Poisson limit of bumping routes in the Robinson-Schensted correspondence

We consider the Robinson-Schensted-Knuth algorithm applied to a random input and investigate the shape of the bumping route (in the vicinity of the $y$-axis) when a specified number is inserted into a large Plancherel-distributed tableau. We show that after a projective change of the coordinate system the bumping route converges in distribution to the Poisson process.


INTRODUCTION
1.1. Notations. The set of Young diagrams will be denoted by Y; the set of Young diagrams with n boxes will be denoted by Y n . The set Y has a structure of an oriented graph, called Young graph; a pair µ Õ λ forms an oriented edge in this graph if the Young diagram λ can be created from the Young diagram µ by addition of a single box.
We will draw Young diagrams and tableaux in the French convention with the Cartesian coordinate system Oxy, cf. Figures 1 and 2a. We index the rows and the columns of tableaux by non-negative integers from N 0 " t0, 1, 2, . . . u. In particular, if l is a box of a tableau, we identify it with the Cartesian coordinates of its lower-left corner: l " px, yq P N 0ˆN0 . For a tableau T we denote by T x,y its entry which lies in the intersection of the row y P N 0 and the column x P N 0 . The position of the box s in the tableau T will be denoted by Pos s pT q P N 0ˆN0 .
Also the rows of any Young diagram λ " pλ 0 , λ 1 , . . . q are indexed by the elements of N 0 ; in particular the length of the bottom row of λ is denoted by λ 0 .
1.2. Schensted row insertion. The Schensted row insertion is an algorithm which takes as an input a tableau T and some number a. The number a is inserted into the first row (that is, the bottom row, the row with the index 0) of T to the leftmost box which contains an entry which is strictly bigger than a. In the case when the row contains no entries which are bigger than a, the number a is inserted into the leftmost empty box in this row and the algorithm terminates.
If, however, the number a is inserted into a box which was not empty, the previous content a 1 of the box is bumped into the second row (that is, the row with the index 1). This means that the algorithm is iterated but this time the number a 1 is inserted into the second row to the leftmost box which contains a number bigger than a 1 . We repeat these steps of row insertion and bumping until some number is inserted into a previously empty box. This process is illustrated on Figures 1b and 1c. The outcome of the Schensted insertion is defined as the result of the aforementioned procedure; it will be denoted by T Ð a.
Note that this procedure is well defined also in the setup when T is an infinite tableau (see Figure 2a for an example), even if the above procedure does not terminate after a finite number of steps.
1.3. Robinson-Schensted-Knuth algorithm. For the purposes of this article we consider a simplified version of the Robinson-Schensted-Knuth algorithm; for this reason we should rather call it the Robinson-Schensted algorithm. Nevertheless, we use the first name because of its well-known acronym RSK. The RSK algorithm associates to a finite sequence w " pw 1 , . . . , w ℓ q a pair of tableaux: the insertion tableau P pwq and the recording tableau Qpwq.

The insertion tableau
(1) P pwq "´`pH Ð w 1 q Ð w 2˘Ð¨¨¨¯Ð w ℓ is defined as the result of the iterative Schensted row insertion applied to the entries of the sequence w, starting from the empty tableau H. The recording tableau Qpwq is defined as the standard Young tableau of the same shape as P pwq in which each entry is equal to the number of the iteration of (1) in which the given box of P pwq stopped being empty; in other words the entries of Qpwq give the order in which the entries of the insertion tableau were filled.
The tableaux P pwq and Qpwq have the same shape; we will denote this common shape by RSKpwq and call it the RSK shape associated to w.
The RSK algorithm is of great importance in algebraic combinatorics, especially in the context of the representation theory [Ful97].
1.4. Plancherel measure, Plancherel growth process. Let S n denote the symmetric group of order n. We will view each permutation π P S n as a sequence π " pπ 1 , . . . , π n q which has no repeated entries, and such that π 1 , . . . , π n P t1, . . . , nu. The restriction of RSK to the symmetric group is a bijection which to a given permutation from S n associates a pair pP, Qq of standard Young tableaux of the same shape, consisting of n boxes. A fruitful area of study concerns the RSK algorithm applied to a uniformly random permutation from S n , especially asymptotically in the limit n Ñ 8, see [Rom15] and the references therein.
The Plancherel measure on Y n , denoted Plan n , is defined as the probability distribution of the random Young diagram RSKpwq for a uniformly random permutation w P S n .
An infinite standard Young tableau [Ker99, Section 2.2] is a filling of the boxes in a subset of the upper-right quarterplane with positive integers, such that each row and each column is increasing, and each positive integer is used exactly once. There is a natural bijection between the set of infinite standard Young tableaux and the set of infinite paths in the Young graph (2) λ p0q Õ λ p1q Õ¨¨¨with λ p0q " H; this bijection is given by setting λ pnq to be the set of boxes of a given infinite standard Young tableau which are ď n.
If w " pw 1 , w 2 , . . . q is an infinite sequence, the recording tableau Qpwq is defined as the infinite standard Young tableau in which each non-empty entry is equal to the number of the iteration in the infinite sequence of Schensted row insertions`p H Ð w 1 q Ð w 2˘Ð¨¨¨ in which the corresponding box stopped being empty, see [RŚ15,Section 1.2.4]. Under the aforementioned bijection, the recording tableau Qpwq corresponds to the sequence (2) with λ pnq " RSKpw 1 , . . . , w n q.
Let ξ " pξ 1 , ξ 2 , . . . q be an infinite sequence of independent, identically distributed random variables with the uniform distribution Up0, 1q on the unit interval r0, 1s. The Plancherel measure on the set of infinite standard Young tableaux is defined as the probability distribution of Qpξq. Any sequence with the same probability distribution as (2) with (3) λ pnq " RSKpξ 1 , . . . , ξ n q will be called the Plancherel growth process [Ker99]. It turns out that the Plancherel growth process is a Markov chain [Ker99, Sections 2.2 and 2.4].
For a more systematic introduction to this topic we recommend the monograph [Rom15, Section 1.19].
1.5. Bumping route. The bumping route consists of the boxes the entries of which were changed by the action of Schensted insertion, including the last, newly created box, see Figures 1b and 1c. The bumping route will be denoted by T ø a or by T øa depending on current typographic needs. In any row y P N 0 there is at most one box from the bumping route T ø a; we denote by T øa pyq its x-coordinate. We leave T øa pyq undefined if such a box does not exist. In this way (4) T ø a " !`T øa pyq, y˘: y P N 0 ) .
For example, for the tableau T from Figure 1a and a " 18 we have T øa pyq " # 1 for y P t0, 1u, 0 for y P t2, 3u.
The bumping route T ø a can be visualized either as a collection of its boxes or as a plot of the function cf. the thick red line on Figure 2a.
1.6. Bumping routes for infinite tableaux. Any bumping route which corresponds to an insertion to a finite tableau is, well, also finite. This is disadvantageous when one aims at the asymptotics of such a bumping route in a row of index y in the limit y Ñ 8. For such problems it would be preferable to work in a setup in which the bumping routes are infinite; we present the details in the following.  An example of an infinite standard Young tableau T sampled with the Plancherel distribution. The highlighted boxes form a bumping route obtained by adding the entry m`1{2 for m " 3. The thick red line is the corresponding plot of the function xpyq " T øm`1{2 ptyuq.
(b) The same tableau shown in the projective coordinates Oxz with z " 2m y . The thick red line is the plot of the function xpzq " T øm`1{2 pt 2m z uq.
Let us fix the value of an integer m P N 0 . Now, for an integer n ě m we consider a real number 0 ă α n ă 1 and a finite sequence ξ " pξ 1 , . . . , ξ n q of independent, identically distributed random variables with the uniform distribution Up0, 1q on the unit interval r0, 1s. In order to remove some randomness from this picture we will condition the choice of ξ in such a way that there are exactly m entries of ξ which are smaller than α n ; heuristically this situation is similar to a scenario without conditioning, for the choice of (5) α n " m n .
We will study the bumping route (6) P pξ 1 , . . . , ξ n q ø α n in the limit as n Ñ 8 and m is fixed.
Without loss of generality we may assume that the entries of the sequence ξ are all different. Let π P S n be the unique permutation which encodes the relative order of the entries in the sequence ξ, that is π i ă π j˘ð ñ`X i ă X jf or any 1 ď i, j ď n. Since the algorithm behind the Robinson-Schensted-Knuth correspondence depends only on the relative order of the involved numbers and not their exact values, it follows that the bumping route (6) coincides with the bumping route (7) P pπ 1 , . . . , π n q ø m`1{2.
The probability distribution of π is the uniform measure on S n ; it follows that the probability distribution of the tableau P pπ 1 , . . . , π n q which appears in (7) is the Plancherel measure Plan n on the set of standard Young tableaux with n boxes. Since such a Plancherel-distributed random tableau with n boxes can be viewed as a truncation of an infinite standard Young tableau T with the Plancherel distribution, the bumping routes (6) and (7) can be viewed as truncations of the infinite bumping route 1.7. The main problem: asymptotics of infinite bumping routes. The aim of the current paper is to investigate the asymptotics of the infinite bumping route (8) in the limit m Ñ 8. Heuristically, this corresponds to investigation of the asymptotics of the finite bumping routes (6) in the simplified setting when we do not condition over some additional properties of ξ, in the scaling in which α n does not tend to zero too fast (so that lim nÑ8 α n n " 8, cf. (5)), but on the other hand α n should tend to zero fast enough so that the bumping route is long enough that our asymptotic questions are well defined. We will not pursue in this direction and we will stick to the investigation of the infinite bumping route (8).
Even though Romik and the last named author [RŚ16] considered the asymptotics of finite bumping routes, their discussion is nevertheless applicable in our context. It shows that in the balanced scaling when we focus on the part of the bumping route with the Cartesian coordinates px, yq of magnitude x, y " O`?m˘, the shape of the bumping route (scaled down by the factor 1 ? m ) converges in probability towards an explicit curve, which we refer to as the limit bumping curve, see Figure 3 for an illustration.
In the current paper we go beyond the scaling used by Romik and the last named author and investigate the part of the bumping route with the Cartesian coordinates of order x " Op1q and y " ? m. This part of the bumping curves was not visible on Figure 3; in order to reveal it one can use the semi-logarithmic plot, cf. Figures 4 and 5. 1.8. The naive hyperbola. The first step in this direction would be to stretch the validity of the results of Romik and the last named author [RŚ16] beyond their limitations and to expect that the limit bumping curve describes the asymptotics of the bumping routes also in this new scaling. This would correspond to the investigation of the asymptotics of the (nonrescaled) limit bumping curve`xpyq, y˘in the regime y Ñ 8. The latter analysis was performed by the first named author [Mar21]; one of his results is that lim yÑ8 xpyqy " 2; in other words, for y Ñ 8 the non-rescaled limit bumping curve can be approximated by the hyperbola xy " 2 while its rescaled counterpart which we consider in the current paper by the hyperbola which is shown on Figure 3 as the dashed line. At the very end of Section 1.10 we will discuss the extent to which this naive approach manages to confront the reality.
1.9. In which row a bumping route reaches a given column? Let us fix some (preferably infinite) standard Young tableau T . The bumping route in each step jumps to the next row, directly up or to the left to the original column; in other words T øm`1{2 p0q ě T øm`1{2 p1q ě¨¨ï s a weakly decreasing sequence of non-negative integers. For x, m P N 0 we denote by the index of the first row in which the bumping route T ø m`1{2 reaches the column with the index x (or less, if the bumping route skips the column x completely). For example, for the tableau T from Figure 2a we have If such a row does not exist we set Y x " 8; the following result shows that we do not have to worry about such a scenario. . Four sample bumping routes corresponding to an insertion T Ð m`1{2 for m " 100 and a random infinite standard Young tableau T which was sampled according to the Plancherel measure. In order to improve visibility, each bumping route is visualized as the plot of the corresponding function y Þ Ñ T øm`1{2 ptyuq, cf. Figure 2a, and not as a collection of boxes. Colour and thickness were added in order to help identify the routes. The solid line is the (rescaled) limit bumping curve. The dashed line is the hyperbola xy " 2m, cf. Equation (9).   Figure 4. In this kind of scaling when x " Op1q and y " ? m the results of Romik and the last named author [RŚ16] are not applicable and the rescaled limit bumping curve (the solid line) is shown for illustration purposes only. (11) p0 ă ξ 0 ă ξ 1 ă¨¨¨q on R`can be viewed concretely as the sequence of partial sums for a sequence pψ i q of independent, identically distributed random variables with the exponential distribution Expp1q. Thus a concrete way to express the convergence in Theorem 1.2 is to say that for each l P N 0 the joint distribution of the finite tuple of random variables , 1q.
In particular, for x " 0 it follows that the random variable Y rms 0 2m which measures the (rescaled) number of steps of the bumping route to reach the leftmost column converges in distribution, as m Ñ 8, to the Fréchet distribution of shape parameter α " 1: The Fréchet distribution has a heavy tail; in particular its first moment is infinite which provides a theoretical explanation for a bad time complexity of some of our Monte Carlo simulations. Equivalently, the random variable e´2 m Y 0 converges in distribution, as m Ñ 8, to the uniform distribution Up0, 1q on the unit interval. Figure 6 presents the results of Monte Carlo simulations which illustrate this result.
1.10. Projective convention for drawing Young diagrams. Usually in order to draw a Young diagram we use the French convention and the Oxy coordinate system, cf. Figure 2a. For our purposes it will be more convenient to change the parametrization of the coordinate y by setting m Y 0 . The thick red line corresponds to m " 1 (sample size is equal to 10 000); the blue line corresponds to m " 6 (sample size 3 000); the thin green line corresponds to m " 25 (sample size 2 500). The dashed line corresponds to the cumulative probability distribution function of the uniform distribution Up0, 1q on the unit interval. Due to constraints on computation time it was not possible to get Monte Carlo data for all values of u. The staircase feature of the plots is due to the discrete nature of the probability distribution of Y 0 .
This convention allows us to show an infinite number of rows of a given tableau on a finite piece of paper, cf. Figure 2b. We will refer to this way of drawing Young tableaux as the projective convention; it is somewhat reminiscent of the English convention in the sense that the numbers in the tableau increase along the columns from top to bottom.  Figure 2b). The dashed line x " z corresponds to the hyperbola (9); it is tangent in 0 to the rescaled limit bumping curve (the solid line); it is also the plot of the mean value of the Poisson process z Þ Ñ ENpzq.
In the projective convention the bumping route can be seen as the plot of the function (13) x proj T ,m pzq " T øm`1{2˜Z 2m z^¸f or z P Rs hown on Figure 2b as the thick red line.
With these notations Theorem 1.2 allows the following convenient reformulation. For an illustration see Figure 7.
Remark 1.6. In Theorem 1.5 above, the convergence in distribution for stochastic processes is understood as follows: for any finite collection z 1 , . . . , z l ą 0 we claim that the joint distribution of the tuple of random variableś x proj T ,m pz 1 q, . . . , x proj T ,m pz l qc onverges in the weak topology of probability measures, as m Ñ 8, to the joint distribution of the tuple of random variables Npz 1 q, . . . , Npz l q˘.
Proof of Theorem 1.5. The process (14) is a counting process. By the definition (13), the time of its k-th jump (for an integer k ě 1) is directly related to the number of the row in which the bumping route reaches the column with the index k´1. By Theorem 1.2 the joint distribution of the times of the jumps converges to the Poisson point process; it follows therefore that (14) converges to the Poisson counting process, as required.
The plot of the mean value of the standard Poisson process z Þ Ñ ENpzq is the straight line x " z which is shown on Figure 7 as the dashed line. Somewhat surprisingly it coincides with the hyperbola (9) shown in the projective coordinate system; a posteriori this gives some justification to the naive discussion from Section 1.8.
1.11. The main result with the right-to-left approach. Theorem 1.2 was formulated in a compact way which may obscure the true nature of this result. Our criticism is focused on the left-to-right approach from Remark 1.3 which might give a false impression that the underlying mechanism for generating the random variable 2m of the bumping route to the column number x`1 is based on generating first the random variable 2m Y rms x related to the previous column (that is the column directly to the left), and adding some 'waiting time' for the transition.
In fact, such a mechanism is not possible without the time travel because the chronological order of the events is opposite: the bumping route first visits the column x`1 and then lands in the column x. In the following we shall present an alternative, right-to-left viewpoint which explains better the true nature of Theorem 1.2.
For the Poisson point process (11) and an integer l ě 1 we consider the collection of random variables which consists of ξ l and the ratios R i :" ξ i`1 ξ i of consecutive entries of pξ i q. Then (15) are independent random variables with the distributions that can be found easily. This observation can be used to define ξ 0 , . . . , ξ l from the Poisson point process by setting With this in mind we may reformulate Theorem 1.2 as follows.
Theorem 1.7 (The main result, reformulated). For any integer l ě 0 the joint distribution of the tuple of random variableŝ converges, as m Ñ 8, to the joint distribution of the random variableŝ where ξ l , R l´1 , . . . , R 0 are independent random variables, the distribution of ξ l is equal to Erlangpl`1, 1q, and for each i ě 0 the distribution of the ratio R i is supported on r1, 8q with the power law The order of the random variables in (16) reflects the chronological order of the events, from left to right. Heuristically, (17) states that the transition of the bumping route from the column x`1 to the column x gives a multiplicative factor R x to the total waiting time, with the factors R 0 , R 1 , . . . independent.
It is more common in mathematical and physical models that the total waiting time for some event arises as a sum of some independent summands, so the multiplicative structure in Theorem 1.7 comes as a small surprise. We believe that this phenomenon can be explained heuristically as follows: when we study the transition of the bumping route from row y to the next row y`1, the probability of the transition from column x`1 to column x seems asymptotically to be equal to x`1 y`oˆ1 y˙f or fixed value of x, and for y Ñ 8.
This kind of decay would explain both the multiplicative structure ('if a bumping route arrives to a given column very late, it will stay in this column even longer') as well as the power law (18). We are tempted therefore to state the following conjecture which might explain the aforementioned transition probabilities of the bumping routes.
or fixed x ě 2 and y Ñ 8.
Furthermore, for each x P t1, 2, . . . u the set of points Numerical experiments are not conclusive and indicate interesting clustering phenomena for the random set (19).
1.12. Asymptotics of fixed m. The previous results concerned the bumping routes T ø m`1 2 in the limit m Ñ 8 as the inserted number tends to infinity. In the following we concentrate on another class of asymptotic problems which concern the fixed value of m.
The following result shows that (12) gives asymptotically a very good approximation for the distribution tail of Y rms 0 in the scaling when m is fixed and the number of the row y Ñ 8 tends to infinity.
This result is illustrated on Figure 6 in the behavior of each of the cumulative distribution functions in the neighborhood of u " 1. The proof is postponed to Section 5.1. Question 1.10. What can we say about the other columns, that is the tail for fixed values of x P N 0 and m ě 1, in the limit y Ñ 8?
1. 13. More open problems. Let T be a random Plancherel-distributed infinite standard Young tableau. We consider the bumping tree [Duz19] which is defined as the collection of all possible bumping routes for this tableaù which can be visualized, for example, as on Figure 8. Computer simulations suggest that the set of boxes which can be reached by some bumping route for a given tableau T is relatively 'small'. It would be interesting to state this vague observation in a meaningful way. We conjecture that the pictures such as Figure 8 which use the logarithmic scale for the y coordinate converge (in the scaling when x " Op1q is bounded and y Ñ 8) to some meaningful particle jump-and-coalescence process.
1.14. Overview of the paper. Sketch of the proof of Theorem 1.2. As we already mentioned, the detailed proof of Theorem 1.2 is postponed to Section 5.3. In the following we present an overview of the paper and a rough sketch of the proof.
1.14.1. Trajectory of infinity. Lazy parametrization of the bumping route. Without loss of generality we may assume that the Plancherel-distributed infinite tableau T from the statement of Theorem 1.2 is of the form T " Qpξ 1 , ξ 2 , . . . q for a sequence ξ 1 , ξ 2 , . . . of independent, identically distributed random variables with the uniform distribution Up0, 1q.
We will iteratively apply Schensted row insertion to the entries of the infinite sequence which is the initial sequence ξ with our favorite symbol 8 inserted at the position m`1. At step m`1 the symbol 8 is inserted at the end of the bottom row; as further elements of the sequence (20) are inserted, the symbol 8 stays put or is being bumped to the next row, higher and higher. In Proposition 3.1 we will show that the trajectory of 8 in this infinite sequence of Schensted row insertions oincides with the bumping route T ø m`1{2. Thus our main problem is equivalent to studying the time evolution of the position of 8 in the infinite sequence of row insertions (21). This time evolution also provides for a given Plancherel-distributed random infinite standard Young tableau. The y axis is shown using the logarithmic scale. In order to improve visibility, each bumping route was drawn as a piecewise-affine function connecting the points (4) and not as a jump function as in Figure 2a.
a convenient alternative parametrization of the bumping route, called lazy parametrization.
1.14.2. Augmented Young diagrams. For t ě m we consider the insertion tableau which appears at an intermediate step in (21) after some finite number of row insertions was performed. By removing the information about the entries of the tableau T ptq we obtain the shape of T ptq , denoted by sh T ptq , which is a Young diagram with t`1 boxes. In the following we will explain how to modify the notion of the shape of a tableau so that it better fits our needs. Let us remove from the tableau T ptq the numbers ξ 1 , . . . , ξ t and let us keep only the information about the position of the box which contains the symbol 8. The resulting object, called augmented Young diagram (see Figure 9 for an illustration), can be regarded as a pair Λ ptq " pλ, lq which consists of: ‚ the Young diagram λ with t boxes which keeps track of the positions of the boxes with the entries ξ i , i P t1, . . . , tu, in T ptq ; ‚ the outer corner l of λ which is the position of the box with 8 in T ptq . We will say that sh˚T ptq " Λ ptq is the augmented shape of T ptq .
The set of augmented Young diagrams, denoted Y˚, has a structure of an oriented graph which is directly related to Schensted row insertion, as follows. For a pair of augmented Young diagrams Λ, r Λ P Y˚we say that Λ Õ r Λ if there exists a tableau T (which contains exactly one entry equal to 8) such that Λ " sh˚T and there exists some number x such that r Λ " sh˚pT Ð xq, see Figure 10 and Section 3.4 for more details.
With these notations the time evolution of the position of 8 in the sequence of row insertions (21) can be extracted from the sequence of the corresponding augmented shapes 1.14.3. Augmented Plancherel growth processes. The random sequence (23) is called the augmented Plancherel growth process initiated at time m; in Section 3.6 we will show that it is a Markov chain with dynamics closely related to the usual (i.e., non-augmented) Plancherel growth process. Since we have a freedom of choosing the value of the integer m P N 0 , we get a whole family of augmented Plancherel growth processes. It turns out that the transition probabilities for these Markov chains do not depend on the value of m.
Our strategy is to use the Markov property of augmented Plancherel growth processes combined with the following two pieces of information.
‚ Probability distribution at a given time t. In Proposition 3.9 we give an asymptotic description of the probability distribution of Λ ptq in the scaling when m, t Ñ 8 in such a way that t " Θpm 2 q. ‚ Averaged transition probabilities. In Proposition 4.2 we give an asymptotic description of the transition probabilities for the augmented Plancherel growth processes between two moments of time n and n 1 (with n ă n 1 ) in the scaling when n, n 1 Ñ 8.
Thanks to these results we will prove Theorem 4.3 which gives an asymptotic description of the probability distribution of the trajectory of the symbol 8 or, equivalently, the bumping route in the lazy parametrization. Finally, in Section 5 we explain how to translate this result to the non-lazy parametrization of the bumping route in which the boxes of the bumping route are parametrized by the index of the row; this completes the proof of Theorem 1.2.
The main difficulty lies in the proofs of the aforementioned Proposition 3.9 and Proposition 4.2; in the following we sketch their proofs.
1.14.4. Probability distribution of the augmented Plancherel growth process at a given time. In order to prove the aforementioned Proposition 3.9 we need to understand the probability distribution of the augmented shape of the insertion tableau T ptq given by (22) in the scaling when m " O`?t˘. Thanks to some symmetries of the RSK algorithm, the tableau T ptq is equal to the transpose of the insertion tableau The remaining difficulty is therefore to understand the probability distribution of the augmented Plancherel growth process initiated at time m 1 , after additional m steps of Schensted row insertion were performed. We are interested in the asymptotic setting when m 1 Ñ 8 and the number of additional steps m " O`?m 1˘i s relatively small. This is precisely the setting which was considered in our recent paper about the Poisson limit theorem for the Plancherel growth process [MMŚ20]. We summarize these results in Section 2; based on them we prove in Proposition 3.6 that the index of the row of the symbol 8 in the tableau (24) is asymptotically given by the Poisson distribution.
By taking the transpose of the augmented Young diagrams we recover Proposition 3.9, as desired.
1.14.5. Averaged transition probabilities. We will sketch the proof of the aforementioned Proposition 4.2 which concerns an augmented Plancherel growth process for which the initial probability distribution at time n is given by Λ pnq " λ pnq , l pnq˘, where λ pnq is a random Young diagram with n boxes distributed (approximately) according to the Plancherel measure and l pnq is its outer corner located in the column with the fixed index k. Our goal is to describe the probability distribution of this augmented Plancherel growth process at some later time n 1 , asymptotically as n, n 1 Ñ 8.
Our first step in this direction is to approximate the probability distribution of the Markov process (25) by a certain linear combination (with real, positive and negative, coefficients) of the probability distributions of augmented Plancherel growth processes initiated at time m. This linear combination is taken over the values of m which are of order O`?n˘. Finding such a linear combination required the results which we discussed above in Section 1.14.4, namely a good understanding of the probability distribution at time n of the augmented Plancherel growth process initiated at some specified time m " O`?n˘.
The probability distribution of Λ pn 1 q is then approximately equal to the aforementioned linear combination of the laws (this time evaluated at time n 1 ) of the augmented Plancherel growth processes initiated at some specific times m. This linear combination is straightforward to analyze because for each individual summand the results from Section 1.14.4 are applicable. This completes the sketch of the proof of Proposition 4.2.

GROWTH OF THE BOTTOM ROWS
In the current section we will gather some results and some notations from our recent paper [MMŚ20, Section 2] which will be necessary for the purposes of the current work.
2.1. Total variation distance. Suppose that µ and ν are (signed) measures on the same discrete set S. Such measures can be identified with real-valued functions on S. We define the total variation distance between the measures µ and ν (26) δpµ, νq :" 1 2 }µ´ν} ℓ 1 as half of their ℓ 1 distance as functions. If X and Y are two random variables with values in the same discrete set S, we define their total variation distance δpX, Y q as the total variation distance between their probability distributions (which are probability measures on S). Usually in the literature the total variation distance is defined only for probability measures. In such a setup the total variation distance can be expressed as In the current paper we will occasionally use the notion of the total variation distance also for signed measures for which (26) and (27) are not equivalent.
2.2. Growth of rows in Plancherel growth process. Let λ p0q Õ λ p1q Õ¨¨b e the Plancherel growth process. For integers n ě 1 and r P N 0 we denote by E pnq r the random event which occurs if the unique box of the skew diagram λ pnq {λ pn´1q is located in the row with the index r.
Let us fix an integer k P N 0 . We define N " t0, 1, . . . , k, 8u. For n ě 1 we define the random variable R pnq P N which is given by occurs for some r ą k.
Let ℓ " ℓpnq be a sequence of non-negative integers such that ℓ " O`?n˘.
For a given integer n ě pk`1q 2 we focus on the specific part of the Plancherel growth process (28) λ pnq Õ¨¨¨Õ λ pn`ℓq .
We will encode some partial information about the growths of the rows as well as about the final Young diagram in (28) by the random vector (29) V pnq "´R pn`1q , . . . , R pn`ℓq , λ pn`ℓq¯P N ℓˆY .
We also consider the random vector which is defined as a sequence of independent random variables; the random variables R pn`1q , . . . , R pn`ℓq have the same distribution given by 1´k`1 ? n and λ pn`ℓq is distributed according to the Plancherel measure Plan n`ℓ ; in particular the random variables λ pn`ℓq and λ pn`ℓq have the same distribution.
Heuristically, the following result states that when the Plancherel growth process is in an advanced stage and we observe a relatively small number of its additional steps, the growths of the bottom rows occur approximately like independent random variables. Additionally, these growths do not affect too much the final shape of the Young diagram.
Theorem 2.2 ([MMŚ20, Theorem 2.2]). With the above notations, for each fixed k P N 0 the total variation distance between V pnq and V pnq converges to zero, as n Ñ 8; more specifically δ´V pnq , V pnq¯" oˆℓ ? n˙.

AUGMENTED PLANCHEREL GROWTH PROCESS
In this section we will introduce our main tool: the augmented Plancherel growth process which keeps track of the position of a very large number in the insertion tableau when new random numbers are inserted.
3.1. Lazy parametrization of bumping routes. Our first step towards the proof of Theorem 1.2 is to introduce a more convenient parametrization of the bumping routes. In (4) we used y, the number of the row, as the variable which parametrizes the bumping route. In the current section we will introduce the lazy parametrization.
Let us fix a (finite or infinite) standard Young tableau T and an integer m P N 0 . For a given integer t ě m we denote by l lazy T ,m ptq "´x lazy T ,m ptq, y lazy T ,m ptqt he coordinates of the first box in the bumping route T ø m`1{2 which contains an entry of T which is bigger than t. If such a box does not exists, this means that the bumping route is finite, and all boxes of the tableau T which belong to the bumping route are ď t. If this is the case we define l lazy T ,m ptq to be the last box of the bumping route, i.e. the box of the bumping route which lies outside of T . We will refer to (31) t Þ Ñ`x lazy T ,m ptq, y lazy T ,m ptqȃ s the lazy parametrization of the bumping route.
Clearly, the set of values of the function (31) coincides with the bumping route understood in the traditional way (4).
We denote by T | ďt the outcome of keeping only these boxes of T which are at most t. Note that the element of the bumping route (32) l lazy T ,m ptq " sh`T | ďt Ð m`1{2˘{ sh`T | ďtȋ s the unique box of the difference of two Young diagrams on the right-hand side.
3.2. Trajectory of 8. Let ξ " pξ 1 , ξ 2 , . . . q be a sequence of independent, identically distributed random variables with the uniform distribution Up0, 1q on the unit interval r0, 1s and let m ě 0 be a fixed integer. We will iteratively apply Schensted row insertion to the entries of the infinite sequence ξ 1 , . . . , ξ m , 8, ξ m`1 , ξ m`2 , . . . The following result shows a direct link between the above problem and the asymptotics of bumping routes. This result also shows an interesting link between the papers [RŚ16] and [Mar21].
Proposition 3.1. Let ξ 1 , ξ 2 , . . . be a (non-random or random) sequence and T " Qpξ 1 , ξ 2 , . . . q be the corresponding recording tableau. Then for each m P N the bumping route in the lazy parametrization coincides with the trajectory of 8 as defined in (33): (34) l lazy T ,m ptq " l traj m ptq for each integer t ě m.
We will provide two proofs of Proposition 3.1. The first one is based on the following classic result of Schützengerger. The first proof of Proposition 3.1. Let π " pπ 1 , . . . , π t q P S t be the permutation generated by the sequence pξ 1 , . . . , ξ t q, that is the unique permutation such that for any choice of indices i ă j the condition π i ă π j holds true if and only if ξ i ď ξ j . Let π´1 " pπ´1 1 , . . . , π´1 t q be the inverse of π. Since RSK depends only on the relative order of entries, the restricted tableau T | ďt is equal to (35) T | ďt " Qpξ 1 , . . . , ξ t q " Qpπq " P pπ´1q.
The above proof has an advantage of being short and abstract. The following alternative proof highlights the 'dynamic' aspects of the bumping routes and the trajectory of infinity.
The second proof of Proposition 3.1. We use induction over the variable t.
The induction base t " m is quite easy: l lazy T ,m pmq is the leftmost box in the bottom row of T which contains a number which is bigger than m. This box is the first to the right of the last box in the bottom row in the tableau Qpξ 1 , . . . , ξ m q. On the other hand, since this recording tableau has the same shape as the insertion tableau P pξ 1 , . . . , ξ m q, it follows that l traj m pmq " l lazy T ,m pmq and the proof of the induction base is completed. We start with an observation that 8 is bumped in the process of calculating the row insertion if and only if the position of 8 at time t, that is l traj m ptq, is the unique box which belongs to the skew diagram RSKpξ 1 , . . . , ξ t`1 q{ RSKpξ 1 , . . . , ξ t q.
The latter condition holds true if and only if the entry of T located in the box l traj m ptq fulfills T l traj m ptq " t`1. In order to make the induction step we assume that the equality (34) holds true for some t ě m. There are the following two cases.

Case 1. Assume that the entry of T located in the box l lazy
T ,m ptq is strictly bigger than t`1. In this case the lazy bumping route stays put and l lazy T ,m pt`1q " l lazy T ,m ptq.
By the induction hypothesis, the entry of T located in the box l traj m ptq " l lazy T ,m ptq is bigger than t`1. By the previous discussion, 8 is not bumped in the process of calculating the row insertion (36) hence l traj m pt`1q " l traj m ptq and the inductive step holds true.
Case 2. Assume that the entry of T located in the box l lazy T ,m ptq is equal to t`1. In this case the lazy bumping route moves to the next row. It follows that l lazy T ,m pt`1q is the leftmost box of T in the row above l lazy T ,m ptq which contains a number which is bigger than T l lazy T ,m ptq " t`1. By the induction hypothesis, T l traj m ptq " T l lazy T ,m ptq " t`1, so 8 is bumped in the process of calculating the row insertion (36) to the next row r. The box l traj m pt`1q is the first to the right of the last box in the row r in RSKpξ 1 , . . . , ξ t , ξ t`1 q. Clearly, this is the box in the row r of T which has the least entry among those which are bigger than t`1, so it is the same as l lazy T ,m pt`1q.
3.3. Augmented Young diagrams. Augmented shape of a tableau. For the motivations and heuristics behind the notion of augmented Young diagrams see Section 1.14.2. A pair Λ " pλ, lq will be called an augmented Young diagram if λ is a Young diagram and l is one of its outer corners, see Figure 9b. We will say that λ is the regular part of Λ and that l is the special box of Λ.
The set of augmented Young diagrams will be denoted by Y˚and for n P N 0 we will denote by Yn the set of augmented Young diagrams pλ, lq with the additional property that λ has n boxes (which we will shortly denote by |λ| " n).
Suppose T is a tableau with the property that exactly one of its entries is equal to 8. We define the augmented shape of T sh˚T "´sh`T zt8u˘, Pos 8 T¯P Y˚, as the pair which consists of (a) the shape of T after removal of the box with 8, and (b) the location of the box with 8 in T , see Figure 9.
3.4. Augmented Young graph. The set Y˚can be equipped with a structure of an oriented graph, called augmented Young graph. We declare that a pair Λ Õ r Λ forms an oriented edge (with Λ " pλ, lq P Y˚and The augmented shape sh˚T " pλ, lq of the tableau T . The regular part λ " p4, 2, 1q is shown as the Young diagram drawn with solid lines, the position of the special box l " px l , y l q " p2, 1q is marked as the decorated box drawn with the dotted lines. With the notations of Section 4.2 this augmented Young diagram corresponds to px l , λq " p2, λq P N 0ˆY . r Λ " p r λ, r lq P Y˚) if the following two conditions hold true: (37) λ Õ r λ and r l " the outer corner of r λ which is in the row above l if r λ{λ " tlu , l otherwise, see Figure 9 for an illustration. If Λ Õ r Λ (with Λ " pλ, lq P Y˚and r Λ " p r λ, r lq P Y˚) are such that l ‰ r l (which corresponds to the first case on the right-hand side of (37)), we will say that the edge Λ Õ r Λ is a bump. The above definition was specifically tailored so that the following simple lemma holds true.

Lemma 3.3. Assume that T is a tableau which has exactly one entry equal to 8 and let x be some finite number. Theǹ
sh˚T˘Õ`sh˚pT Ð xq˘.
The position r l of 8 in T Ð x is either: ‚ in the row immediately above the position l of 8 in T (this happens exactly if 8 was bumped in the insertion T Ð x; equivalently if r λ{λ " tlu), or ‚ the same as the position l of 8 in T (this happens exactly when 8 was not bumped; equivalently if r λ{λ ‰ tlu). Clearly these two cases correspond to the second condition in (37) which completes the proof.
3.5. Lifting of paths. We consider the 'covering map' p : Y˚Ñ Y given by taking the regular part Lemma 3.4. For any Λ pmq P Y˚and any path in the Young graph with a specified initial element λ pmq " p`Λ pmq˘t here exists the unique lifted path Λ pmq Õ Λ pm`1q Õ¨¨¨P Yi n the augmented Young graph with the specified initial element Λ pmq , and such that λ ptq " p`Λ ptq˘h olds true for each t P tm, m`1, . . . u.
Proof. From (37) it follows that for each pλ, lq P Y˚and each r λ such that λ Õ r λ there exists a unique r l such that pλ, lq Õ p r λ, r lq. This shows that, given Λ piq , the value of Λ pi`1q is determined uniquely. This observation implies that the lemma can be proved by a straightforward induction.
3.6. Augmented Plancherel growth process. We keep the notations from the beginning of Section 3.2, i.e., we assume that ξ " pξ 1 , ξ 2 , . . . q is a sequence of independent, identically distributed random variables with the uniform distribution Up0, 1q on the unit interval r0, 1s and m ě 0 is a fixed integer. We consider a path in the augmented Young graph  ith the initial condition that the special box l pmq m is the outer corner of λ pmq which is located in the bottom row. It follows that for any augmented Young diagrams Σ m , . . . , Σ t`1 P Y˚with the regular parts σ m , . . . , σ t`1 P Y The sequence of the regular parts (41) forms the usual Plancherel growth process (with the first m entries truncated) hence it is a Markov chain (the proof that the usual Plancherel growth process is a Markov chain can be found in [Ker99, Sections 2.2 and 2.4]). It follows that the probability on the top of the right-hand side of (42) can be written in the product form in terms of the probability distribution of λ pmq and the transition probabilities for the Plancherel growth process.
We compare (42) with its counterpart for t :" t´1; this shows that the conditional probability s equal to the right-hand side of (40) for r Λ :" Σ t`1 and Λ :" Σ t . In particular, this conditional probability does not depend on the values of Σ m , . . . , Σ t´1 and the Markov property follows.
The special box in the augmented Plancherel growth process can be thought of as a test particle which provides some information about the local behavior of the usual Plancherel growth process. From this point of view it is reminiscent of the second class particle in the theory of interacting particle systems or jeu de taquin trajectory for infinite tableaux [RŚ15]. 3.7. Probability distribution of the augmented Plancherel growth process. Proposition 3.6 and Proposition 3.9 below provide information about the probability distribution of the augmented Plancherel growth process at time t for t Ñ 8 in two distinct asymptotic regimes: very soon after the augmented Plancherel process was initiated (that is when t " mÒ p ? mq, cf. Proposition 3.6) and after a very long time after the augmented Plancherel process was initiated (that is when t " Θpm 2 q " m, cf. Proposition 3.9).
Proposition 3.6. Let z ą 0 be a fixed positive number and let t " tpmq be a sequence of positive integers such that tpmq ě m and with the property Õ¨¨¨be the augmented Plancherel growth process initiated at time m. We denote Λ

c) The total variation distance between
‚ the probability distribution of the random vector (43)´λ ptq , y ptq m¯P YˆN 0 and ‚ the product measure Plan tˆP oispzq converges to 0, as m Ñ 8.
Let us fix an integer k ě 0. We use the notations from Section 2.2 for n :" m and ℓ " t´m so that n`ℓ " t; we assume that m is big enough so that m ě pk`1q 2 . Our general strategy is to read the required information from the vector V pnq given by (29) and to apply Theorem 2.2. Before the proof of Proposition 3.6 we start with the following auxiliary result.
For s ě m we define the random variable y We also define the random variable F s P t0, . . . , ku by   By iteratively applying this observation to the previous values (47) it is possible to express the value of y psq m Ó N purely in terms of (48). Also the value of F s " F s´R pm`1q , . . . , R psqc an be expressed as a function of the entries of the sequence R related to the past, as required.
Proof of Proposition 3.6. Lemma 3.7 shows that the event y ptq m " k can be expressed in terms of the vector V pnq given by (29). We apply Theorem 2.2; it follows that the probability P ! y ptq m " k ) is equal, up to an additive error term op1q, to the probability that there are exactly k values of the index u P tm`1, . . . , tu with the property that We denote by A u the random event that the equality (49) holds true.
Let i 1 ă¨¨¨ă i l be an increasing sequence of integers from the set tm`1, . . . , tu for l ě 1. We will show that Indeed, by Lemma 3.7, the event A i 1 X¨¨¨X A i l´1 is a disjoint finite union of some random events of the form over some choices of r m`1 , r m`2 , . . . , r j P N , where j :" i l´1 . Since the random variables`R piq˘a re independent, it follows that By summing over the appropriate values of r m`1 , . . . , r j P N the equality (50) follows. By iterating (50) it follows that the events A m`1 , . . . , A t are independent and each has equal probability 1 ? m . By the Poisson limit theorem [Dur10, Theorem 3.6.1] the probability of k successes in ℓ Bernoulli trials as above converges to the probability of the atom k in the Poisson distribution with the intensity parameter equal to which concludes the proof of part a).
The above discussion also shows that the conditional probability distribution considered in point b) is equal to the conditional probability distribution of the last coordinate λ ptq of the vector V pnq under certain condition which is expressed in terms of the coordinates R pm`1q , . . . , R ptq . By Theorem 2.2 this conditional probability distribution is in the distance op1q (with respect to the total variation distance) to its counterpart for the random vector V pnq .
The latter conditional probability distribution, due to the independence of the coordinates of V pnq , is equal to the Plancherel measure Plan t , which concludes the proof of b).
Part c) is a direct consequence of parts a) and b).
Proposition 3.9. Let z ą 0 be a fixed real number. Let t " tpmq be a sequence of positive integers such that tpmq ě m and with the property that Õ¨¨¨be the augmented Plancherel growth process initiated at time m. We denote Λ The total variation distance between ‚ the probability distribution of the random vector (52)´x ptq m , λ ptq¯P N 0ˆY and ‚ the product measure PoispzqˆPlan t converges to 0, as m Ñ 8.
Proof. By Lemma 3.8 the probability distribution of (52) coincides with the probability distribution of (53)´y ptq m 1 , " λ ptq ‰ Tf or m 1 :" t´m. The random vector (53) can be viewed as the image of the vector`y ptq m 1 , λ ptq˘u nder the bijection idˆT : py, λq Þ Ñ py, λ T q.
By Proposition 3.6 it follows that the total variation distance between (53) and the push-forward measure pidˆT q`PoispzqˆPlan t˘" PoispzqˆPlan t converges to zero as m Ñ 8; the last equality holds since the Plancherel measure is invariant under transposition.
3.8. Lazy version of Proposition 1.9. Proof of Proposition 1.1. In Section 1.9 we parametrized the shape of the bumping route by the sequence Y 0 , Y 1 , . . . which gives the number of the row in which the bumping route reaches a specified column, cf. (10). With the help of Proposition 3.1 we can define the lazy counterpart of these quantities: for x, m P N 0 we denote by T rms the time it takes for the bumping route (in the lazy parametrization) to reach the specified column.
The following result is the lazy version of Proposition 1.9.
Lemma 3.10. For each integer m ě 1 Proof. By Lemma 3.8, for any u P N 0 ) .
In the special case m " 1 the proof is particularly easy: the right-hand side is equal to P´E puq 0¯a nd Proposition 2.1 provides the necessary asymptotics.
For the general case m ě 1 we use the notations from Section 2.2 for k " 0, and n " u´m, and ℓ " m. The event y puq u´m ě 1 occurs if and only if at least one of the numbers R pn`1q , . . . , R pn`ℓq is equal to 0. We apply Theorem 2.2; it follows that the probability of the latter event is equal, up to an additive error term of the order o´m ?
u´m¯" o´1 ? u¯, to the probability that in m Bernoulli trials with success probability 1 ? n there is at least one success. In this way we proved that as desired.
ě¨¨¨is a weakly decreasing sequence, it is enough to consider the case x " 0. We apply Lemma 3.10 in the limit u Ñ 8. It follows that the probability that the bumping route T ø m`1{2 does not reach the column with the index 0 is equal to as required.

TRANSITION PROBABILITIES FOR THE AUGMENTED PLANCHEREL GROWTH PROCESS
Our main result in this section is Theorem 4.3. It will be the key tool for proving the main results of the current paper.
and then to consider its k-th derivative with respect to the parameter z for z " 0. This is not exactly a solution to the original question (the derivative is not a linear combination), but since the derivative can be approximated by the forward difference operator, this naive approach gives a hint that an expression such as (54) in the special case p " 1 might be, in fact, a good answer. is a probability measure on N 0 . As h Ñ 0, the measure ν k,p,h converges (in the sense of total variation distance) to the binomial distribution Binompk, pq.
Proof. The special case p " 1. For a function f on the real line we consider its forward difference function ∆rf s given by ∆rf spxq " f px`1q´f pxq.
It follows that the iterated forward difference is given by A priori, ν k,1,h is a signed measure with the total mass equal to The right-hand side of (55) is equal to 1, since the forward difference of an exponential function is again an exponential: The atom of ν k,1,h at an integer m ě 0 is equal to Note that the monomial x m can be expressed in terms of the falling factorials x p with the coefficients given by the Stirling numbers of the second kind: When we evaluate the above expression at x " 0, there is only one non-zero summand and the above expression is non-zero only for m ě k. All in all, ν k,1,h is a probability measure on N 0 , as required. It follows that the total variation distance between Binompk, 1q " δ k and ν k,1,h is equal to ÝÝÑ 0, as required.
The general case. For a signed measure µ which is supported on N 0 and 0 ď p ď 1 we define the signed measure C p rµs on N 0 by In the case when µ is a probability measure, C p rµs has a natural interpretation as the probability distribution of a compound binomial random variable BinompM, pq, where M is a random variable with the probability distribution given by µ.
It is easy to check that for any 0 ď q ď 1 the image of a binomial distribution C p rBinompn, qqs " Binompn, pqq is again a binomial distribution, and for any λ ě 0 the image of a Poisson distribution C p rPoispλqs " Poisppλq is again a Poisson distribution. Since C p is a linear map, by the very definition (54) it follows that (56) C p rν k,1,h s " ν k,p,h ; in particular the latter is a probability measure, as required. By considering the limit h Ñ 0 of (56) we get lim hÑ0 ν k,p,h " C p " Binompk, 1q ‰ " Binompk, pq in the sense of total variation distance, as required.

4.2.
The inclusion Y˚Ă N 0ˆY . We will extend the meaning of the notations from Section 3.3 to a larger set. The map where l " px l , y l q, allows us to identify Y˚with a subset of N 0ˆY . For a pair px, λq P N 0ˆY we will say that λ is its regular part.
We define the edges in this larger set N 0ˆY Ą Y˚as follows: we declare that px, λq Õ pr x, r λq if the following two conditions hold true: (58) λ Õ r λ and r x " if the unique box of r λ{λ is located in the column x, x otherwise.
In this way the oriented graph Y˚is a subgraph of N 0ˆY . An analogous lifting property as in Lemma 3.4 remains valid if we assume that the initial element Λ pmq P N 0ˆY and the elements of the lifted path Λ pmq Õ Λ pm`1q Õ¨¨¨P N 0ˆY are allowed to be taken from this larger oriented graph.
With these definitions the transition probabilities (40) also make sense if Λ, r Λ P N 0ˆY are taken from this larger oriented graph and can be used to define Markov chains valued in N 0ˆY . The latter encourages us to consider a general class of Markov chains (61)´`x ptq , λ ptq˘¯t ěn valued in N 0ˆY Ą Y˚, for which the transition probabilities are given by (40) and for which the initial probability distribution of`x pnq , λ pnqc an be arbitrary. We will refer to each such a Markov chain as augmented Plancherel growth process.
Proposition 4.2. Let an integer k P N 0 and a real number 0 ă p ă 1 be fixed, and let n 1 " n 1 pnq be a sequence of integers such that n 1 ě n and lim nÑ8 c n n 1 " p. For a given integer n ě 0 let (61) be an augmented Plancherel growth process with the initial probability distribution at time n given by δ kˆP lan n .
Proof. Let ǫ ą 0 be given. By Lemma 4.1 there exists some h ą 0 with the property that for each q P t1, pu the total variation distance between the measure ν k,q,h defined in (54) and the binomial distribution Binompk, qq is bounded from above by ǫ.
Let T be a map defined on the set of probability measures on N 0ˆYn in the following way. For a probability measure µ on N 0ˆYn consider the augmented Plancherel growth process (61) with the initial probability distribution at time n given by µ and define T µ to be the probability measure on N 0ˆYn 1 which gives the probability distribution of`x pn 1 q , λ pn 1 q˘a t time n 1 .
It is easy to extend the map T so that it becomes a linear map between the vector space of signed measures on N 0ˆYn and the vector space of signed measures on N 0ˆYn 1 . We equip both vector spaces with a metric which corresponds to the total variation distance. Then T is a contraction because of Markovianity of the augmented Plancherel growth process.
For m P t0, . . . , nu and t ě n we denote by µ m ptq the probability measure on N 0ˆY , defined by the probability distribution at time t of the augmented Plancherel growth process`x ptq m , λ ptq˘i nitiated at time m. For the aforementioned value of h ą 0 we consider the signed measure on N 0ˆYt given by the linear combination p´1q k´jˆk j˙e jh µ t jh ? nu ptq (which is well-defined for sufficiently big values of n which assure that kh ? n ă n ď t). We apply Proposition 3.9; it follows that for any j P t0, . . . , ku the total variation distance between µ tjh ? nu pnq and the product measure PoispjhqˆPlan n converges to 0, as n Ñ 8; it follows that the total variation distance between Ppnq and the product measure (63) ν k,1,hˆP lan n converges to 0, as n Ñ 8. On the other hand, the value of h ą 0 was selected in such a way that the total variation distance between the probability measure (63) and the product measure (64) δ kˆP lan n is smaller than ǫ. In this way we proved that An analogous reasoning shows that lim sup nÑ8 δ ! Ppn 1 q, Binompk, pqˆPlan n 1 ) ď ǫ.
The image of Ppnq under the map T can be calculated by linearity of T : By the triangle inequality and the observation that the map T is a contraction, (62) ď δ "´x pn 1 q , λ pn 1 q¯, Ppn 1 q holds true for sufficiently big values of n, as required.

4.4.
Bumping route in the lazy parametrization converges to the Poisson process. Let`Nptq : t ě 0˘denote the Poisson counting process which is independent from the Plancherel growth process λ p0q Õ λ p1q Õ¨¨¨. The following result is the lazy version of Theorem 1.5. For each 1 ď i ď l let t i " t i pmq be a sequence of positive integers such that We assume that t 1 ď¨¨¨ď t l . Then the total variation distance between ‚ the probability distribution of the vector (65)´x pt 1 q m , . . . , x pt l q m , λ pt l q¯, and ‚ the probability distribution of the vector (66)´Npz 1 q, . . . , Npz l q, λ pt l qc onverges to 0, as m Ñ 8.
Proof. We will perform the proof by induction over l. Its main idea is that the collection of the random vectors (65) over l P t1, 2, . . . u forms a Markov chain; the same holds true for the analogous collection of the random vectors (66). We will compare their initial probability distributions (thanks to Proposition 3.9) and -in a very specific sense -we will compare the kernels of these Markov chains (with Proposition 4.2). We present the details below.
The induction base l " 1 coincides with Proposition 3.9.
In the light of the general definition (61) of the augmented Plancherel growth process, the measures (67) and (68) on N l`1 0ˆY can be viewed as applications of the same Markov kernel (which correspond to the last factors on the right-hand side of (67) and (68)) P "´x pt l`1 q m , λ pt l`1 q¯" px l`1 , λqˇˇˇˇ´x pt l q m , λ pt l q¯" px l , µq to two specific initial probability distributions. Since such an application of a Markov kernel is a contraction (with respect to the total variation distance), we proved in this way that the total variation distance between (67) and (68) is bounded from above by the total variation distance between the initial distributions, that is the random vectors (65) and (66). By the inductive hypothesis the total variation distance between the measures P and Q converges to zero as m Ñ 8. The remaining difficulty is to understand the asymptotic behavior of the measure Q.
It is easy to check that P`Npz l`1 q " x l`1ˇN pz l q " x l˘" Binomˆx l , z l`1 z l˙p x l`1 q.
Hence the probability of the binomial distribution which appears as the last factor on the right-hand side of (70) can be interpreted as the conditional probability distribution of the Poisson process in the past, given its value in the future. We show that the Poisson counting process with the reversed time is also a Markov process. Since the Poisson counting process has independent increments, the probability of the event Npz 1 q, . . . , Npz l q˘" px 1 , . . . , x l q can be written as a product; an analogous observation is valid for l :" l`1. Due to cancellations of the factors which contribute to the numerator and the denominator, the following conditional probability can be simplified: P´Npz l`1 q " x l`1ˇ`N pz 1 q, . . . , Npz l q˘" px 1 , . . . , x l q¯" " P !`N pz 1 q, . . . , Npz l`1 q˘" px 1 , . . . , x l`1 q ) P !`N pz 1 q, . . . , Npz l q˘" px 1 , . . . , x l q ) " " P`Npz l q " x l^N pz l`1 q " x l`1P`N pz l q " x l˘" " P`Npz l`1 q " x l`1ˇN pz l q " x l˘.
Let N denote the Poisson point process with the uniform unit intensity on R 2 . For s, t P R`we denote by N s,t " N`r0, ssˆr0, tst he number of sampled points in the specified rectangle. Note that the results of the current paper show the convergence of the marginals which correspond to (a) fixed value of s and all values of t ą 0 (cf. Theorem 4.3), or (b) fixed value of t and all values of s ą 0 (this is a corollary from the proof of Proposition 3.9).
It is a bit discouraging that the contour curves obtained in computer experiments (see Figure 11) do not seem to be counting the number of points from some set which belong to a specified rectangle, see Figure 12 for comparison. On the other hand, maybe the value of m used in our experiments was not big enough to reveal the asymptotic behavior of these curves.

REMOVING LAZINESS
Most of the considerations above concerned the lazy parametrization of the bumping routes. In this section we will show how to pass to the parametrization by the row number and, in this way, to prove the remaining claims from Section 1 (that is Theorem 1.2 and Proposition 1.9).  5.1. Proof of Proposition 1.9. Our general strategy in this proof is to use Lemma 3.10 and to use the observation that a Plancherel-distributed random Young diagram with n boxes has approximately 2 ? n columns in the scaling when n Ñ 8.
Proof of Proposition 1.9. We denote by c pnq the number of rows (or, equivalently, the length of the leftmost column) of the Young diagram λ pnq . Our proof will be based on an observation (recall Proposition 3.1) that Let ǫ ą 0 be fixed. Since c pnq has the same distribution as the length of the bottom row of a Plancherel-distributed random Young diagram with n boxes, the large deviation results [DZ99;Sep98] show that there exists a constant C ǫ ą 0 such that (75) P¨sup n 0¯ď oˆ1 n 0i n the limit as n 0 Ñ 8.
Consider an arbitrary integer y ě 1. Assume that (i) the event on the lefthand side of (75) does not hold true for n 0 :" y, and (ii) Y By considering two possibilities: either the event on the left-hand side of (75) holds true for n 0 :" y or not, it follows that For the lower bound, assume that (i) the event on the left-hand side of (75) does not hold true for Regretfully, the ideas used in the proof of Proposition 1.9 (cf. Section 5.1 above) are not directly applicable for the proof of Proposition 5.1 when x ě 1 because we are not aware of suitable large deviation results for the lower tail of the distribution of a specific row a Plancherel-distributed Young diagram, other than the bottom row.
Our general strategy in this proof is to study the length µ ptq x of the column with the fixed index x in the Plancherel growth process λ ptq , as t Ñ 8. Since we are unable to get asymptotic uniform bounds for (77)ˇˇˇˇµ ptq x ? t´2ˇˇˇˇ over all integers t such that t m 2 belongs to some compact subset of p0, 8q in the limit m Ñ 8, as a substitute we consider a finite subset of p0, 8q of the form ! cp1`ǫq, . . . , cp1`ǫq l ) for arbitrarily small values of c, ǫ ą 0 and arbitrarily large integer l ě 0 and prove the appropriate bounds for the integers t i pmq for which t i m 2 are approximately elements of this finite set. We will use monotonicity in order to get some information about (77) also for the integers t which are between the numbers tt i pmqu.
Proof. Let ǫ ą 0 be fixed. Let δ ą 0 be arbitrary. By Proposition 4.4 the law of the random variable m ?
T rms x converges to the Erlang distribution which is supported on R`and has no atom in 0. Let W be a random variable with the latter probability distribution; in this way the law of T rms x m 2 converges to the law of W´2. Let c ą 0 be a sufficiently small number such that P`W´2 ă c˘ă δ. Now, let l P N 0 be a sufficiently big integer so that P´cp1`ǫq l ă W´2¯ă δ.
With these notations there exists some m 1 with the property that for each m ě m 1 (78) P´t 0 ă T rms x ď t l¯ą 1´2δ.
Let µ pnq " " λ pnq ‰ T be the transpose of λ pnq ; in this way µ pnq x is the number of the boxes of T which are in the column x and contain an entry ď n. The probability distribution of µ pnq is also given by the Plancherel measure. The monograph of Romik [Rom15, Theorem 1.22] contains that proof that holds true in the special case of the bottom row i " 0; it is quite straightforward to check that this proof is also valid for each i P t0, . . . , lu, for the