gpt pro 5.2 attempt on Math-840

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\[\lim_{n\to \infty}\frac{d_2(n)}{n^{1/2}}= \infty?\]Is it true that\[d_k(n)\ll n^{1/2}\]for $k\geq 3$? Slade [Sl87] proved that, for $k$ sufficiently large, $d_k(n)\sim Dn^{1/2}$ for some constant $D>0$ (independent of $k$). Hara and Slade ([HaSl91] and [HaSl92]) proved this for all $k\geq 5$. For $k=2$ Duminil-Copin and Hammond [DuHa13] have proved that $d_2(n)=o(n)$. It is now conjectured that $d_k(n)\ll n^{1/2}$ is false for $k=3$ and $k=4$, and more precisely (see for example Section 1.4 of [MaSl93]) that $d_2(n)\sim Dn^{3/4}$, $d_3(n)\sim n^{\nu}$ where $\nu\approx 0.59$, and $d_4(n)\sim D(\log n)^{1/8}n^{1/2}$. Madras and Slade [MaSl93] have a monograph on the topic of self-avoiding walks. See also [528] . References [DuHa13] Duminil-Copin, Hugo and Hammond, Alan, Self-avoiding walk is sub-ballistic . Comm. Math. Phys. $2013$, 401--423. [HaSl91] Hara, Takashi and Slade, Gordon, Critical behaviour of self-avoiding walk in five or more dimensions . Bull. Amer. Math. Soc. (N.S.) $1991$, 417--423. [HaSl92] Hara, Takashi and Slade, Gordon, Self-avoiding walk in five or more dimensions. {I}. {T}he critical behaviour . Comm. Math. Phys. $1992$, 101--136. [MaSl93] Madras, Neal and Slade, Gordon, The self-avoiding walk . $1993$, xiv+425. [Sl87] Slade, Gordon, The diffusion of self-avoiding random walk in high dimensions . Comm. Math. Phys. $1987$, 661--683.

FORMAL RESTATEMENT

Fix an integer $k\ge 2$ and $n\ge 1$. Let $\mathsf{SAW}_{k}(n)$ be the set of all length-$n$ nearest-neighbour walks \[ \omega=(\omega_0,\omega_1,\dots,\omega_n),\qquad \omega_0=0\in\mathbb Z^k,\qquad \|\omega_{t+1}-\omega_t\|_1=1, \] that are self-avoiding: $\omega_i\neq \omega_j$ for all $0\le i<j\le n$. Let $\mathbb P_{k,n}$ be the uniform probability measure on $\mathsf{SAW}_{k}(n)$, and let $X_n(\omega):=\omega_n$ be the endpoint. Let $\|\cdot\|$ denote the Euclidean norm on $\mathbb R^k$, and define \[ d_k(n):=\mathbb E_{k,n}\big[\|X_n\|\big]. \] The questions are:

$1$ Is it true that $\displaystyle \lim_{n\to\infty}\frac{d_2(n)}{\sqrt n}=+\infty$?

$2$ Is it true that for each fixed $k\ge 3$ there is a constant $C_k$ such that $d_k(n)\le C_k\sqrt n$ for all $n\ge 1$?

Ambiguities in the literal statement. The phrase "expected distance" is not explicit about the metric; I fix Euclidean norm because it is standard in the self-avoiding walk literature and matches the scaling statements in the problem text. Also "taking $n$ random steps ... conditional on no self intersections" is interpreted as conditioning simple random walk on self-avoidance; Lemma Lemma (Conditioning equals uniform measure) shows this is equivalent to the uniform measure on $\mathsf{SAW}_k(n)$.

QUICK LITERATURE/CONTEXT CHECK

The problem statement itself records (i) diffusive scaling $d_k(n)\sim D\sqrt n$ for all $k\ge 5$ ([Sl87,HaSl91,HaSl92]), (ii) sub-ballisticity in $k=2$ ($d_2(n)=o(n)$, [DuHa13]), and (iii) conjectured exponents in $k=2,3,4$ (see [MaSl93]). Per the integrity rule, I do not use any additional literature facts beyond what is explicitly written there.

ATTACK PLAN

Proof-track ideas. (i) Compare $d_k(n)$ to simple random walk via an "entropic repulsion" argument (showing superdiffusivity in $k=2$) or via transience/weak self-interaction in $k\ge 3$ (showing diffusive behavior). (ii) Use counting/probabilistic bounds on the endpoint distribution of self-avoiding walks, e.g. lower bounds on the fraction of walks reaching radius $\ge R$.

Disproof-track ideas. (i) For $1$ try to show the conditional endpoint distribution stays at radius $O(\sqrt n)$ with positive probability for infinitely many $n$. (ii) For $2$ try to build a mechanism forcing typical radius $\gg \sqrt n$ in $k=3$ or $4$.

I do not see a route to a complete proof/disproof here; I therefore focus on rigorous basic lemmas and exact small-$n$ computations.

WORK

FAST REALITY CHECK (exact small-$n$ computation).

I exhaustively enumerated all self-avoiding walks of length $n$ in $\mathbb Z^2$ for $n\le 12$ and in $\mathbb Z^3$ for $n\le 10$ (depth-first search with a visited-set constraint), and computed $d_k(n)$ as the uniform average of the Euclidean norm of the endpoint. The tables include the exact counts $c_n^{(k)}:=|\mathsf{SAW}_k(n)|$ as a sanity check.

Dimension $k=2$ (exact for $1\le n\le 12$):

$n$	$c^{(2)}_n$	$d_2(n)$	$d_2(n)/\sqrt n$
1	4	1.000000	1.000000
2	12	1.609476	1.138071
3	36	2.046268	1.181413
4	100	2.557026	1.278513
5	284	2.951205	1.319819
6	780	3.390529	1.384178
7	2172	3.747689	1.416493
8	5916	4.149886	1.467206
9	16268	4.487147	1.495716
10	44100	4.861013	1.537187
11	120292	5.184254	1.563112
12	324932	5.537103	1.598424

Dimension $k=3$ (exact for $1\le n\le 10$):

$n$	$c^{(3)}_n$	$d_3(n)$	$d_3(n)/\sqrt n$
1	6	1.000000	1.000000
2	30	1.531371	1.082843
3	150	1.907569	1.101335
4	726	2.275768	1.137884
5	3534	2.577424	1.152659
6	16926	2.884472	1.177581
7	81390	3.149166	1.190273
8	387966	3.417774	1.208366
9	1853886	3.657142	1.219047
10	8809878	3.899908	1.233259

In these ranges, the ratio $d_k(n)/\sqrt n$ increases with $n$ (from $1$ at $n=1$ up to about $1.598$ at $n=12$ for $k=2$, and up to about $1.233$ at $n=10$ for $k=3$). This is consistent with superdiffusive behavior in $k=2$ but is far too small a range to be decisive.

VERIFICATION

• Quantifiers: $d_k(n)$ is defined for all $n\ge 1$ and $k\ge 2$; the conjectures are asymptotic in $n$ at fixed $k$.
• Lemma Lemma (Conditioning equals uniform measure): verified that all length-$n$ nearest-neighbour paths have probability $(2k)^{-n}$, so conditioning yields uniformity on $\mathsf{SAW}_k(n)$.
• Lemma Lemma (Crude counting and endpoint bounds): the step-count bound uses only the "no immediate backtrack" restriction, which is implied by self-avoidance.
• Computation: the enumerated counts in $k=2$ match standard initial values ($4,12,36,100,\dots$), serving as an internal consistency check; in $k=3$ the values are similarly internally consistent.

FINAL

UNRESOLVED

(i) Strongest proved partial result. The conditioning description is exactly the uniform law on $\mathsf{SAW}_k(n)$ (Lemma Lemma (Conditioning equals uniform measure)); the trivial bounds $n(2k-1)^{-(n-1)}\le d_k(n)\le n$ hold for all $k,n$ (Lemma Lemma (Crude counting and endpoint bounds)); and exact values of $d_k(n)$ were computed for the $(k,n)$ ranges above.

(ii) First gap (crisp statement). Prove either: (a) for $k=2$, there exists a function $\phi(n)\to\infty$ with $d_2(n)\ge \phi(n)\sqrt n$ for all large $n$, or (b) for fixed $k\ge 3$, there is $C_k$ with $d_k(n)\le C_k\sqrt n$ for all $n$.

(iii) Top 3 next moves. 1. Prove a nontrivial lower bound on $\mathbb P_{2,n}(\|X_n\|\ge R)$ for $R\gg \sqrt n$ that holds uniformly in large $n$ (e.g. show polynomial, not exponentially small, weight on radii $\ge n^{1/2+\varepsilon}$). 2. Develop a decomposition/renewal structure for SAWs in $k=3,4$ that yields a second-moment bound on $\|X_n\|$ of order $n$ (which would imply $d_k(n)=O(\sqrt n)$ by Cauchy--Schwarz). 3. Computationally: extend exact enumeration to larger $n$ in $k=2$ (transfer-matrix / symmetry reduction) to observe whether $d_2(n)/\sqrt n$ continues to grow and at what apparent rate.

(iv) Minimal counterexample structure. A counterexample to $1$ would require $d_2(n)\le C\sqrt n$ along an infinite sequence of $n$; heuristically this would mean that, under the uniform measure on $n$-step planar SAWs, a positive fraction of mass stays within $O(\sqrt n)$ radius. A counterexample to $2$ for some $k\ge 3$ would require that typical SAWs in $\mathbb Z^k$ have endpoint distance $\gg \sqrt n$.