Arctic Circle Law for the Aztec Diamond. Monte-Carlo Simulations

Supplementary material for the article "Domain wall fluctuations of the six-vertex model at the ice point"
by Michael Prähofer and Herbert Spohn (2023), arxiv.org/abs/2305.09502

Domino tilings of the Aztec diamond can be mapped to the six-vertex model of $//<![CDATA[ \begin{array}{l}(N{+}1)\times(N{+}1)\end{array} //]]>$ tiles with domain wall boundary condition (DWBC). It can be further mapped configurationwise to an integer-valued height function $//<![CDATA[ \begin{array}{l}\Phi:[0,N{+}2]^2\to\mathbb{Z}\end{array} //]]>$  which is constant on $//<![CDATA[ \begin{array}{l}\mathbb{\R}^2\setminus(\mathbb{R}{\times}\mathbb{Z}\,\cup\,\mathbb{Z}{\times}\mathbb{R})\end{array} //]]>$ and has fixed values in the outside frame of tiles as  $//<![CDATA[ \begin{array}{l}\Phi(x,y)=0\end{array} //]]>$ for  $//<![CDATA[ \begin{array}{l}x\in[0,1)\end{array} //]]>$or $//<![CDATA[ \begin{array}{l}y\in[0,1)\end{array} //]]>$ and $//<![CDATA[ \begin{array}{l}\Phi(x,y)=\lfloor x\rfloor\end{array} //]]>$ for $//<![CDATA[ \begin{array}{l}y\in(N+1,N+2]\end{array} //]]>$, $//<![CDATA[ \begin{array}{l}\Phi(x,y)=\lfloor y\rfloor\end{array} //]]>$ for $//<![CDATA[ \begin{array}{l}x\in(N+1,N+2]\end{array} //]]>$. On this page we determined for some $//<![CDATA[ \begin{array}{l}N\in\mathbb{N}\end{array} //]]>$ the probability $//<![CDATA[ \begin{array}{l}(P_{N})_{k,l}\end{array} //]]>$,that $//<![CDATA[ \begin{array}{l}\Phi(k+\tfrac12,l+\tfrac12)=0\end{array} //]]>$, $//<![CDATA[ \begin{array}{l}k,l=1,\dots,N\end{array} //]]>$, which sometimes is called the emptiness probability matrix.
For example $//<![CDATA[ \begin{array}{l}P_1=\begin{pmatrix}\tfrac12\end{pmatrix}\in\R^{1\times1}\end{array} //]]>$, $//<![CDATA[ \begin{array}{l}P_2=\begin{pmatrix}\tfrac34&\tfrac14\\\tfrac14&0\end{pmatrix}\in\R^{2\times2}\end{array} //]]>$, $//<![CDATA[ \begin{array}{l}P_3=\frac1{16}\begin{pmatrix}14&8&2\\8&1&0\\2&0&0\end{pmatrix}\in\R^{3\times3}\end{array} //]]>$, $//<![CDATA[ \begin{array}{l}P_4=\frac1{64}\begin{pmatrix}60&44&20&4\\44&13&1&0\\20&1&0&0\\4&0&0&0\end{pmatrix}\in\R^{4\times4}\end{array} //]]>$.
Note that due to symmetries of the model we have $//<![CDATA[ \begin{array}{l}(P_N)_{1,k}+(P_N)_{1,N+1-k}=1=(P_N)_{k,1}+(P_N)_{N+1-k,1}\end{array} //]]>$ for $//<![CDATA[ \begin{array}{l}k=1,\dots,N\end{array} //]]>$.

In general, the total number of domino tilings of an Aztec diamond of generation $//<![CDATA[ \begin{array}{l}N\end{array} //]]>$ is $//<![CDATA[ \begin{array}{l}2^{N(N+1)/2}\end{array} //]]>$. Nevertheless $//<![CDATA[ \begin{array}{l}d_N=2^{\lfloor(N{+}1)/2\rfloor\lceil(N{+}1)/2\rceil}\end{array} //]]>$ is the smallest denominator, which renders  $//<![CDATA[ \begin{array}{l}d_N P_N\end{array} //]]>$integer-valued. The following file contains in each line  $//<![CDATA[ \begin{array}{l}\{N,\ d_N,\ d_NP_N\}\end{array} //]]>$ for $//<![CDATA[ \begin{array}{l}N=1,\dots,32\end{array} //]]>$, with the matrix $//<![CDATA[ \begin{array}{l}d_NP_N\in\mathbb{N}_0^{N\times N}\end{array} //]]>$ written in the format $//<![CDATA[ \begin{array}{l}\{\{\dots,\dots\},\dots,\{\dots,\dots\}\}\end{array} //]]>$:

aztec_occupation_exact_1_32.txt (265K)

The exact values were obtained by numerically evaluating the corresponding infinite dimensional Fredholm determinants with Meixner kernel for the Aztec diamond to a high enough precision, to accurately guess the numerators in $//<![CDATA[ \begin{array}{l}P_N\end{array} //]]>$.

The following files contain the results from Monte-Carlo simulations, the first lines starting with '#' and containing header information like system size and number of samples. Each data line contains the numbers $//<![CDATA[ \begin{array}{l}k, l,(P_N)_{k,l}\end{array} //]]>$ separated by spaces, with the probability $//<![CDATA[ \begin{array}{l}(P_N)_{k,l}\in(0,1)\end{array} //]]>$
guessed from the Monte-Carlo results. Only entries of $//<![CDATA[ \begin{array}{l}P_N\end{array} //]]>$which are different from $//<![CDATA[ \begin{array}{l}0\end{array} //]]>$ and $//<![CDATA[ \begin{array}{l}1\end{array} //]]>$ are recorded.  

aztec_occupation_sparse_00004.txt
aztec_occupation_sparse_00008.txt
aztec_occupation_sparse_00016.txt
aztec_occupation_sparse_00032.txt
aztec_occupation_sparse_00064.txt
aztec_occupation_sparse_00128.txt
aztec_occupation_sparse_00256.txt
aztec_occupation_sparse_00512.txt (1.8M)
aztec_occupation_sparse_01024.txt (4.6M)
aztec_occupation_sparse_02048.txt (12M)
aztec_occupation_sparse_04096.txt (26M)
aztec_occupation_sparse_08192.txt (65M)
aztec_occupation_sparse_16384.txt (124M)

They have been produced with the help of the C program "ArcticCircle.c" and a primitive python script contained in

ArcticCircle.zip

Michael Prähofer, 17.04.2023