Combinatorics and Graph Theory

On a Function Modeling an L-Step Self Avoiding Walk

Authors: Theophilus Agama
Comments: 14 Pages. A few technicalities resolved regarding the scale of compression and the inequality in the notion of points contained in a compression ball has been made strict. This is because the case of equality is treated separately as admissible points.

We introduce and study the needle \begin{align}(\Gamma_{\vec{a}_1} \circ \mathbb{V}_m)\circ \cdots \circ (\Gamma_{\vec{a}_{\frac{l}{2}}}\circ \mathbb{V}_m):\mathbb{R}^n\longrightarrow \mathbb{R}^n.\nonumber \end{align} By exploiting the geometry of compression, we prove that this function is a function modeling an $l$-step self avoiding walk for $l\in \mathbb{N}$. We show that the total length of the $l$-step self-avoiding walk modeled by this function is of the order \begin{align}\ll \frac{l}{2}\sqrt{n}\bigg(\mathrm{\max}\{\mathrm{sup}(x_{j_k})\}_{\substack{1\leq j\leq \frac{l}{2}\\1\leq k\leq n}}+\mathrm{\max}\{\mathrm{sup}(a_{j_k})\}_{\substack{1\leq j\leq \frac{l}{2}\\1\leq k\leq n}}\bigg)\nonumber \end{align}and at least \begin{align}\gg \frac{l}{2}\sqrt{n}\bigg(\mathrm{\min}\{\mathrm{Inf}(x_{j_k})\}_{\substack{1\leq j\leq \frac{l}{2}\\1\leq k\leq n}}+\mathrm{\min}\{\mathrm{Inf}(a_{j_k})\}_{\substack{1\leq j\leq \frac{l}{2}\\1\leq k\leq n}}\bigg).\nonumber \end{align}
Category: Combinatorics and Graph Theory