The oriented swap process and last passage percolation

We present new probabilistic and combinatorial identities relating three random processes: the oriented swap process on $n$ particles, the corner growth process, and the last passage percolation model. We prove one of the probabilistic identities, relating a random vector of last passage percolation times to its dual, using the duality between the Robinson-Schensted-Knuth and Burge correspondences. A second probabilistic identity, relating those two vectors to a vector of 'last swap times' in the oriented swap process, is conjectural. We give a computer-assisted proof of this identity for $n\le 6$ after first reformulating it as a purely combinatorial identity, and discuss its relation to the Edelman-Greene correspondence. The conjectural identity provides precise finite-$n$ and asymptotic predictions on the distribution of the absorbing time of the oriented swap process, thus conditionally solving an open problem posed by Angel, Holroyd and Romik.


INTRODUCTION
Randomly growing Young diagrams, and the related models known as Last Passage Percolation (LPP) and the Totally Asymmetric Simple Exclusion Process (TASEP), are intensively studied stochastic processes. Their analysis has revealed many rich connections to the combinatorics of Young tableaux, longest increasing sub-In this article we discuss a new and surprising meeting point between the aforementioned subjects. In an attempt to address an open problem from [AHR09] concerning the absorbing time of the OSP, we discovered elegant distributional identities relating the oriented swap process to last passage percolation, and last passage percolation to itself. We will prove one of the two main identities; the other one is a conjecture that we have been able to verify for small values of a parameter n. The analysis relies in a natural way on well-known notions of algebraic combinatorics, namely the RSK, Burge, and Edelman-Greene correspondences.
Our conjectured identity apparently requires new combinatorics to be explained, and has far-reaching consequences for the asymptotic behavior of the OSP as the number of particles grows to infinity, as will be explained in Subsection 1.3.
Most of the results in this paper were obtained in 2019 and announced in the proceedings of the 32nd Conference on Formal Power Series and Algebraic Combinatorics [BCGR20]. The present paper contains complete proofs, as well as additional material including: • more detailed information about the RSK and Burge correspondences for random tableaux and their connection to distributional symmetries in last passage percolation; • some explicit formulas related to the conjectural identity and its connection to the largest eigenvalue of certain random matrices and Tracy-Widom distributions; • more details about the Edelman-Greene correspondence and its relation to the conjectural identity. We define the vector U n = (U n (1), . . . , U n (n − 1)) of last swap times by U n (k) := the last time t at which a swap occurs between positions k and k + 1.
As explained in [AHR09], the last swap times are related to the particle finishing times: it is easy to see that max{U n (n − k), U n (n − k + 1)} is the finishing time of particle k (with the convention that U n (0) = U n (n) = 0); see the equation on the last line of page 1988 of [AHR09].
RANDOMLY GROWING A STAIRCASE SHAPE YOUNG DIAGRAM. This process is a variant of the corner growth process. Starting from the empty Young diagram, boxes are successively added at random times, one box at each step, to form a larger diagram until the staircase shape δ n = (n − 1, n − 2, . . . , 1) is reached. We identify each box of a Young diagram λ with the position (i, j) ∈ N 2 , where i and j are the row and column index respectively. All boxes are assigned independent Poisson clocks. Each box (i, j) ∈ δ n , according to its Poisson clock, attempts to add itself to the current diagram λ, succeeding if and only if λ ∪ {(i, j)} is still a Young diagram. Notice that the randomly growing Young diagram model can be thought of as a continuous-time random walk, starting from ∅ and ending at δ n , on the graph of Young diagrams contained in δ n (regarded in the obvious way as a directed graph). See Fig. 1B. Furthermore, note that every such random walk path is encoded by a standard Young tableau of shape δ n , where the box added after m steps is filled with m, for all m = 1, . . . , n 2 . For more details on this, see Subsection 3.1 and, in particular, (22).
We define V n = (V n (1), . . . , V n (n − 1)) as the vector that records when boxes along the (n − 1)th anti-diagonal are added: V n (k) := the time at which the box at position (n − k, k) is added. THE LAST PASSAGE PERCOLATION MODEL. This process describes the maximal time spent travelling from one vertex to another of the two-dimensional integer lattice along a directed path in a random environment. Let (X i,j ) i,j 1 be an array of independent and identically distributed (i.i.d.) non-negative random variables, referred to as weights. For (a, b), (c, d) ∈ N 2 , define a directed lattice path from (a, b) to (c, d) to be any sequence (i k , j k ) m k=0 of minimal length |c − a| + |d − b| such that (i 0 , j 0 ) = (a, b), (i m , j m ) = (c, d), and |i k+1 − i k | + |j k+1 − j k | = 1 for all 0 k < m. We then define the Last Passage Percolation (LPP) time from (a, b) to where the maximum is over all directed lattice paths π from (a, b) to (c, d). It is immediate to see that LPP times starting at a fixed point, say (1, 1), satisfy the recursive relation with the boundary condition L(1, 1; i, j) If the weights X i,j are i.i.d. exponential random variables of rate 1, the LPP model has a precise connection (see [Rom15,Ch. 4]) with the corner growth process, whereby each random variable L(1, 1; i, j) is the time when box (i, j) is added to the randomly growing Young diagram. We can thus equivalently define V n in terms of the last passage times between the fixed vertex (1, 1) and the vertices (i, j) along the anti-diagonal line i + j = n: V n = (L(1, 1; n − 1, 1), L(1, 1; n − 2, 2), . . . , L(1, 1; 1, n − 1)) .
We refer to this as the point-to-line LPP vector (see the illustration in Fig. 2A and the discussion in Subsection 1.3 below).
In this case, the starting and ending points for each last passage time vary simultaneously along the two lines i = 1 and j = 1, respectively. We then refer to this vector W as the line-to-line LPP vector (see Fig. 2B).

MAIN RESULTS.
We can now state our results.
for n 3, with the convention that y 0 = y n−1 = 0, with the initial condition . Surprisingly, formula (5) also holds for the line-to-line LPP vector W n (as it must, by virtue of Theorem 1.1); Conjecture 1.2 says that the joint density of U n should also satisfy the same recursive relation. However, we know of no simple recursive structure in the corresponding models to make possible such a direct proof.
Theorem 1.1 and Conjecture 1.2 imply the equality of the one-dimensional mar- , for all 1 k n − 1, n 2.
The identity U n (k) D = V n (k) was proved by Angel, Holroyd and Romik [AHR09] using a connection between the oriented swap process, the TASEP and the corner growth model. The identity V n (k) D = W n (k) follows immediately from the observation that these two variables are the LPP times, on the same i.i.d. environment (X i,j ) i,j 1 , between two pairs of opposite vertices of the same rectangular lattice It is also easy to see that the following two-dimensional marginals coincide for all n 2. The second equality actually holds almost surely, since V n and W n are LPP vectors on the same environment (X i,j ) i,j 1 . To check the first identity, observe that U n (n − 1) and U n (1) are the finishing times of the first and last particle in the OSP, respectively. Particle labelled 1 (resp. n) jumps n − 1 times only to the right (resp. to the left), always with rate 1. All these jumps are independent of each other, except the one that occurs when particles 1 and n are adjacent and swap. Hence, (U n (1), U n (n − 1)) is jointly distributed as (Γ + X, Γ + X) where Γ , Γ are independent with Gamma(n − 2, 1) distribution and X has Exp(1) distribution and is independent of Γ , Γ . This is the same joint distribution of the LPP times (V n (1), V n (n − 1)).
Theorem 1.1 is proved in Section 2. As we will see, the distributional identity On the other hand, the conjectural equality in distribution between U n and V n remains mysterious, but we made some progress towards understanding its meaning by reformulating it as an algebraic-combinatorial identity that is of independent interest.
Precise definitions and examples will be given in Section 3, where we will prove the equivalence between Conjectures 1.2 and 1.3. For the moment, we only remark that the sums on the left-hand and right-hand sides of (8) range over the sets of staircase shape standard Young tableaux t and sorting networks s of order n, respectively; f t and g s are certain rational functions, and σ t , π s are permutations in the symmetric group S n−1 that are associated with t and s.
The identity (8) reduces the proof of U n D = V n for fixed n to a concrete finite computation. This enabled us to provide a computer-assisted verification of Conjecture 1.2 for 4 n 6 (the cases n = 2, 3 can be checked by hand) and thus prove the following: i.e. the absorbing time of the OSP on n particles.
Observe first that the random variable where (X i,j ) i,j 1 are i.i.d. exponential random variables of rate 1, represents the time until the staircase shape δ n is reached in the corner growth process. As the last expression in (10) points out, it can also be seen as the maximal time spent On the other hand, modulo Conjecture 1.2, we have that The precise knowledge of the (finite n and asymptotic) distribution of V max n thus extends to U max n .
Corollary 1.5. Let U max n be the absorbing time of the OSP on n particles, as in (9). Then, assuming Conjecture 1.2: (i) for any n 2, t 0, where C n is a normalization constant; (ii) the following limit in distribution holds: where F 1 is the β = 1 Tracy-Widom law.
The integral formula in (12)  Bufetov, Gorin and Romik found a way to derive (11) (and therefore deduce (12) and (13)) by proving a weaker version of our Conjecture 1.2 that equates the joint distribution functions of the random vectors U n and V n for 'diagonal points', i.e.
points (t, t, . . . , t) ∈ R n−1 . This is of course sufficient to imply equality in distribution of the maxima of the coordinates of the respective vectors. Thus, the open problem from [AHR09] is now settled.

ALONG BORDER STRIPS
The goal of this section is to prove Theorem 1.1. We will in fact prove a more general statement (Theorem 2.2), which establishes the joint distributional equality between LPP times and dual LPP times along the so-called 'border strips'.

LPP AND DUAL LPP TABLEAUX.
We first fix some terminology. We say that is the last box of its diagonal. We refer to the set of border boxes of λ as the border Note that every corner is a border box. We refer to any array x = {x i,j : (i, j) ∈ λ} of non-negative real numbers as a tableau of shape λ. We call such an x an interlacing tableau if its diagonals interlace, in the sense that for all (i, j) ∈ λ, or equivalently if its entries are weakly increasing along rows and columns. As a reference, see the tableaux in Throughout this section, λ will denote an arbitrary but fixed Young diagram.
Let now X be a random tableau of shape λ with i.i.d. non-negative random entries X i,j . We can then define the associated LPP time L(a, b; c, d) on X between two boxes (a, b), (c, d) ∈ λ as in (1). We will mainly be interested in the special λ- ∈λ , which we respectively call the LPP tableau and the dual LPP tableau, defined by and It is easy to see from the definitions that L and L * are both (random) interlacing tableaux.
Now, it is evident that, for each (i, j) ∈ λ, the distributions of L i,j and L * i,j coincide. However, the joint distributions of L and L * do not coincide in general.
Proposition 2.1. Let X be a Young tableau of shape λ with i.i.d. non-deterministic † entries. Then the corresponding LPP and dual LPP tableaux L and L * follow the same law if and only if λ is a hook shape (a Young diagram with at most one row of length > 1).
Proof. If λ is a hook shape, then L = L * almost surely; in particular, the two tableaux have the same law. Suppose now that λ is not a hook shape, i.e. (2, 2) ∈ λ.
The main result of this section is that certain distributional identities between LPP and dual LPP do hold as long as the common distribution of the weights is geometric or exponential: Theorem 2.2. Let X be a Young tableau of shape λ with i.i.d. geometric or i.i.d. exponential weights. Then the border strip entries (and in particular the corner entries) of the corresponding LPP and dual LPP tableaux L and L * have the same joint distribution.
Theorem 1.1 immediately follows from Theorem 2.2 applied to tableaux of staircase shape (n − 1, n − 2, . . . , 1), since in this case the coordinates of V n and W n are precisely the corner entries of L and L * , respectively. Remark 2.3. In a similar vein to how Proposition 2.1 illustrates the limits of what types of identities in distribution might be expected to hold, note as well that, in general, Theorem 2.2 fails to hold if the weights are not geometric nor exponential. For example, consider the square shape λ = (2, 2) and assume the X i,j 's are uniformly distributed on {0, 1}. Then, we have that P(L 1,2 = 2, L 2,2 = 3, L 2,1 = 1) = P(X 1,1 = X 1,2 = X 2,2 = 1, X 2, Thus L and L * , even when restricted to the border strip B = {(2, 1), (2, 2), (1, 2)} of λ, are not equally distributed.

RSK AND BURGE CORRESPONDENCES.
We will prove Theorem 2.2 via an extended version of two celebrated combinatorial maps, the Robinson-Schensted-Knuth and Burge correspondences, acting on arrays of arbitrary shape λ.
We denote by Tab Z 0 (λ) the set of tableaux of shape λ with non-negative integer entries, and by IntTab Z 0 (λ) the subset of interlacing tableaux, in the sense of (14). Let Π (k) m,n be the set of all unions of k disjoint non-intersecting directed lattice paths π 1 , . . . , π k with π i starting at (1, i) and ending at (m, n − k + i). Similarly, let Π * (k) m,n be the set of all unions of k disjoint non-intersecting directed lattice paths π 1 , . . . , π k with π i starting at (m, i) and ending at (1, n − k + i).
called the Robinson-Schensted-Knuth and Burge correspondences, that are characterized (in fact defined) by the following relations: for any (m, n) ∈ B and 1 k min(m, n), The For the proof of Theorem 2.2, we will be using the extremal cases k = 1 and (16) and (17).
The case k = 1 explains the connection between the outputs of the RSK (respectively, Burge) correspondence and the LPP (respectively, dual LPP) times. More precisely, we have that for all (m, n) on the border strip B on λ.
On the other hand, taking k = min(m, n) in Theorem 2.4, it is easy to see that the maxima in (16) and (17) become both equal to the same 'rectangular sum' Rec m,n (x) of inputs: Let now (m 1 , n 1 ), . . . , (m l , n l ) be the corners of a partition λ, ordered so that m 1 > · · · > m l and n 1 < · · · < n l . Then, (19) holds for (m, n) = (m k , n k ) and, if k > 1, also for (m, n) = (m k , n k−1 ) (both are border boxes by construction We thus deduce a fact crucial for our purposes: for any shape λ with corners (m 1 , n 1 ), . . . , (m l , n l ) as above, define {ω i,j : (i, j) ∈ λ} by setting We then have that for all x ∈ Tab Z 0 (λ), where r := RSK(x) and b := Bur(x).
Example 2.5. In Fig. 3 we give a reference example of the RSK and Burge maps.

EQUIDISTRIBUTION OF RANDOM RSK AND BURGE TABLEAUX.
We now formulate as a lemma the key identity in the proof of Theorem 2.2. In a broad sense, we will say that a random variable G is geometrically distributed (with support Z k , for some integer k 0, and parameter p ∈ (0, 1)) if Lemma 2.6. If X is a random tableau of shape λ with i.i.d. geometric entries, then Proof. Assume first that X has i.i.d. geometric entries with support Z 0 and any parameter p ∈ (0, 1). Fix a tableau t ∈ IntTab Z 0 (λ) and let y := RSK −1 (t) and z := Bur −1 (t). It then follows from (20) that where |λ| := i 1 λ i is the size of λ. This proves that RSK(X) and Bur(X) are equal in distribution.
The proof in the case of tableaux with i.i.d. geometric entries with support in Z k , k 0, follows immediately from the following observation: if we shift all the entries of a tableau by a constant k, i.e. set Y i,j := X i,j + k, then from (16)- (17) we have By combining this lemma with (18), we derive the announced conclusion. Using this property, the argument used to prove Lemma 2.6 can then be adapted to establish the distributional equality between RSK(X) and Bur(X) also when the input tableau X has exponential i.i.d. entries. The proof of Theorem (2.2) in the exponential case would then be akin to the geometric case, with no need to take a scaling limit.
Remark 2.8. Let X be a random tableau of shape λ. The proof of Lemma 2.6 suggests a sufficient condition on the joint distribution of X in order for (21) (and, hence, Theorem 2.2) to hold. Such a condition is the property that the P(X = y) = P(X = z) whenever y, z ∈ Tab Z 0 (λ) have equal global sum, i.e.
(i,j)∈λ y i,j = (i,j)∈λ z i,j . If we further assume the entries of X to be independent, this property forces the entries of X to be i.i.d. with a geometric distribution. The latter claim follows from the fact that, if f, g 1 , . . . , g k are probability mass functions on Z 0 such that g 1 (x 1 )g 2 (x 2 ) · · · g k (x k ) is proportional to f(x 1 + · · · + x k ) for all x 1 , . . . , x k ∈ Z 0 , then f, g 1 , . . . , g k are necessarily all geometric with the same parameter.

FROM A PROBABILISTIC TO A COMBINATORIAL CONJECTURE
In this section we reformulate Conjecture 1.2 by showing its equivalence to Conjecture 1.3. We start by discussing the two families of combinatorial objects and defining the relevant associated quantities appearing in identity (8).

STAIRCASE SHAPE YOUNG TABLEAUX.
Let δ n denote the partition (n − 1, n − 2, . . . , 1) of N = n(n − 1)/2; as a Young diagram we will refer to δ n as the staircase shape of order n. Let SYT(δ n ) denote the set of standard Young tableaux of shape δ n . We associate with each t ∈ SYT(δ n ) several parameters, which we denote by cor t , σ t , deg t , and f t . (Note: these definitions are somewhat technical; refer to Example 3.1 below for a concrete illustration that makes them easier to follow.) First, we define cor t := (t n−1,1 , t n−2,2 , . . . , t 1,n−1 ) to be the vector of corner entries of t read from bottom-left to top-right. Second, we define σ t ∈ S n−1 to be the permutation encoding the ordering of the entries of cor t , so that cor t (j) < cor t (k) if and only if σ t (j) < σ t (k) for all j, k. The vector cor t will denote the increasing rearrangement of cor t , so that cor t (k) := cor t (σ −1 t (k)) for all k. For later convenience we also adopt the notational convention that cor t (0) = 0.
Notice that a tableau t ∈ SYT(δ n ) encodes a growing sequence of Young diagrams that starts from the empty diagram, ends at δ n , and such that each λ (k) is obtained from λ (k−1) by adding the box (i, j) for which t i,j = k. We Notice that the randomly growing Young diagram model introduced in Subsection 1.1 is nothing but a continuous-time simple random walk on Y(δ n ) that starts from the empty diagram (and necessarily ends at δ n ). Let T be the (random) standard Young tableau that encodes the path of such a random walk, i.e.
the associated sequence of growing diagrams (22); then, for all t ∈ SYT(δ n ) .
Finally, we define the generating factor of t as the rational function Recall from Section 1 that the vector V n records the times when the corner boxes of the shape δ n are added in the randomly growing Young diagram model / random walk on Y(δ n ). The generating factor f t (x 1 , . . . , x n−1 ) is, essentially, the joint Fourier transform of the vector V n , conditioned on the random walk path encoded by the tableau t; see Subsection 3.5. FIGURE 4. A staircase shape standard Young tableau t of order 6, shown in 'English notation', and the associated sorting network s = EG(t) of order 6 (illustrated graphically as a wiring diagram) with swap sequence (5, 1, 2, 4, 1, 3, 5, 4, 2, 1, 5, 3, 2, 4, 3).
For example, deg t (5) = 3, because λ (5) , the sixth Young diagram in the growth sequence associated with the tableau t, is the partition (3, 1, 1), which has 3 external corners lying within δ 5 , that is, its out-degree in the graph Y(δ 5 ) is 3.
Here, we have used colors to illustrate how the entries of cor t determine a decomposition of deg t into blocks, which correspond to different variables x k in the definition of the generating factor f t .

SORTING NETWORKS.
Recall that a sorting network of order n is a synonym for a reduced word decomposition of the reverse permutation rev n = (n, n − 1, . . . , 1) in terms of the Coxeter generators τ j = (j j + 1), 1 j < n, of the symmetric group S n . Formally, a sorting network is a sequence of indices s = (s 1 , . . . , s N ) of length N = n(n − 1)/2, such that 1 s j < n for all j and rev n = τ s N · · · τ s 2 τ s 1 .
We denote by SN n the set of sorting networks of order n. The elements of SN n can be portrayed graphically using wiring diagrams, as illustrated in Fig. 4. They can also be interpreted as maximal length chains in the weak Bruhat order or, equivalently, shortest paths in the poset lattice (which is the Cayley graph of S n with the adjacent transpositions τ j as generators, see Fig. 1A) connecting the identity permutation id n to the permutation rev n . We refer to [BB05;Hum90] for details on this terminology.
We associate with a sorting network s ∈ SN n the parameters last s , π s , deg s , and g s that will play a role analogous to the parameters cor t , σ t , deg t , and f t for t ∈ SYT(δ n ).
We define the vector last s = (last s (1), last s (2), . . . , last s (n − 1)) by setting last s (k) := max{1 j N : s j = k} to be the index of the last swap occurring between positions k and k + 1. We define π s ∈ S n−1 to be the permutation encoding the ordering of the entries of last s , so that last s (j) < last s (k) if and only if π s (j) < π s (k). We denote by last s the increasing rearrangement of last s , and use the notational convention last s (0) = 0. Notice that the oriented swap process on n particles introduced in Subsection 1.1 is a continuous-time simple random walk on this graph that starts from id n (and necessarily ends at rev n ). The (random) sorting network S that encodes the path of the OSP is then distributed as follows: Finally, the generating factor g s of s is defined, analogously to (24), as the rational function Recall from Section 1 that the vector U n records the times when the last swap between particles in any two neighboring positions occurs in the oriented swap process / random walk on the graph defined above. The generating factor g s (x 1 , . . . , x n−1 ) is, essentially, the joint Fourier transform of the vector U n , conditioned on the random walk path encoded by the sorting network s; see Subsection 3.5.
sorting network where Φ m denotes the m-th iterate of Φ. See Fig. 5.
The following result is easy to guess from Examples 3.1 and 3.2.
Proposition 3.3. If t ∈ SYT n and s = EG(t) ∈ SN n , then last s = cor t and π s = σ t .
Proof. The second relation follows trivially from the first. This first identity is an easy consequence of the definition of the Edelman-Greene correspondence, and specifically of the way the map EG : SYT(δ n ) → SN n can be visualized as 'emptying' the tableau t (see the discussion above and Fig. 5) by repeatedly applying the Schützenberger operator:

THE COMBINATORIAL IDENTITY.
Let C n−1 x S n−1 denote the free vector space generated by the elements of S n−1 over the field of rational functions The above limit is equivalent to the statement |{t ∈ SYT(δ n ) : σ t = γ}| = |{s ∈ SN n : π s = γ}| for all γ ∈ S n−1 , which is true by Proposition 3.3.

Remark 3.5.
It is natural to wonder if there exists a bijection φ : SYT(δ n ) → SN n (necessarily different from EG), such that f t = g φ(t) for all t ∈ SYT(δ n ), thus leading to a proof of Conjecture 1.2. However, already for n = 4, one can verify using Fig. 6 that the two sets of generating factors {f t } t∈SYT(δ n ) and {g s } s∈SN n are different. Therefore, no bijection between SYT(δ n ) and SN n has the desired property.

EQUIVALENCE OF COMBINATORIAL AND PROBABILISTIC CONJECTURES.
We now prove the equivalence between Conjectures 1.2 and 1.3. Conjecture 1.2 can be viewed as claiming the equality p U n = p V n of the joint density functions of U n and V n . We thus aim to derive explicit formulas for p U n and p V n . DECOMPOSITION OF THE DENSITIES. As discussed in Subsections 3.1 and 3.2, both the randomly growing Young diagram model and the oriented swap process can be interpreted as continuous-time random walks. The idea is then to write the density function of the last swap times U n (resp. V n ) as a weighted average of the conditional densities conditioned on the path that the process takes to get from the initial state id n (resp. ∅) to the final state rev n (resp. δ n ): p U n (u 1 , . . . , u n−1 ) = s∈SN n P(S = s) p U n |S=s (u 1 , . . . , u n−1 ) , Here, s (resp. t) can be viewed as a realization of a simple random walk S (resp. T ) on the Cayley graph of S n (resp. on the directed graph Y(δ n )). The probabilities CONDITIONAL DENSITIES. We will now show that the conditional densities p U n |S=s (u 1 , . . . , u n−1 ) and p V n |T =t (v 1 , . . . , v n−1 ) are completely determined by the vectors last s and cor t and their corresponding orderings σ t and π s in the simple random walks, and the sequences of out-degrees deg t and deg s along the paths (which correspond to the exponential clock rates to leave each vertex in the graph where the random walk is taking place).
In the case of the OSP conditioned on the path S = s, take a sequence of independent random variables ξ 1 , . . . , ξ N , where ξ j has exponential distribution with rate deg s (j). Once the OSP has reached the state τ s k · · · τ s 2 τ s 1 , there are deg s (j) Poisson clocks running in parallel, so, by standard properties of Poisson clocks (see [Rom15,Ex. 4.1, p. 264]) the time until a swap occurs is distributed as ξ j and is independent of the choice of the swap actually occurring. Let then η t be defined as Thanks to the remarks above, this construction gives the correct distribution for the process (η t ) t 0 as an oriented swap process on n particles.
The last piece of information needed to compute the conditional density is the vector of integers last s that encodes, for each k, the point along the path wherein the last swap between positions k and k + 1 occurred. Denote by U n the increasing rearrangement of U n , so that U n (1) U n (2) . . . U n (n − 1) are the order statistics of U n . Conditioned on S = s, we have that U n (k) = U n (π −1 s (k)) and U n (1) = ξ 1 + · · · + ξ last s (1) , U n (2) − U n (1) = ξ last s (1)+1 + · · · + ξ last s (2) , . . .
In particular, conditioned on the event S = s, the variables U n (k) − U n (k − 1), k = 1, . . . , n − 1, are independent and have density where the notation m * j=1 f j is a shorthand for the convolution f 1 * . . . * f m of onedimensional densities and E ρ (x) = ρe −ρx 1 [0,∞) (x) is the exponential density with parameter ρ > 0. We conclude that the density of U n conditioned on S = s is with the convention that u 0 := 0 and, for any γ ∈ S n−1 , γ(0) := 0.
An analogous construction holds for the continuous-time random walk on Y(δ n ). Mutatis mutandis, we thus obtain that with the convention that v 0 := 0.
PROBABILITY DENSITIES OF U n AND V n . Putting together (25) with (33) and (23) with (34), the formulas for the density functions of U n and of V n take the form Notice that the indicator functions of the Weyl chambers may be dropped, due to the support [0, ∞) of the exponential densities; however, we keep them in the formulas for later convenience.
Similarly, one can compute p V 4 , using the data cor t , σ t and deg t (or, alternatively, using the recursion (5)) and check that p U 4 = p V 4 . FOURIER TRANSFORMS AND WEYL CHAMBERS. The conjectural equality p U n = p V n of the joint density functions of U n and V n is equivalent to the equality p U n = p V n of their corresponding Fourier transforms. In turn, the latter can be manipulated and recast as the combinatorial identity (8) of Conjecture 1.3. We now outline the calculations.
Recalling the notation W γ for the Weyl chamber associated to a permutation γ ∈ S n−1 , as in (32), we observe that the identity p U n = p V n is equivalent to the (n − 1)! equalities Introduce the change of variables defined by setting ζ k = z 1 + · · · + z γ(k) for 1 k n − 1 .
Applying the convolution theorem and the fact that the Fourier transform of the exponential density is Similarly, the expression for the density of V n yields q γ V n (x 1 , . . . , x n−1 ) = q γ V n (z 1 , . . . , z n−1 ) Replacing each x k with −ix k in the expressions for q γ U n and q γ V n , we recognize the generating factors g s and f t from (26) and (24), respectively. We thus conclude that the equality p U n = p V n is equivalent to the (n − 1)! identities 1 {σ t =γ} f t (x 1 , . . . , x n−1 ) , γ ∈ S n−1 .
These can be written more compactly as the equality of the generating functions F n and G n defined in (28)-(29), that is, the relation (8).
starting at R and ending at D. For instance, the shape of the tableaux in Fig. 3 is encoded as the sequence RDRRRDDRD.
Given a partition λ associated with a D-R sequence w = w 1 . . . w k , [Kra06, Theorem 7] describes the RSK map as a bijection between Young tableaux x of shape λ with non-negative integers entries and sequences (∅ = µ 0 , µ 1 , . . . , µ k = ∅) of partitions such that µ i /µ i−1 is a horizontal strip if w i = R and µ i−1 /µ i is a horizontal strip if w i = D. One can easily verify that, for 1 i k − 1, the partition µ i is of length p i := min(m i , n i ) at most. We can then form a new Young tableau r = {r i,j : (i, j) ∈ λ} by setting the diagonal of r that contains the border box (m i , n i ) to be (r m i ,n i , r m i −1,n i −1 , . . . , r m i −p i +1,n i −p i +1 ) := µ i for 1 i k − 1.