Grothendieck Shenanigans: Permutons from pipe dreams via integrable probability

Abstract

We study random permutations arising from reduced pipe dreams. Our main model is motivated by Grothendieck polynomials with parameter $β=1$ arising in K-theory of the flag variety. The probability weight of a permutation is proportional to the principal specialization (setting all variables to 1) of the corresponding Grothendieck polynomial. By mapping this random permutation to a version of TASEP (Totally Asymmetric Simple Exclusion Process), we describe the limiting permuton and fluctuations around it as the order $n$ of the permutation grows to infinity. The fluctuations are of order $n^{\frac13}$ and have the Tracy-Widom GUE distribution, which places this algebraic (K-theoretic) model into the Kardar-Parisi-Zhang universality class. We also investigate non-reduced pipe dreams and make progress on a recent open problem on the asymptotic number of inversions of the resulting permutation. Inspired by Stanley's question for the maximal value of principal specializations of Schubert polynomials, we resolve the analogous question for $β=1$ Grothendieck polynomials, and provide bounds for general $β$.

Figures (16)

• Figure 1: Left: A pipe dream $D$ of order $6$. Right: A reduction of the pipe dream leading to the permutation $w(D)=241653$. The right image appears in color online. Dashing is added for the printed version and accessibility.
• Figure 2: Left and center: A sample of a Grothendieck random permutation of order $n=2000$ with $p=\frac{4}{5}$ and $p=\frac{1}{2}$, respectively. Right: An average of Grothendieck random permutations with $n=2000$ and $p=\frac{1}{2}$ over $2000$ samples. We take a sum of permutation matrices, and coarse-grain the result into $8\times 8$ blocks. The plot is the heat map of the resulting matrix, which approximates the Grothendieck permuton. The C code for these simulations is available on the arXiv as an ancillary file. An interactive simulation is available at Grothendieck_Simulations_1.
• Figure 3: Simulations of $\mathbf{w}^{(q)}$, from left to right: (1) An average over $2000$ samples with $n=1000$, $p=0.7$, and $q=0.5$. We see curved boundary in the top right, similarly to $q=0$. (2) An average over $2000$ samples with $n=1000$, $p=0.5$, and $q=1$. The random permutation $\mathbf{w}^{(1)}$ is close to the identity (and converges to it, see \ref{['thm:non_reduced_intro']}). (3) An average over $2000$ samples with $n=1000$, $p=0.9985$, and $q=1$. The permutation has a positive mass close to the anti-diagonal. (4) A single sample with $n=1000$, $p=0.9985$, and $q=1$, with the anti-diagonal clearly visible. The latter two simulations suggest a permuton limit as $p=p(n)\to 1$. The C code for these simulations is available on the arXiv as an ancillary file.
• Figure 4: Left: Permutation matrix of a layered permutation $w(b)\in S_{877}$, where the composition is $b=(256, 182, 128, 91, 64, 46, 32, 23, 16, 12, 8, 6, 4, 3, 2, 2, 1, 1)$. Note that $b_i /b_{i+1} \approx 1/\sqrt{2}$. Right: Table of exact values for $3\leqslant k\leqslant 19$ of layered permutations $w(b)$ with $b_i/b_{i+1} \approx 1/\sqrt{2}$. The third column is $f(n)\coloneqq \frac{1}{n^2} \log_2 \Upsilon_{w(b)}(1)$, where $n=\sum_i b_i$.
• Figure 5: Permutation matrix of $w=(2,4,1,6,5,3)$ coming from the pipe dream in \ref{['fig:pipe_dream']}, right (dots indicate 1's). The highlighted rectangle has $H(4,3)=2$ entries.

Theorems & Definitions (80)

• Definition 1.1: Reduction of a pipe dream
• Definition 1.2: Permutation from a pipe dream
• Remark 1.3
• Theorem 1.4: Grothendieck polynomials as sums over pipe dreams
• Theorem 1.5
• Proposition 1.6: \ref{['prop:Grothendieck_permuton_inversions']}
• Remark 1.7
• Definition 1.8: $q$-Reduction of a pipe dream
• Remark 1.9
• Remark 1.10