Chapter summary. In 1997, Kurt Johansson discovered that the corner growth process we studied in the previous chapter is directly related to longest increasing subsequences in generalized permutations. This connection can be studied via the RSK algorithm, which is an extension of the Robinson-Schensted algorithm discussed in Chapter 1, leading to a remarkable explicit representation for the distribution of the passage times, that is itself related to Wishart matrices from random matrix theory. Applying ideas from the theory of orthogonal polynomials and asymptotic analysis techniques, we prove Johansson's result that the distribution of the passage times converges to the Tracy-Widom distribution F2.

The fluctuations of G(m, n) and the Tracy–Widom distribution

In previous chapters we studied two natural processes of randomly growing Young diagrams, and derived the limiting shapes for both: the Plancherel growth process, which was used in Chapter 1 to solve the Ulam-Hammersley problem of deriving the (first-order) asymptotics of the maximal increasing subsequence length in a random permutation; and the corner growth process, which we analyzed in Chapter 4, where we also saw it bears an interesting relation to other natural random processes such as the totally asymmetric simple exclusion process and random domino tilings.

As many students of mathematics know, mathematical problems that are simple to state fall into several classes: there are those whose solutions are equally simple; those that seem practically impossible to solve despite their apparent simplicity; those that are solvable but whose solutions nonetheless end up being too complicated to provide much real insight; and finally, there are those rare and magical problems that turn out to have rich solutions that reveal a fascinating and unexpected structure, with surprising connections to other areas that lie well beyond the scope of the original problem. Such problems are hard, but in the most interesting and rewarding kind of way.

The problems that grew out of the study of longest increasing subsequences, which are the subject of this book, belong decidedly in the latter class. As readers will see, starting from an innocent-sounding question about random permutations we will be led on a journey touching on many areas of mathematics: combinatorics, probability, analysis, linear algebra and operator theory, differential equations, special functions, representation theory, and more. Techniques of random matrix theory, a sub-branch of probability theory whose development was originally motivated by problems in nuclear physics, will play a key role. In later chapters, connections to interacting particle systems, which are random processes used to model complicated systems with many interacting elements, will also surface.

Chapter summary. We continue our study of longest increasing subsequences in permutations by considering a special class of permutations called Erdőos-Szekeres permutations, which have the property that their longest monotone subsequence is the shortest possible and are thus extremal cases demonstrating sharpness in the Erdőos-Szekeres theorem. These permutations are related via the Robinson-Schensted correspondence to an especially well-behaved class of standard Young tableaux, the square Young tableaux. We use the tools developed in Chapter 1 to analyze the behavior of random square Young tableaux, and this leads us to an interesting result on the limiting shape of random Erdőos-Szekeres permutations. We also find a mysterious arctic circle that appears when we interpret some of the results as describing the asymptotic behavior of a certain interacting particle system.

Erdős–Szekeres permutations

In the previous two chapters we studied the statistical behavior of the permutation statistic L(σ) for a typical permutation σ chosen at random from among all permutations of given order. In this chapter we focus our attention instead on those permutations σ whose behavior with regard to longest increasing subsequences, or more precisely longest monotone subsequences, is atypical in the most extreme way possible. We refer to these permutations as Erdős–Szekeres permutations, because of their role as extremal cases demonstrating the sharpness in the Erdős-Szekeres theorem (Theorem 1.2).

In this chapter, our goal is to collect some of the facts and theorems we have seen so far in order to conclude that percolation crossings are indeed noise sensitive. Recall from the “BKS” Theorem (Theorem 1.21) that it is enough for this purpose to prove that influences are “small” in the sense that ΣkIk(fn)2 goes to 0. If we want to use only what we have actually proved in this book, namely Proposition 5.6, then we need to demonstrate (5.6) in this proposition.

In the first section, we deal with a careful study of influences in the case of percolation crossings on the triangular lattice. Then, we treat the case of ℤ2, where conformal invariance is not known. Finally, we speculate to what “extent” percolation is noise sensitive.

This whole chapter should be considered somewhat of a “pause” in our program, where we take the time to summarize what we have achieved so far in our understanding of the noise sensitivity of percolation, and what remains to be done if one wishes to obtain the exact “noise sensitivity exponent” as well as the existence of exceptional times for dynamical percolation.

Bounds on influences for crossing events in critical percolation on the triangular lattice

6.1.1 Setup

Fix a, b > 0, let us consider some rectangle [0, a·n]×[0, b·n], and let Rn be the set of hexagons in T that intersect [0, a·n]×[0, b·n]. Let fn be the event that there is a left-to-right crossing event in Rn. (This is the same event as in Example 1.22 in Chapter 1, but with ℤ2 replaced by T.) By the RSW Theorem 2.1, we know that {fn} is nondegenerate. Conformal invariance tells us that E[fn] = ℙ[fn = 1] converges as n → ∞. This limit is given by the so-called Cardy's formula.