Limit shape of the MERW pyramidal process

Prove that, for the suitably scaled maximal-entropy random walk pyramidal process \((P_n)_{n\geq 0}\), the distance between the boundary of \(P_n\) and the explicitly defined symmetric Plancherel-type curve \(\mathcal G=T(\mathcal H)\cup ST(\mathcal H)\) converges to zero almost surely, where \(\mathcal H\) is the graph of \(\Omega(u)=\frac{2}{\pi}[u\arcsin(u)+\sqrt{1-u^2}]\\) on \([-1,1]\), \(T(u,v)=((u+v)/2,(v-u)/2)\), and \(S(x,y)=(-x,y)\).

Background

The paper introduces a Monte Carlo procedure based on Knuth’s algorithm to approximate transition probabilities of maximal-entropy random walks when direct path enumeration is infeasible. The method is applied to a pyramidal growth model, and simulations are performed for pyramids of size 500.

The resulting empirical shapes are compared with a symmetric version of the classical limit shape for Young tableaux under Plancherel measure. On the basis of these numerical experiments and the analogy with Plancherel growth, the paper leaves unresolved whether the scaled random pyramids converge almost surely to the proposed curve.

References

In light of our simulations, and similarly to the Plancherel growth process, we propose the following conjecture: Let $(P_n)_{n \geq 0}$ be the suitably scaled MERW Pyramidal process. The distance between the boundary of ${P_n}$ and $\mathcal{G}$ tends to zero with probability one.

— Maximal entropy random walks and central Markov chains  (2503.08172 - Offret et al., 11 Mar 2025) in Conjecture in Section 6.2, “Numerical Simulations and Conjecture,” immediately after Figure \ref{fig:MERW}