Let $d_k(n)$ be the expected distance from the origin after taking $n$ random steps from the origin in ${\mathbb{Z}}^k$ (conditional on no self intersections) - that is, a self-avoiding walk. Is it true that$$\underset{n\to \infty }{\lim }\frac{d_2(n)}{n^{1/2}}=\infty ?$$Is it true that$$d_k(n)\ll n^{1/2}$$for $k\ge 3$?

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Let (\mathrm{SAW}*n) be the uniform measure on $n$-step self‑avoiding walks (\omega=(\omega(0),\dots,\omega(n))) in (\mathbb Z^k) with (\omega(0)=0), and write (X_n=\omega(n)). Your quantity is [ d_k(n)=\mathbb E*{\mathrm{SAW}_n}\big[|X_n|*2\big]. ] Much of the rigorous literature instead studies the **root mean square…

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.