Domain Wall Fluctuations of the Six-Vertex Model at the Ice Point (Simulation Results)

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

The total number of $//<![CDATA[ \begin{array}{l}(N+1)\times(N+1)\end{array} //]]>$ ASMs is given by $//<![CDATA[ \begin{array}{l}Z_{N}=\displaystyle\prod\limits_{k=0}^N \frac{(3k+1)!}{(N+k+1)!}\end{array} //]]>$, eg. $//<![CDATA[ \begin{array}{l}Z_0=1\end{array} //]]>$, $//<![CDATA[ \begin{array}{l}Z_1=2\end{array} //]]>$, $//<![CDATA[ \begin{array}{l}Z_2=7\end{array} //]]>$, $//<![CDATA[ \begin{array}{l}Z_3=42\end{array} //]]>$, $//<![CDATA[ \begin{array}{l}Z_3=42\end{array} //]]>$, $//<![CDATA[ \begin{array}{l}Z_4=429\end{array} //]]>$, $//<![CDATA[ \begin{array}{l}Z_5=7436\end{array} //]]>$, $//<![CDATA[ \begin{array}{l}Z_6=218348\end{array} //]]>$.
Below we indicate the emptiness probability matrix $//<![CDATA[ \begin{array}{l}P_N\end{array} //]]>$, which is the sum of all $//<![CDATA[ \begin{array}{l}N\times N\end{array} //]]>$ occupation matrices divided by $//<![CDATA[ \begin{array}{l}Z_N\end{array} //]]>$. for $//<![CDATA[ \begin{array}{l}N=1,\dots,6\end{array} //]]>$:

$//<![CDATA[ \begin{array}{l}P_1=\begin{pmatrix}\frac12\end{pmatrix}\end{array} //]]>$, $//<![CDATA[ \begin{array}{l}P_2=\begin{pmatrix}\frac57&\frac27\\\frac27&0\end{pmatrix}\end{array} //]]>$, $//<![CDATA[ \begin{array}{l}P_3=\displaystyle\frac1{42}\begin{pmatrix}35&21&7\\21&4&0\\7&0&0\end{pmatrix}\end{array} //]]>$, $//<![CDATA[ \begin{array}{l}P_4=\frac1{429}\begin{pmatrix}387 & 282 & 147 & 42 \\ 282 & 102 & 14 & 0 \\ 147 & 14 & 0 & 0 \\ 42 & 0 & 0 & 0 \end{pmatrix}\end{array} //]]>$,

$//<![CDATA[ \begin{array}{l}P_5=\frac1{7436}\begin{pmatrix}7007 & 5720 & 3718 & 1716 & 429 \\  5720 & 2889 & 805 & 84 & 0 \\  3718 & 805 & 49 & 0 & 0 \\  1716 & 84 & 0 & 0 & 0 \\  429 & 0 & 0 & 0 & 0 \end{pmatrix}\end{array} //]]>$, $//<![CDATA[ \begin{array}{l}P_6=\frac1{218348}\begin{pmatrix}210912 & 184884 & 137567 & 80783 & 33463 & 7436 \\  184885 & 115089 & 46785 & 10414 & 859 & 0 \\  137565 & 46788 & 7007 & 294 & 0 & 0 \\  80781 & 10414 & 294 & 0 & 0 & 0 \\  33464 & 859 & 0 & 0 & 0 & 0 \\  7436 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}\end{array} //]]>$

The above exact values have been obtained by guessing the numerators from sufficiently precise Monte-Carlo simulations. By brute force enumeration one might be able to extend this table up to $//<![CDATA[ \begin{array}{l}N=10\end{array} //]]>$. 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 $//<![CDATA[ \begin{array}{l}k=1,\dots,N\end{array} //]]>$.

The following files contain the results from Monte-Carlo simulations with the coupling-from-the-past algorithm, taking advantage of the possibility to choose all local flip probabilities to be $//<![CDATA[ \begin{array}{l}\tfrac12\end{array} //]]>$ using 64 multispin encoding of the gradients of the height matrix for system sizes $//<![CDATA[ \begin{array}{l}N=2^k-2\end{array} //]]>$, $//<![CDATA[ \begin{array}{l}k=2,\dots,11\end{array} //]]>$. In the files the first lines starting with '#' 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.

ASM_occupation_sparse_00002.txt
ASM_occupation_sparse_00006.txt
ASM_occupation_sparse_00014.txt
ASM_occupation_sparse_00030.txt
ASM_occupation_sparse_00062.txt
ASM_occupation_sparse_00126.txt
ASM_occupation_sparse_00254.txt
ASM_occupation_sparse_00510.txt
ASM_occupation_sparse_01022.txt
ASM_occupation_sparse_02046.txt

These files have been produced with the help of the C program "squareice.c" and a simple python script contained in 

squareice.zip

We have also recorded the edge functions  $//<![CDATA[ \begin{array}{l}h_N\end{array} //]]>$for each realization when $//<![CDATA[ \begin{array}{l}N\geq30\end{array} //]]>$. Due to large memory requirement we provide these data only on request (mail to praehofer@ma.tum.de).

Michael Prähofer, 2023-04-17