wt}t≥1 be an i.i.d. sequence. The graph process is defined so that, at each integer time t, a new vertex with Wt edges attached to it, is added to the graph. The new edges added at time t are then preferentially connected to older vertices, i.e., conditionally on G(t-1), the probability that a given edge of vertex t is connected to vertex i is proportional to di(t-1)+δ, where di(t-1) is the degree of vertex i at time t-1, independently of the other edges. The main result is that the asymptotical degree sequence for this process is a power
law with exponent τ=min{τW,τP}, where τW is the power-law exponent of the initial degrees {Wt}t≥1 and τP the exponent predicted by pure preferential attachment. This result extends previous work by Cooper and Frieze. 相似文献
The lace expansion is a powerful tool for analysing the critical behaviour of self-avoiding walks and percolation. It gives
rise to a recursion relation which we abstract and study using an adaptation of the inductive method introduced by den Hollander
and the authors. We give conditions under which the solution to the recursion relation behaves as a Gaussian, both in Fourier
space and in terms of a local central limit theorem. These conditions are shown elsewhere to hold for sufficiently spread-out
models of networks of self-avoiding walks in dimensions d > 4, and for sufficiently spread-out models of critical oriented percolation in dimensions d + 1 > 5, providing a unified approach and an essential ingredient for a detailed analysis of the branching behaviour of these
models.
Received: 13 September 2000 / Revised version: 16 May 2001 / Published online: 20 December 2001 相似文献
The viscosity-temperature relation is determined for the water models SPC/E, TIP4P, TIP4P/Ew, and TIP4P/2005 by considering Poiseuille flow inside a nano-channel using molecular dynamics. The viscosity is determined by fitting the resulting velocity profile (away from the walls) to the continuum solution for a Newtonian fluid and then compared to experimental values. The results show that the TIP4P/2005 model gives the best prediction of the viscosity for the complete range of temperatures for liquid water, and thus it is the preferred water model of these considered here for simulations where the magnitude of viscosity is crucial. On the other hand, with the TIP4P model, the viscosity is severely underpredicted, and overall the model performed worst, whereas the SPC/E and TIP4P/Ew models perform moderately. 相似文献
Hierarchical self‐assembly of transient composite hydrogels is demonstrated through a two‐step, orthogonal strategy using nanoparticle tectons interconnected through metal–ligand coordination complexes. The resulting materials are highly tunable with moduli and viscosities spanning many orders of magnitude, and show promising self‐healing properties, while maintaining complete optical transparency.