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How Yang-Baxter unravels �Kardar-Parisi-Zhang

Ivan Corwin

Columbia University

Based on arXiv:2308.11908, 2402.06868, 2403.01341, 2412.18117

Amol Aggarwal

(Columbia, CMI)

Milind Hegde

(Columbia)

Alexei Borodin

(MIT)

Jiaoyang Huang

(U. Penn.)

The purpose of this lecture will be to describe a method that should prove convergence for all known integrable representatives of the KPZ class to this universal scaling limit. The method has been fully realized for the Asymmetric Simple Exclusion Process and the Stochastic Six Vertex Model. It relies on the Yang-Baxter equation as its only input and unravels the rich complexity of the KPZ class and its asymptotics from first principles. This approach is probabilistic in nature, relying only on the one-point asymptotics along with certain Gibbs properties. This is based on a few works involving various subsets of the authors pictured on the right, and me.

There is a broader goal of this talk – to show how by embedding a problem in a higher dimensional and seemingly more complicated setting, one sometimes is able to gain new structure that unravels the problem.

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Colored stochastic six vertex (S6V) model

‘Quantum’ parameter q

‘Spectral’ parameter z

0 0 0 0

[Petrov]

[Kulish-Reshetikhin-Sklyanin ’81] [Bazhanov ‘85], [Jimbo ‘86], [Kuniba-Mangazeev-Maruyama-Okado ‘16], [Borodin-Wheeler ‘18]

h(x;y,t) := # color ≥ x arrows below y at time t

The special “square-ice” case of the six vertex model goes back to Pauling in 1935. In general, vertex models involve weights on certain types of vertices that are required to fit together with each other and specified boundary conditions; like puzzle pieces. The measure on configurations is proportional to the product of all weights. In 1967, Lieb computed the residual entropy of the square-ice model, and his methods illuminated a path that ultimately led to the introduction of the Yang-Baxter equation and the overarching theory of quantum groups.

[click] The colored stochastic six vertex model was introduced by Kulish, Reshetikhin and Sklyanin in the early 80s and is parameterized by a quantum parameter q and a spectral parameter z. Ever edge (or arrow) has a color and at vertices, there is conservation of colors. The weights defining this model are shown here, assuming i<j. The first configuration assigns weight 1 to (i,i) 🡪 (i,i). The other four configurations involve (i,j) or (j,i) staying the same or swapping. The term stochastic refers to the fact that the weights given the input, always sum to 1. Thus, if q and z are chosen so all weights are in [0,1], these weights are probabilities of the output of arrows given the inputs.

[click] In particular, if we start with a rainbow of arrow colors on the left and, say, all color 0 on the bottom, then we can sequentially update vertex-by-vertex to get the measure on the rectangle. In other words, this defines a Markov process and the arrows are its trajectory.

[click] Here was a simulation for a large system size; there is a lot of interesting behavior to observe.

[click] If you take the spectral parameter z🡪1 the crossing weights tend to 0 so colors tend to zigzag. However, if we speed up time in the diagonal direction by 1/(1-z) and move into a diagonal frame, the arrows stay put except occasionally, at Poisson times, swap. This exactly recovers colored ASEP! Hence, anything said or proved about this model can be transferred through this limit to ASEP.

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Mesoscopic scaling limit of colored S6V

time t

S6V sheet

scale x,y by t^2/3

h by t^1/3

time 0

Thm: The S6V sheet converges to the Airy sheet .

h(x;y,t) := # color ≥ x arrows below y at time t

Some corollaries:

Temporal evolution
Arbitrary initial data
Jointly invariant measures
Fluctuating hydrodynamics
Similar results for ASEP limit

Some extensions:

Fusion to other KPZ models
Mixing times
Periodic, half-space and open boundary conditions
…

To fix notation, let h(x;y,t) equal the number of particles of color >=x to the right of y at time t; or equivalently, the height function for the colored merged process at color x. We start under packed initial data. Here is a graph of the height function at time 0. Clearly it increases linearly to the left of the line x+y=0, and is 0 to the right. This 2d height surface evolves as a Markov process via the colored ASEP dynamics.

After running a simulation to some large time t, the crease has relaxed with a parabolic interpolation. This is a manifestation of the macroscopic and deterministic hydrodynamic limit for the model, governed by a quadratic flux Hamilton Jacobi PDE. It is the mesoscopic fluctuations around that we seek to understand.

KPZ scaling predicts that for any given x, we should consider y varying in a t^{2/3} window and scale the height function (after centering) by t^{1/3}. In a sense it also indicates that we should scale the location of the step x by t^{2/3} too.

This is what one sees after doing that – a random two-dimensional height surface whose law depends on the time t, and the q parameter of colored ASEP. We call this the ASEP_{t,q} sheet.

Our main theorem is that as t->infinity, all ASEP_{t,q} sheets converge to a single limit regardless of the value of q<1 – the Airy sheet. Recall here that I assumed that we measure time in units of 1-q which is important to ensure that we do not need any q-dependent tuning parameters for this matching.

The limiting Airy sheet was first described as a central object in the KPZ class in my work with Quastel, Remenik and first constructed explicitly and shown to arise in a special model of random directed metrics by Dauvergne, Orthman and Virag. It has a very explicit and interesting construction in terms of infinitely many non-intersecting Brownian motions, as I’ll describe a bit later.

For now, what does this result give us?

A universal scaling limit that should arise in many other contexts.
It recovers all previously known distributional results for the past 50 years about ASEP as marginals.
The ridge structures visible in these figures and the scaling limit related to a phenomena of intermittency which ultimately should be useable to prove various conjectures about the mixing time for ASEP.
There are several other implications or corollaries. [click] For instance, we can use this result to actually describe how the height function evolves with a varying scaled time, and also what happens for completely general initial/boundary data. You should think of the Airy sheet convergence as the kernel result from which essentially all asymptotics follow with some work.

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Structure revealed through embedding

????

If there is one message you get from this talk it should be that there is sometimes great benefit in embedding a hard problem into a higher dimensional setting. Indeed, new structures may arise from which the original problem can be unwound. We just saw this in action -- by embedding colored ASEP with packed initial data into the ASEP random dynamical system, we were able to extract a full space-time limit of that system, and consequently access the full space-time limit of colored ASEP with arbitrary initial data.

Let me describe now a much simpler yet still conceptually useful example of this idea in motion.

Feynman’s approach to integrals often involved introducing auxiliary parameters and solving a seemingly more complicated problem.

[click] The integral of the sinc function is such an example. [click] Adding an exponential for some t>=0 seems to complicate the problem. Yet, upon differentiation, [click] the integrand becomes pure exponentials (recalling the formula for sin(x)) which [click] can be evaluated explicitly. All that remains is to integrate and use the boundary condition that as t🡪infinity, H(t)🡪0 to give a general formula [click]. Specializing back to t=0 gives [click] the answer.

In a sense, the inclusion of the exponential transfers the calculation of the oscillatory integral for t=0 to the trivial integral of 0 when t=infinity. Another example of embedding comes when studying asymptotics of special functions such as the Airy function that are defined via oscillatory integrals. By rewriting these functions in terms of complex integrals and transforming to new controls, one have the method of stationary phase or Laplaces method that localizes the asymptotics of the integral around a critical point.

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Structure revealed through embedding

Airy line ensemble

[Prahofer-Spohn ‘02]

[C-Hammond ‘14]

N^2/3

N^1/3

Gaussian Unitary Ensemble

large

[Tracy-Widom ’93]

Dyson Brownian Motion

[Dyson ’62]

Non-intersecting Brownian Gibbs property

Strong characterization

[Aggarwal-Huang ‘23]

Random matrix integrals and distributions offers a considerably more complicated example of this idea in action, and also moves us towards our ultimate goal.

[click] The Gaussian unitary ensemble is a Gaussian measure on NxN Hermitian symmetric matrices. [click] Here is a 2x2 matrix. In general all diagonals are real Gaussian of variance 2, and off-diagonals are the sum of real Gaussian and imaginary Gaussian of variance 1. All entries are independent up to the symmetry constraint. [click] Due to this symmetry, all eigenvalues are real and for large N their histogram converges to a limit shape – the Wigner semi-circle. Consider the largest eigenvalue – how does it behave as N🡪infinity? How random is it?

Looking at this top eigenvalue alone neglects some given structure, namely the GUE is unitarily invariant and hence its eigenvalues and eigenfunctions are independent. Thus, by a version of the Weyl integration formula, we can evaluate the joint density of all eigenvalues. [click] The formula has many nice interpretations and implications, for instance connecting the problem to electrostatics of N particles in a quadratic potential with log repulision. The presence of a squared vandermonde determinant along with the Cauchy-Binet formula and orthogonal polynomial theory allows for integrating out all put the top eigenvalue and leads to a remarkable asymptoticin N formula [click], first proved by Tracy and Widom in 1993.

While looking at all eigenvalues is not such a remarkable embedding, there is a more powerful embedding found by Dyson. [click] Replace the static Gaussians by dynamic Brownian motions. [click] Here we draw the temporal evolution of the N eigenvalues. Quite remarkably, these form a Markov process, namely they evolve according to a system of stochastic differential equations whose generator is given by the Laplacian on the Weyl chamber of ordered particle locations with Dirichlet boundary condition, conjugated by its ground state eigenfunction, the vandermonde determinant. Said otherwise, the law of the eigenvalue evolution is that of N Brownian motions conditioned to never intersect for all time.

The Karlin-McGregor formula along with the earlier mentioned formula shows that the determinantal structure persists with this extra dimension, and this enables a more robust asymptotic limit. Namely, Prahofer and Spohn essentially showed that if you normalize the limit shape to scale N, and zoom into an N^{2/3} temporal region and N^{1/3} fluctuation region, then for finitely many time slices, the edge points have a limit described via determinantal formulas.

But there is a much nicer way to take and describe this limit. The non-intersecting description of DBM means that it satisfies a Gibbs property; namely if you look at a given curve on a time interval and resample it according to a Brownian bridge conditioned to avoid the curve above and below, the law of the ensemble is unchanged. Let me illustrate this on the bottom picture. [click]. The point of my work with Hammond was that this Gibbs property can be quite powerful for asymptotics. For instance, if we accept the N^{1/3} fluctuation of the top curve, it implies the whole ensemble of edge curves have non-trivial yet “tight” fluctuations in the specified scale. The N^{2/3} comes simply from the scaling of Brownian motion. In fact, even the N^{1/3} scale can be understood as coming from the Gibbs property – this is the scaling of a Brownian motion conditioned to avoid the parabolic limit shape of the ensemble.

Over the last two decades, a substantial theory has evolved to understand the origins and uses of such Gibbsian line ensembles. For instance, Okounkov and Sheffield, about 20 years ago, put forward a uniqueness conjecture that was recently proved in an even stronger form by Aggarwal and Huang. It says that up to an overall height shift, the Airy line ensemble is the unique parabolically shaped, natural number indexed ensemble of random curves which enjoys the non-intersecting Brownian Gibbs property. If the top curve has GUE TW fluctuations, this pins down the ensemble exactly.

To summarize, by adding a temporal dimension, the GUE was embedded in a Gibbsian line ensemble to great value. The next example shows how a random metric problem can be embedded in DBM in a manner which leads to the definition of the Airy sheet that featured in our main result.

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The Airy sheet via RSK

Airy sheet

[Dauvergne-Orthmann-Virag ‘18]

Dyson BM

[Baryshnikov ’01],

[Gravner-Tracy-Widom ’01]

Rest of talk: The Airy sheet via Yang-Baxter equation

Airy line ensemble

[Prahofer-Spohn ‘02],

[C-Hammond ‘14]

Brownian LPP

RSK

[Greene ‘74],

[Noumi-Yamada ‘02]

N^2/3

N^1/3

Let’s consider the following random metric model, known as Brownian last passage percolation. The environment is N independent Brownian motions and paths start at level 1 and stay flat or jump to the next level, ending at level N. The length of a path is the sum of the corresponding Brownian increments along the horizontal components (or the integral of the associated white noises along the path). The metric is the maximum over all such paths of the length. And, as before, we want to capture a scaling limit as the starting and ending points vary in an N^{2/3} scale with the number of levels N going to infinity.

[click] The Robinson-Schensted-Knuth correspondence – introduced between 60 to 90 years ago – provides a path transformation of any environment into an ensemble of non-intersecting paths.

[click] Quite remarkably the push-forward of Brownian measure is precisely the DBM discussed on the last slide! Greene’s theorem implies that the last passage time from the origin to the top level N is the top curve under RSK and also relates lower curves to multipath maximization problems. Noumi and Yamada proved a generalization of this identity that shows that the last passage time from another starting time on level 1 to ending time on level N in the environment on the left is deterministically equal to the same last passage time problem but in the transformed environment on the right. Said otherwise, to understand the scaling limit of the random metric model, we must understand the limit of the DBM along with the variational problem within it.

[click] Of course, we know the Airy line ensemble limit already. Because the DBM has most of its randomness near the top edge, the variational problem ends up only relying on that randomness and hence the Airy sheet arises as a variational problem in the Airy line ensemble. This is how it was defined in the work of Dauvergne, Orthmann and Virag.

The applicability of RSK is limited to a small subset of the known integrable models and the occurrence of DBM seems a bit ad hoc. The general mechanism for integrability is the Yang-Baxter equation, and the purpose of the rest of this talk is to show how this simple combinatorial identity can be developed into a general theory for extracting such space-time scaling limits of all integrable models. Indeed, RSK can even be seen as a suitable limit of Yang-Baxter. I should also note here that this example is determinantal as described on the last slide. However, most integrable models are not determinantal, and thus new methods, often based on Yang-Baxter as well have become central to their analysis.

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Key ideas behind the theorem

Yang-Baxter equation embeds S6V into a colored Gibbsian line ensemble (i.e., colored q-Boson model)

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Yang-Baxter equation for colored S6V

Here are the weights for the colored stochastic six-vertex model again. What is shown below is the Yang-Baxter equation. It deserves some explanation.

Here and in what follows, when I draw a picture with labeled arrows, it represents a number, or rather rational function, given by the product of all vertex weights given those labeled arrows. So, on the left, we have colors I, a_1 and i_1 coming in, J, b_2 and j_2 coming out, and K, b_1 and j_1 internal colors. The x and y label the spectral variables for the two left-vertices, and when they cross, it means to take the spectral variable given by their ratio y/x. The quantum parameter q is fixed for all vertices here and in what follows. The summation on the left means to sum over all internal colors; of course most terms are 0 since they don’t fit together. In general there are at most 2 non-zero terms. The picture on the right has involved moving the cross through the upright. This is a different sum. The Yang-Baxter equation is the equality of these 3-vertex partition functions over all possible boundary values. It is non-trivial that there even exist solution weights, let alone such a nice one. Of course, checking this is solution works can be done easily by hand.

There is a procedure, known as fusion, which takes this basic solution to Yang-Baxter and produces more involved solutions, with additional parameters.

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Fusion

[Kulish-Reshetikhin-Sklyanin ’81]

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Fusion

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Yang-Baxter intertwines S6V and q-Boson weights

S6V weights q-Boson weights

As noted before, we can produce more complicated solutions to Yang-Baxter via fusion. In fact, through gauge transforms we can produce non-stochastic weight solutions too, or solutions which involve fused and unfused vertices. What is shown here is the outcome of fusing in the horizontal direction. The thick lines carry arbitrary numbers of colored arrows and hence can be thought of as Bosonic. The vertex weights, given above, are called the colored q-Boson weights. There is no need to look closely at the weights. I will say in a moment all you need to know about them, and this will justify their use.

Before that, let me note one graphical convention that I will use in the next two slides. If I specify (by label or color) some boundary conditions but leave internal arrows unlabeled, you should interpret a picture as being the partition function given by the sum over all choices of internal arrows of the weight of the corresponding pictures. For instance, in this way this Yang-Baxter equation can be rewritten as [click]. The next slide shows the efficiency of this graphical approach.

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Relating the colored q-Boson and S6V models

Uncolored: [Borodin-Bufetov-Wheeler ‘16], Colored:[Aggarwal-Borodin ‘24]

trivial weight color S6V model trivial weight colored q-Boson model

Here N and M will ultimately represent the dimensions of the six vertex model, and the sigma represents the different arrow colors (illustrated here with three colors, blue, orange and red). The spectral variables on the top N rows are 1’s and on the bottom M rows are z which we will take in (0,1). As before, spectral variables follow the horizontal lines and when they cross on the right part, the spectral variable is the ratio, z/1. There is also a specified configuration of outgoing arrows on the right portion. No arrows enter or exit from below or above. The … represent a currently finite number of intermediate columns that we will eventually take to infinity. Thus, all boundary conditions are set and the picture represents the partition function of all configuration weights summed over all configurations. Notice, however, that since all colored arrows start on the top N rows and the q-Boson model only allows one color per horizontal arrow, the only admissible configuration has these colors going to the right. Thus, the partition function is the product of this trivial weight, times that of the color S6V model with specified initial and terminal condition, which is itself proportional to the probability of that output configuration of the model. So, the LHS here is something we care about.

The YBE equation provides us with a way to transform the picture without changing the partition function. In particular, we can pull the X’s through to the left sequentially. For finitely many columns (…), the paths can traverse the six vertex and q-Boson portion in an arbitrary manner. However – and here is where the q-Boson weights come in – if you take the number of columns to diverge, all vertices in the top M rows of the q-Boson side that go horizontal get weight z<1. In the bottom N rows, they get weight 1. Thus, in the large column limit, the only paths that contribute are those that go infinitely far horizontally in the bottom N rows, and then separate in the finitely many last few columns. This trivializes the six vertex weight and thus we see that the color S6V probability of a given set of outgoing arrows is given in terms of the colored q-Boson model partition function with specified boundary condition in the last column.

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Colored q-Boson (i.e. Hall-Littlewood) line ensemble

The colored q-Boson weights actually define a probability measure on all colored arrow configurations on the right. If we look at the marginal distribution in the first column, the last slide showed that this precisely matches the probability distribution of colored S6V output configurations. In other words, the YBE has embedded a 2-d height function – the colored S6V one – into a 3d object. Indeed, one dimension here is the color, and the other two are the column and row.

The hidden structure that this embedding reveals is the natural Gibbs property associated with vertex models. Namely, if you erase a region, and resample according to the weights conditioned on the boundary, the measure is invariant. In fact, there are some more subtle implications of this, and to explain them it helps to rephrase the picture on the right in terms of a colored line ensemble.

For each color we have a line ensemble defined as follows. Group all colors of an equal or higher priority together and then read off in the first column, starting from the bottom, the number of lines at or above you. As you move up the column, this number diminishes. This defines the top curve of the corresponding line ensemble; the second curve etc comes from doing the same for the second column, etc.

For instance, the bottom picture is the uncolored ensemble where I count all colors, here 1 and 2. Since there are 4 arrows on the LHS, the height starts at 4. The top picture is the color 2 ensemble which starts at height 2 since there are only 2 arrows of color 2 (or higher).

I want to pause and answer a blaring question – the colored S6V model also provides a natural 3d embedding of the colored S6V height function and has a Gibbs property. So, what have we gained in this transformation?

The answer is that while the colored S6V model’s entire bulk impacts the height function on its boundary, this will not be true for the colored q-Boson model. This is because the fluctuations at the top edge of these colored line ensembles end up being far larger than in the bulk. This is very much in the spirit of the simple sinc example or Laplaces method. YBE transferred from a bulk problem to a boundary problem to great benefit. Let’s see how this works.

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Key ideas behind the theorem

Yang-Baxter equation embeds S6V into a colored Gibbsian line ensemble (i.e., colored q-Boson model)
Intercolor Gibbs property yields approximate variational representation.

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Intercolor q-Boson Gibbs property

satisfies a variational formula at q=0:

and an approximate variational formula at q>0

Given ,

This is the intercolor Gibbs property. To illustrate, assume you are given the uncolored line ensermble L1. Assume for the moment q=0 – this is closest to the last passage setting. Then the color 2 line ensemble’s k^th curve is determined entirely from the k^th curve of the uncolored ensemble and the k+1^st curve of the color 2 ensemble via something called a Pitman transform. I’ll say more about this in a moment, but first, what happens for q>0?

It turns out, and this can be proved by explicit analysis of the weights, the deterministic Pitman transform does not exactly recover the kth curve of the 2^nd ensemble. However, the randomness around the transform is well controlled so that the probability of deviation m over all y decays like q^{cm^2}. We will soon talk about scaling limit where the elements of the line ensemble have growing in N and M fluctuations. Thus, after scaling this result show that the variational problem is recovered exactly in the scaling limit!

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Iterating intercolor Gibbs property yields (q=0)

Given ,

satisfies

where

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Key ideas behind the theorem

Yang-Baxter equation embeds S6V into a colored Gibbsian line ensemble (i.e., colored q-Boson model)
Intercolor Gibbs property yields approximate variational representation.
Uncolored Gibbs property, one-point GUE Tracy-Widom asymptotics and strong characterization of Airy line ensemble yields Airy line ensemble limit.

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Uncolored q-Boson Gibbs property

Non-crossing Bernoulli bridges

q=0

Marginal ∝

q>0

Dyson BM

[Baryshnikov ’01],

[Gravner-Tracy-Widom ’01]

N^2/3

N^1/3

Here is the uncolored q-Boson Gibbs property enjoyed by L1. Again, it is easiest when q=0. In that case, the law of the blue highlighted curve in the uncolored ensemble conditioned on the starting and ending points and above and below curve is simply that of a Bernoulli random walk bridge conditioned on non-crossing. This generalizes to multiple curves easily and is an exact discrete analog of the non-intersecting Brownian Gibbs property of the Airy line ensemble.

When q>0 things get weirder. Now, the measure is reweighted by these factors. Every time the distance between two curves drops, you get a weight 1-q^{initial distance}. Of course, if you fix q and imagine that the curves separate, these weird factors are all very close to 1 in which case it is quite reasonable to expect Airy line ensemble convergence.

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Airy limit of uncolored q-Boson line ensemble

Lack of FKG inequality requires complete reworking of theory of Gibbsian line ensembles to only use ‘weak monotonicity’

GUE Tracy-Widom fluctuations & HL Gibbs property

Identification of limit as Airy line ensemble

[Borodin-C-Gorin ‘16], [C-Dimitrov ‘18]

[Aggarwal-Huang ‘23]

Tightness at edge & Brownian Gibbs property for limits

[Aggarwal-C-Hegde ‘24]

More precisely, for any fixed q<1 we seek to show that as N->infinity, under N^{2/3} transversal scaling and N^{1/3} fluctuation scaling (as with DBM), the ensemble at the edge converges to the Airy line ensemble. For q=0 this was known because the Gibbs property is simple and the model is determinantal, meaning it has determinant correlation functions like those of the Airy line ensemble.

For q>0 the situation is much trickier though we were able to prove it in the following way, which should be a general proof mechanism.

Prove a one-point fluctuation result to the GUE Tracy-Widom distribution and identify the Gibbs property.
Show that these two inputs alone imply that the entire edge is tight in the sense that its probability measure in the space of collections of random curves is compact. Thus, any subsequential limit of this measure will enjoy the non-intersecting Brownian Gibbs property since the 1-q^ factors become 1 in the N^{1/3} scaling for fixed q.
Finally, using the strong characterization result of Aggarwal and Huang, and the one-point GUE TW fluctuations, identify the limit at the �Airy line ensemble.

Let me focus on step 2. This was deceivingly hard. [click] The reason is that the entire theory of Gibbsian line ensembles over the last decade has been built assuming a property of the measures – the FKG inequality – which says that if you lift your boundary conditions on the line ensemble, then the internal measure lifts too. For the pure Bernoulli bridge case, this is true – if you condition to stay above a higher lower curve, the measure lifts. However, when q>0, this fails quite badly. This picture on the right shows that relative weights of two choices for the blue curve given the bottom black one. As k shrinks the black curve rises. Surprisingly, the blue curve’s chance of dropping increases!

Despite this loss of monotonicity, there is a very weak monotonicity that persists. The one-point distribution of the blue curve might move the wrong way, but just by a q-dependent factor. With this weaker input we had to redevelop the theory of Gibbsian line ensembles, and luckily we were able to do so.

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Key ideas behind the theorem

Yang-Baxter equation embeds S6V into a colored Gibbsian line ensemble (i.e., colored q-Boson model)
Intercolor Gibbs property yields approximate variational representation.
Uncolored Gibbs property, one-point GUE Tracy-Widom asymptotics and strong characterization of Airy line ensemble yields Airy line ensemble limit.
Combining yields Airy sheet limit theorem.

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Putting it all together

approximate variational

formula

variational

formula

convergence

Airy line ensemble

convergence

(jointly)

Airy sheet

S6V sheet

Goal

[click] Recall our goal was to show converge of the fluctuations of the colored ASEP or rather colored S6V height function with packed initial condition.

[click] The YBE equation enables us to embed this height function into the colored q-Boson model and correspond colored line ensemble. The colored ensembles are well approximated by variational problems in the uncolored ensemble.

[click] Using the weak monotonicity along with the strong characterization for the Airy line ensemble, we show convergence at the edge of the uncolored ensemble to the Airy line ensemble.

[click] In this limit, the variational problem becomes exact, and hence we recover the joint convergence of both line ensembles to the Airy line ensemble and a variational problem therein.

[click] This is precisely how the Airy sheet is defined.

The approach that we developed should apply to all YBE solvable models, though this needs to be done. There are many implications of the asymptotics, as well as pre-asymptotic descriptions that should be probed, for instance in studying mixing times. And, an analogous story is yet to be developed for models in contact with boundaries, though we now believe we have all of the pieces needed, namely the correct solutions to the boundary version of the Yang-Baxter equations, that is, the reflection equations.

Ok, so I had two goals. One was to describe a fascinating physical picture about the scaling limit of colored ASEP and its associated random dynamical system. The second was to show how, by embedding these models in a larger, richer structure, we acquire new and powerful tools through which to access the scaling limit.