Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userGreens Function for a Hierarchical Self-Avoiding Walk in Four Dimensions
68
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
This is the second of two papers on the end-to-end distance of a weakly self-repelling walk on a four dimensional hierarchical lattice. It completes the proof that the expected value grows as a constant times sqrt{T} log^{1/8}T (1+O((log log T)/log T)), which is the same law as has been conjectured for self-avoiding walks on the simple cubic lattice Z^4. - Apart from completing the program in the first paper, the main result is that the Greens function is almost equal to the Greens function for the Markov process with no self-repulsion, but at a different value of the killing rate beta which can be accurately calculated when the interaction is small. Furthermore, the Greens function is analytic in beta in a sector in the complex plane with opening angle greater than pi.
rate research
Read More
In [BEI] we introduced a Levy process on a hierarchical lattice which is four dimensional, in the sense that the Greens function for the process equals 1/x^2. If the process is modified so as to be weakly self-repelling, it was shown that at the critical killing rate (mass-squared) beta^c, the Greens function behaves like the free one. - Now we analyze the end-to-end distance of the model and show that its expected value grows as a constant times sqrt{T} log^{1/8}T (1+O((log log T)/log T)), which is the same law as has been conjectured for self-avoiding walks on the simple cubic lattice Z^4. The proof uses inverse Laplace transforms to obtain the end-to-end distance from the Greens function, and requires detailed properties of the Greens function throughout a sector of the complex beta plane. These estimates are derived in a companion paper [math-ph/0205028].
We consider a self-avoiding walk model of polymer adsorption where the adsorbed polymer can be desorbed by the application of a force. In this paper the force is applied normal to the surface at the last vertex of the walk. We prove that the appropriate limiting free energy exists where there is an applied force and a surface potential term, and prove that this free energy is convex in appropriate variables. We then derive an expression for the limiting free energy in terms of the free energy without a force and the free energy with no surface interaction. Finally we show that there is a phase boundary between the adsorbed phase and the desorbed phase in the presence of a force, prove some qualitative properties of this boundary and derive bounds on the location of the boundary.
In this paper we study some classes of second order non-homogeneous nonlinear differential equations allowing a specific representation for nonlinear Greens function. In particular, we show that if the nonlinear term possesses a special multiplicativity property, then its Greens function is represented as the product of the Heaviside function and the general solution of the corresponding homogeneous equations subject to non-homogeneous Cauchy conditions. Hierarchies of specific non-linearities admitting this representation are derived. The nonlinear Greens function solution is numerically justified for the sinh-Gordon and Liouville equations. We also list two open problems leading to a more thorough characterizations of non-linearities admitting the obtained representation for the nonlinear Greens function.
We have explained in detail why the canonical partition function of Interacting Self Avoiding Walk (ISAW), is exactly equivalent to the configurational average of the weights associated with growth walks, such as the Interacting Growth Walk (IGW), if the average is taken over the entire genealogical tree of the walk. In this context, we have shown that it is not always possible to factor the the density of states out of the canonical partition function if the local growth rule is temperature-dependent. We have presented Monte Carlo results for IGWs on a diamond lattice in order to demonstrate that the actual set of IGW configurations available for study is temperature-dependent even though the weighted averages lead to the expected thermodynamic behavior of Interacting Self Avoiding Walk (ISAW).
This article is concerned with self-avoiding walks (SAW) on $mathbb{Z}^{d}$ that are subject to a self-attraction. The attraction, which rewards instances of adjacent parallel edges, introduces difficulties that are not present in ordinary SAW. Ueltschi has shown how to overcome these difficulties for sufficiently regular infinite-range step distributions and weak self-attractions. This article considers the case of bounded step distributions. For weak self-attractions we show that the connective constant exists, and, in $dgeq 5$, carry out a lace expansion analysis to prove the mean-field behaviour of the critical two-point function, hereby addressing a problem posed by den Hollander.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
