Signs of Kravchuk matrix asymptotically produce a large circular region with hyperbolic sinks. Why?

Question

The Kravchuk matrix of dimension $n+1$ is such that its entries satisfy $$K_{i,j}^{(n+1)}=[x^i](1+x)^{n-j}(1-x)^j\quad\forall0\le i,j\le n.$$ It enjoys properties such as involution and has various expressions involving combinatorial sums.

Figure. Signs of the transpose of $K_{i,j}^{(1000)}$: Positive entries filled orange and negative entries filled white.

We observe that as odd $n\to\infty$, there is a large circular region, inside of which lie numerous hyperbolic sinks of various sizes.

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A few weeks ago I posted this observation on MSE, and recently got a detailed answer explaining the circular shape. Briefly, using the theory of $_2F_1$-hypergeometric functions, we find that the circular parametrisation $$(1-2\alpha)^2+(1-2\beta)^2=1$$ with $\alpha=i/n$ and $\beta=j/n$ causes equation (2.5) of Paris (2013) to be reduced to the boundary line $z_\ast^+=-1$.

However, the question of why the hyperbolic sinks appear, and their fractal-like appearances remain unanswered. It also appears that the largest sinks lie on the line $i=j$.

Is there an explanation for this behaviour?

P.S. I found only one publication with a plot of the circular shape (Figure 7). Note that the sinks do not appear there since $n$ is too small.

Answer 1

In Section 2 of our 1996 paper "Local statistics for random domino tilings of the Aztec diamond" (https://arxiv.org/abs/math/0008243), Jim Propp, Noam Elkies, and I analyzed this behavior inside the circle, and the same techniques work outside. The short answer is that you can write the coefficient of $x^i$ in $(1+x)^{n-j} (1-x)^j$ as a contour integral and compute asymptotics using the saddle point method. There are two critical points, and a phase mismatch between them give the Moiré effect shown in the picture. What we analyze in Proposition 4 in the paper is not actually your $K_{i,j}$, but rather $K_{i,j}K_{j,i}$. However, $K_{i,j}$ and $K_{j,i}$ differ by some simple factor (involving factorials), so this is essentially the same as analyzing $K_{i,j}$ in isolation.

I don't know of a simple description of the Moiré effect. It amounts to the factor of $\cos^2 \Phi(\ell,m;n)$ in Proposition 4 (but for your case of just $K_{i,j}$, the cosine factor wouldn't be squared, so it would change sign). We don't write out what $\Phi$ is explicitly in our paper, but it's determined by the computations in the proof, and we give a few properties in lemmas later in the section. For our purposes, all we needed to know about this $\cos^2 \Phi$ factor was that it averaged to $1/2$ in certain sums, and that could be proved with some exponential sum estimates.

My memory of these formulas is that they were rather complicated and not so easy to interpret nicely. However, there may well be a conceptually clearer description, which would be really interesting. It's not something we explored carefully, since for us the Moiré effect was more of an obstacle than an object of study itself.

Answer 2

This is not an answer, but too long for a comment. It related to the PS from the OP.

In arXiv:1702.05474, Keesman and I used Monte--Carlo simulations to study a special case of a model from classical statistical mechanics that is equivalent to $x$-enumerations alternating-sign matrices. As a special case it includes domino tilings of the Aztec diamond, which are known to be closely related to Krawtchouk matrices, cf Cohn's answer.

Your pictures remind me of Fig 11 in our paper (reproduced below), see also https://mathoverflow.net/a/178190 -- where this similarity was noticed by Ilia Smilga in a comment. Our figure shows the difference of the ('thermal', i.e. wrt Gibbs measure) average of per-vertex densities for two types of vertex configurations, see the end of Sect IV-D and end of Sect V of our paper for details. In the language of ASM this is the difference of expectation values of occurrences of $+1$ and $-1$ per entry of the matrix.

On the one hand, unlike your picture, our case is symmetric (maybe up to reversing colours) in horizonal and vertical reflections, which is why we only show a quarter of the full domain in our figure. On the other hand, like you, we observe bands and saddles that look very similar to your pictures. Moreover, these features are not just a Moiré effect: if you zoom in, the pixels that you see do really represent individual vertices (ASM entries) -- but see the comment of Szczepan Hołyszewski at https://mathoverflow.net/a/178190.

I hope this is of some use: I don't know if or how it is precisely related, but our figure looks similar enough, and is close enough in terms of topic, that there might be some connection.