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STEM student's favourite source of amusement posted a comic titled "Unsolved Math Problems" one of which looks like something that could actually be tackled.

If I walk randomly on a grid, never visiting any square twice, placing a marble every N steps, on average how many marbles will be in the longest line after NK steps?

Original comic (with ilustration of the problem)

I know there are some results in form of power laws (critical exponents) like the 0.587597... constant for SARW in 3D.

Can we say anything interesting about the process described in the comic?

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    $\begingroup$ I'm no expert on random walks so I probably miss something easy, but why is this process well-defined? I might corner myself in? $\endgroup$
    – Keba
    Commented Oct 18, 2021 at 9:00
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    $\begingroup$ I feel like we are getting nerd sniped again... $\endgroup$
    – Wojowu
    Commented Oct 18, 2021 at 9:20
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    $\begingroup$ The self-avoiding random walk with $N$ steps is usually simply defined as the uniform measure on the set of all self-avoiding walks of $N$ steps. It doesn't have any kind of Markov property, but is conjectured in $2D$ to converge to an SLE curve with self-similarity exponent $3/4$. Rigorously, almost nothing non-trivial is known about its large-$N$ behaviour. $\endgroup$
    – Martin Hairer
    Commented Oct 18, 2021 at 9:21
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    $\begingroup$ What I very much want to know is: is the Euler field manifold hypergroup isomorphic to a Gödel-Klein meta-algebreic $\epsilon<0$ quasimonoid connection under Sondheim calculus? (Also, what in Apollo's name is going on with this curve?) $\endgroup$
    – Gro-Tsen
    Commented Oct 18, 2021 at 12:35
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    $\begingroup$ @SamHopkins A horizontal or vertical line through the origin will have $\sqrt{K/N}$ marbles on average and I would wildly guess that the correct answer is within $(KN)^\epsilon$ of this. It looks conceivable to prove this by bounding for each line the expectation of the $\ell$th power of the number of marbles on the line and summing over all lines passing through at least a few points within a distance $NK$ of the origin. $\endgroup$
    – Will Sawin
    Commented Oct 18, 2021 at 16:35

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