#### Grothendieck Shenanigans: Permutons From Pipe Dreams via Integrable Probability

IPAM - August 2024

If Alice and Bob each take a walk, a random one, what is the chance they will meet? If n

people walk randomly in Manhattan from the Upper East Side going south or west, how often

will two of them meet until they reach the Hudson? If they do not want to see each other more

than once, in what relative order will they most likely arrive at the Hudson shore? If we make

a rhombus like an Aztec diamond from 2 × 1 dominoes, what would it most likely look like? If

water molecules arrange themselves on a square grid, what angles between the hydrogen atoms

will be formed near the boundaries? And if ribosomes are transcribing the mRNA, how do they

hop between the sites (codons)?

The answers to the above questions are all intertwined, bridging algebra with probability,

integrability with stochasticity, random tilings with random permutations, as revealed in the

recent work1[MPPY24].

people walk randomly in Manhattan from the Upper East Side going south or west, how often

will two of them meet until they reach the Hudson? If they do not want to see each other more

than once, in what relative order will they most likely arrive at the Hudson shore? If we make

a rhombus like an Aztec diamond from 2 × 1 dominoes, what would it most likely look like? If

water molecules arrange themselves on a square grid, what angles between the hydrogen atoms

will be formed near the boundaries? And if ribosomes are transcribing the mRNA, how do they

hop between the sites (codons)?

The answers to the above questions are all intertwined, bridging algebra with probability,

integrability with stochasticity, random tilings with random permutations, as revealed in the

recent work1[MPPY24].

**Figure 1**.

**(a)**a pipe dream of the permutation w = 241653,

**(b)**a permuton in the shape of

an ice cream,

**(c)**an example of a layered permutation,

**(d)**illustration of the corresponding

configuration in the 6-vertex model lying inside the frozen region.

The story started with a problem of Richard Stanley from 2017 – what is the (asymptotically)

maximal value of a principal specialization of a

*Schubert polynomial*. Schubert polynomials \(Sw\) in

n variables, indexed by permutations \(w\in S_n\), present cohomology classes of the flag variety and

are central objects of study at the intersection of Algebraic Combinatorics and Algebraic Geometry.

However, the problem can be stated as a simple tiling model: let us tile the triangle in the

square grid given by i+j ≤ n+1, i, j = 1, . . . , n, with crosses or elbows (see Figure 1

**(a)**), which

form a network of lines aka “pipe dreams”. Moreover, impose a special

*noncrossing condition*that

no two pipes can cross more than once. The configuration in Figure 1

**(a)**satisfies the noncrossing

condition, and counting those is a statistical mechanics problem involving long-range interactions.

It is notoriously difficult to study, we do not even know that the number of such configurations

asymptotically grows as \(2^{c n^2}\) for some c. Curiously, the bounds on the upper and lower limits

we have so far give c ∈ [0.29, 0.37]. The lower bound comes from a specific construction with

so-called

*layered permutations*, where explicit product formulas give the count. The upper bound

follows by embedding noncrossing pipe configurations into the so-called “reduced bumpless pipe

dreams”, a particular subset of the Alternating Sign Matrices (known in statistical mechanics as

square ice with domain-wall boundary conditions and uniform weights; it is a particular case of

the Six-Vertex model).

In our quest to resolve Stanley’s problem, we explored the generalization of Schubert polynomials,

the Grothendieck polynomials \(\mathfrak{G}_w^\beta\), which capture the K-theory of the flag variety. They

can also be studied as partition functions (generating functions) of the pipe dreams described

above, except that each double and further intersection is “penalized” by the factors \(\beta\). When

\(\beta\) = 0, we recover the Schubert polynomials; when \(\beta\) = 1, the model corresponds to pipe dreams

that bounce off each other after the first intersection. For which permutations is \(\mathfrak{G}_w^\beta(1,\ldots,1)\)

maximal? What is the asymptotic behavior of the number of \(\beta\)-weighted pipe dreams?

When \(\beta\) = 1, finding the asymptotically maximal value is easy by counting the number of

tilings (pipe dreams) without constraints. There are \(2^{\binom{n}{2}}\) such tilings (two choices of each of

the \(\binom{n}{2}\) tiles) distributed among \(n!\sim e^{n \log(n) }\) permutations. Thus, the maximal value is still

around \(2^{n^2/2 -o(n^2) }\). But what is the maximal permutation? What does a typical permutation

look like? To answer the second question, we realized that the trajectories (pipes) are in bijection

with histories of hopping particles in the

*Totally Asymmetric Simple Exclusion Process*(TASEP),

initially developed in 1968 to understand RNA translation:

Analyzing the behavior of hopping particles utilizes Integrable Probability. That is, we express

certain observables of the particle system via

*Schur functions*(a central object in Algebraic

Combinatorics), then turn to analysis and probability to extract asymptotics of the observables

via contour integrals. This produces a very interesting permuton (a continuous version of a

permutation matrix) in the shape of an... ice cream cone with one scoop. See Figure 1

**(b)**.

So what about layered permutations, which give large Schubert polynomials? In the Grothendieck

model, they are also close to maximal (see Figure 1

**(c)**). In this case, there are no product

formulas, so how do we compute the values and maximize them? The Grothendieck model can

also be related to the Six-Vertex model. For \(\beta\) = 1, it corresponds to the 2-enumeration of

Alternating Sign Matrices, which are in 2-1 correspondence with domino tilings of the Aztec

diamond. A layered permutation has its last part reversed, making it a frozen region of the

2-ASM/Aztec diamond. The existence of the frozen boundary of the Aztec diamond was one of

the early feats at the intersection of combinatorics with statistical mechanics from 1996. So when

our layered permutation has the last layer close to the tangent to the arctic circle, the number of

configurations is still asymptotically maximal, that is, \(2^{n^2/2}\). See Figure 1

**(d)**.

**References:**

[MPPY24] Alejandro H. Morales, Greta Panova, Leonid Petrov, and Damir Yeliussizov, Grothendieck Shenanigans:

Permutons from Pipe Dreams via Integrable Probability, arXiv preprint (2024). arXiv:2407.21653

[math.PR].