Department of Statistics
Nine month report
Exact solvability in
directed random polymer models
PhD student:
Elia Bisi
Supervisor:
Dr. Nikolaos Zygouras
June 25, 2015
Introduction
Random polymer models are the object of intense research within probability theory
and statistical physics. In particular, this report deals with the theory of directed random
polymers immersed in a random potential, which is introduced in section 1.
In the last few decades, a few (1+1)-dimensional models have been shown to be exactly
solvable, meaning that the probability distributions of quantities such as the polymer
partition sum are exactly computable in terms of algebraic objects: links with algebraic
combinatorics and representation theory have therefore acquired increasing importance.
The exact solvability of such models has turned out to be very useful to prove scaling limit
theorems, thus revealing surprising connections to random matrix theory. We analyze two
combinatorial algebraic structures that arise in this setting. In section 2, we consider the
RSK correspondence, which is related to the zero-temperature limit of polymer models. In
section 3, we examine the recently studied geometric RSK correspondence (also referred to
as tropical RSK), which is related to finite-temperature polymer models; in this field, we
set out possible future developments and open problems.
1
Directed polymers in a random potential
The goal of this section is to introduce the statistical mechanics model of directed
random polymers immersed in a random potential: polymers are supposed to live in
the (1 + d)-dimensional lattice N × Zd and to interact with an environment, being either
attracted to or repelled from sites depending on a random potential given by an i.i.d. field
on the lattice. We first describe general polymer models, providing the main definitions;
then we recall some basic facts about the simple random walk model, which represents the
“zero potential” case. Finally, we analyze the phase transition in the model with arbitrary
potential: the main results are stated and some proofs are sketched. The main references
are [4] and [7].
1.1
Random polymers
From a physical point of view, a polymer is a large molecule consisting of many smaller
molecules called monomers and tied together by chemical bonds. Very long concatenated
structures are easy to find in nature because of the multivalency of atoms. A polymer is
called linear if its monomers only have one reactive group, leading to a linear structure
without multiple cross connections: DNA, RNA and proteins are examples of linear
polymers.
Figure 1. A DNA helix.
1
In Mathematics, linear polymers are modelled as random paths in a lattice (typically
Zd ), where vertices represent the monomers and edges represent the chemical bonds
connecting the monomers. In this setting, the typical configuration of a polymer varies
according to the interactions with itself and the environment it is immersed in. Mathematical research asks how such microscopic interactions determine different macroscopic
behaviors, typically in terms of phase transitions, in the limit as the polymer gets long: for
instance, spreading out of the polymer’s endpoint as well as its localization are widely
studied problems.
In each polymer model we consider:
• Λ, the lattice where the polymers live;
• Xn , a finite set of allowed n-step paths in Λ;
• Hn , a (possibly random) Hamiltonian function that associates an energy to each path
in Xn ;
• Pn , the Gibbs measure on Xn associated to the Hamiltonian Hn and defined by
Pn (x) =
1 −Hn (x)
e
Zn
∀x ∈ Xn ,
where Zn is its normalizing partition sum. We call Pn polymer measure and we denote
by En its associated expectation.
The Hamiltonian Hn determines the interaction of the polymer with itself and the
environment, and often depends on some parameters such as temperature (or its inverse,
denoted by β). According to the intuition, the higher the energy of a path, the lower its
probability.
In case Hn is chosen to be random, it will depend on a random environment, given by
a set of random variables defined on a probability space (Ω, F , P, E). We have then two
polymer measures:
• the quenched polymer measure, which depends on a fixed configuration ω ∈ Ω of
the environment:
ω
1
x ∈ Xn ;
Pnω (x) = ω e−Hn (x) ,
Zn
• the annealed polymer measure, which is averaged w.r.t. the environment:
R
ω
e−Hn (x) P(dω)
E[e−Hn (x) ]
ΩR
Pn (x) =
=
,
x ∈ Xn ,
E[Zn ]
Znω P(dω)
Ω
where E[Zn ] is called annealed partition sum.
A number of random polymer models have been studied in literature (see [4] for an
overview). Here, we will concentrate on directed polymers in a random potential: in this
setting, the very first introductory model is the simple random walk on Zd .
2
1.2
The simple random walk model on Zd
Let us consider the set Zd with its usual lattice structure: two vertices x and y are
connected if the l1 -norm of their difference is 1 (we will write x ∼ y). We set
n
o
Xn = x = (xi )ni=0 ∈ (Zd )n+1 : x0 = 0, xi−1 ∼ xi ∀1 ≤ i ≤ n .
We consider Hn = 0, so that Pn is the uniform distribution on Xn , i.e. Pn (x) = (2d)−n for all
x. These polymers clearly correspond to the paths of a simple random walk (SRW) in Zd .
Using the fact that the endpoint Xn of the polymer is a sum of n i.i.d. increments under
Pn , one can easily compute its mean and variance:
En [Xn ] = 0,
En [|Xn |2 ] = n
∀n ≥ 1 .
The behavior of these polymers is diffusive: the continuous-time process interpolating the
polymer linearly and defined by
Xt := Xbtc + (t − btc)(Xbtc+1 − Xbtc )
∀t ∈ [0, ∞) ,
if suitably rescaled, converges in distribution to a standard Brownian Motion:
!
n→∞
1
−−−−−* (Bt )0≤t≤1
√ Xbntc
n
0≤t≤1
on the Banach space C[0, 1] of continuous functions on [0, 1]. This follows from the
well-known Donsker’s Invariance Principle (see [17, §5.3]).
For the SRW model we also have a Local Limit Theorem, which ensures that the position
Xn of the polymer’s endpoint spreads out as n → ∞:
1
max Pn (Xn = x) = O d/2 .
(1.1)
n
x∈Zd
This can be deduced from the following facts:
• If d = 1, then



= 0
q
Pn (Xn = x) 

∼ 2
πn
if x . n mod 2
as n → ∞
if x ≡ n mod 2 ,
where the asymptotics follow from a simple combinatorial argument and the Stirling
approximation.
(1)
(d)
• For all d ≥ 1, by independence of the components Xn , . . . , Xn
max Pn (Xn = x) =
x∈Zd
d
Y
i=1
(i)
max Pn Xn = x(i) ,
x(i) ∈Z
and each component is distributed as a SRW on Z.
3
1.3
Directed polymers in a random potential
In this section we extend the SRW model by allowing the polymers to be immersed in
an environment characterized by the presence of a random potential depending on both
time and space. For instance, we may think of hydrophilic polymers wafting in water: if
water contains randomly placed hydrophobic molecules that repel the monomers, then
the polymers can be viewed as immersed in a random potential.
We now define the model: polymers now live in the lattice Λ = N × Zd , i.e. in 1 + d
dimensions, where the first coordinate represents time and the others represent space. We
define the set of allowed paths by
n
o
Xn = x = (i, xi )ni=0 ∈ (N × Zd )n+1 : x0 = 0, xi−1 ∼ xi ∀1 ≤ i ≤ n .
The potential is given by a random field of real valued non-degenerate i.i.d. variables
{V (i, x) :
i ∈ N,
x ∈ Zd }
defined on a probability space (Ω, F , P) (expectation denoted by E), with moment generating function
h
i
M(β) = E eβV (1,0) < ∞
∀β ∈ [0, ∞) .
The Hamiltonian is given by
β,ω
Hn (x) = −β
n
X
V (i, xi ) ,
x ∈ Xn ,
i=1
for some choice of β ∈ [0, ∞), which should be thought of as inverse temperature. The
β,ω
associated quenched polymer measure Pn is defined in terms of the law Pn of the n-step
SRW on Zd
β,ω
1
β,ω
Pn (x) = β,ω e−Hn (x) Pn (x) ,
x ∈ Xn ,
Zn
β,ω
so that the partition sum Zn
β,ω
Zn
β,ω
can be expressed as the expectation of e−Hn w.r.t. Pn :
=
β,ω
X
e−Hn
(x)
h
βi
Pn (x) = En e−Hn .
(1.2)
x∈Xn
As the picture of these paths in N × Zd suggests (see Figure 2), these polymers are called
directed.
β
β
β
From now on, for the sake of conciseness we will write Hn (x), Pn (x) and Zn instead
β,ω
β,ω
β,ω
of Hn (x), Pn (x) and Zn ; however, we stress that these quantities depend on the
configuration of the environment, i.e. they are random variables (Ω, F , P) → R.
Remark 1.1. The case β = 0 corresponds to the SRW model we have already gone through.
Let us see the behavior of the polymer partition sum in the zero temperature limit, i.e. as
β → ∞:
β
Zn
−n
= (2d)
X
x∈Xn
exp β
n
X
!
−n
V (i, xi ) (2d)
i=1
4
exp β max
x∈Xn
n
X
i=1
!
V (i, xi ) .
Figure 2. A (1 + 1)-dim. directed polymer in a random environment. Different
vertex colors denote different values of the potential.
This establishes a deep connection between the polymer model we are describing and
another important model in statistical physics: indeed,
Tn∗ := max
x∈Xn
n
X
V (i, xi ) = lim
1
β→∞ β
i=1
β
log Zn
(1.3)
turns out to be the maximal passage time in the context of directed last passage percolation.
We will resume this concept in subsection 2.3.
♦
The main tool in the analysis of these polymers is the so-called normalized partition
sum, i.e. the ratio of the quenched and the annealed partition sum:
β
β
Zn
Zn
β
= En [en ]
Wn := h β i =
M(β)n
E Zn
where
en (x) =
n
Y
eβV (i,xi )
i=1
M(β)
∀n ≥ 0 ,
(1.4)
∀x ∈ Xn .
β
The process (Wn )n≥0 turns out to be a positive martingale w.r.t. the natural filtration of
the environment (Fn )n≥0 , defined by
Fn := σ V (i, x) : 1 ≤ i ≤ n, x ∈ Zd .
This follows from the fact that the environment is given in terms of an i.i.d. random
field: indeed, en is an (Fn )n≥0 -martingale as a product of mean-one i.i.d. random variables
β
on (Ω, F , P), so Wn , which is the average of en w.r.t. En (as (1.4) shows), also is. By the
β
martingale convergence theorem, Wn converges P-a.s., so we can define
β
β
W∞ := lim Wn
n→∞
5
P-a.s. .
β
Since the event {W∞ = 0} belongs to the tail sigma-algebra of (Fn )n≥0 , the Kolmogorov
zero-one law ensures that its probability is either 0 or 1, i.e. one of the two following
possibilities holds:
β
W∞ > 0
P-a.s. ,
(1.5a)
β
W∞
P-a.s. .
(1.5b)
=0
h β 1/2 i
Lemma 1.2 (Comets, Yoshida [8]). The fractional moment E W∞
is non-increasing
on [0, ∞) as a function of β.
h
h
i
i
0 )1/2 = 1, so the latter lemma implies that E W β 1/2
Since Wn0 = 1 for all n, E (W∞
∞
is strictly positive up to a critical value βc and zero from that value on; of course, the
cases βc = 0 and βc = ∞ are also possible. This allows to highlight a phase transition in β
from (1.5a) to (1.5b):
Theorem 1.3. For any fixed d ≥ 1 and any fixed potential distribution, there exists
βc ∈ [0, ∞] such that
β
W∞ > 0
P-a.s.
∀β ∈ [0, βc ) ,
β
W∞
P-a.s.
∀β ∈ (βc , ∞) .
=0
The two phases (1.5a) and (1.5b) are called weak disorder and strong disorder respectively, because the role of the disorder is intuitively significant only when the annealed
β
partition sum grows faster than the quenched partition sum, i.e. when W∞ = 0. A priori,
one cannot deduce from Lemma 1.2 which phase the critical value βc corresponds to.
In the next section we will study the behavior of the polymer in the two phases.
1.4
Analysis of the two phases
This section is devoted to explain the main results about the two phases of the polymer
model introduced in section 1.3. Roughly speaking, for small β, i.e. high temperature, the
behavior is diffusive just as in the SRW model (β = 0); for large β, i.e. low temperature,
the behavior is (expected to be) superdiffusive. In order to quantify how much β should
be “small” or “large” (which of course will depend on the dimension d and the potential
distribution), we introduce the following functions for β ∈ [0, ∞):
γ1 (β) = log M(2β) − 2 log M(β) ,
γ2 (β) = β[log M(β)]0 − log M(β) .
Since the logarithmic moment generating function log M is strictly convex, both γ1 and γ2
are strictly increasing on [0, ∞). We also define the return probability for the SRW in Zd :
πd := P (Xn = 0 for some n ≥ 1) ,
6
where P is the law of the SRW; we recall that πd = 1 if d = 1, 2 (recurrent SRW) and πd < 1
if d ≥ 3 (transient SRW). We now consider the following three conditions:
d≥3
and
γ1 (β) < log(1/πd ) ,
(1.6a)
d≥1
and
γ2 (β) > log(2d) ,
(1.6b)
d = 1, 2
and
β > 0.
(1.6c)
Since γ1 and γ2 are strictly increasing, γ1 (0) = γ2 (0) = 0, πd < 1 for d ≥ 3 and 2d > 1,
condition (1.6a) is satisfied for large values of β, and condition (1.6b) for small values
of β. It has been proven that under condition (1.6a) we have weak disorder phase, and
under condition either (1.6b) or (1.6c) we have strong disorder phase (note that (1.6b)
implies (1.6c) for d = 1, 2).
Theorem 1.4 (Imbrie, Spencer [13], Bolthausen [3], Song, Zhou [21], Carmona, Hu
[5], Comets, Shiga, Yoshida [6]). Under condition (1.6a), the following facts hold.
(i) Weak disorder phase:
β
W∞ > 0
P-a.s.
(ii) Diffusive behavior:
1 β
En [|Xn |2 ] = 1
n→∞ n
lim
P-a.s.
q
(iii) Central Limit Theorem: P-a.s., the distribution of
the d-dimensional standard normal distribution.
d
n Xn
β
under Pn converges to
(iv) Delocalization:
β
lim max Pn (Xn = x) = 0
n→∞
x∈Zd
P-a.s.
β
The proof of (i) follows from the L2 -boundedness of the martingale Wn : this implies
β
the convergence of Wn also in L1 , so that
h βi
h βi
E W∞ = lim E Wn = 1 ,
n→∞
β
excluding (1.5b). For proving that Wn is L2 -bounded, recalling (1.4) the trick is to write
β 2
h (X) (Y ) i
Wn = E ⊗ E en en ,
where E ⊗ E is the expectation w.r.t. the law P ⊗ P of a couple of independent SRWs
(Xi , Yi )∞
i=0 . A series of computations shows then that for all n ≥ 1
 ∞
k
∞ X
k  X
h β 2 i

γ1 (β)
E Wn
≤ 1+
e
− 1 
P ⊗ P (Xi = Yi ) ,
(1.7)
i=1
k=1
and the series over i is actually the expected number of returns to 0 of a SRW in Zd , which
is finite for d ≥ 3 and can be easily computed in terms of πd using the Markov property:
∞
X
P ⊗ P (Xi = Yi ) =
i=1
7
πd
.
1 − πd
It follows that the series over k in (1.7) converges if and only if
πd
< 1,
eγ1 (β) − 1
1 − πd
which is equivalent to condition (1.6a).
The proof of (ii) and (iii) involves the analysis of some classes of martingales that
β
generalize (Wn )n≥0 .
Similarly to the SRW model, for which the Local Limit Theorem (1.1) holds, under
condition (1.6a) the position Xn of the polymer’s endpoint spreads out in the limit as
n → ∞, leading to the delocalization stated in (iv). Let us define for n ≥ 1 the following
random variables on (Ω, F , P):
X β
β
β
β
Jn := max Pn (Xn = x) ,
In :=
Pn (Xn = x)2 .
(1.8)
x∈Zd
x∈Zd
Since clearly
β 2
β
β
0 ≤ Jn ≤ In ≤ Jn ,
β
β
Jn converges to 0 if and only if In does. Thus, the proof of (iv) is carried out by analyzing
β
β
β
β
In instead of Jn . (In )n≥1 and W∞ are connected by the following fundamental fact: for all
β>0
 ∞

β
 X β


P W∞ > 0 = P
In < ∞ ,
n=1
β
whose proof is based on Doob’s decomposition for the process − log Wn . Under conβ
β
dition (1.6a), (i) ensures that the series of In is finite P-a.s., so that In converges to 0
P-a.s..
Theorem 1.5 (Carmona, Hu [5], Comets, Shiga, Yoshida [6]). Under either condition (1.6b) or condition (1.6c), the following facts hold.
(i) Strong disorder phase:
β
W∞ = 0
P-a.s.
(ii) Localization to the favorite sites:
β
lim sup max Pn (Xn = x) ≥ c
n→∞
x∈Zd
P-a.s.
for some c = c(d, β) > 0.
Note that (ii) highlights a qualitative behavior of the polymer in contrast with (iv)
of Theorem 1.4: in the limit as n → ∞, the polymer’s endpoint concentrates on some
favorite sites instead of spreading out. Furthermore,
√ the location of these favorite sites is
predicted to be at a distance of order larger than n, highlighting superdiffusive behavior:
only partial results have been obtained, for example in [20].
Let us sketch the proof of Theorem 1.5 under condition (1.6b). The key is to provide
h β θ i
an estimate on the fractional moment E Wn
for some θ ∈ (0, 1). Using the Markov
8
property of the SRW and the inequality (u + v)θ ≤ u θ + v θ , it is not difficult to prove that
for all θ ∈ (0, 1)
h β θ i
h β θ i
E Wn
≤ r(θ) E Wn−1 ,
1−θ
r(θ) := (2d)
" βV (1,1) !θ #
e
E
.
M(β)
By induction, it follows that
h β θ i
E Wn
≤ r(θ)n .
(1.9)
Since θ → log r(θ) is convex on (0, 1) and log(2d) = log r(0) > log r(1) = 0, there exists
θ ∈ (0, 1) such that log r(θ) < 0 (i.e. r(θ) < 1) if and only if (log r)0 (1) > 0: the latter
h β θ i
condition is indeed equivalent to (1.6b). Choosing such a θ, (1.9) implies E Wn
→0
as n → ∞. By Fatou’s Lemma:
h β θ i
h
β θ i
h β θ i
E W∞
= E lim inf Wn
≤ lim inf E Wn
= 0.
n→∞
n→∞
β
β
β
Since W∞ ≥ 0 P-a.s., this proves (i). A little more work involving estimates on Jn , In (as
β
defined in (1.8)) and − log Wn is needed to prove (ii).
Finally, from Theorems 1.4 and 1.5 we can deduce the following about the critical
value βc introduced in Theorem 1.3:
• βc = 0 for d = 1, 2;
• βc ∈ [βc1 , βc2 ] for d ≥ 3, where βc1 is the maximum β such that γ1 (β) < log(1/πd ) and
βc2 is the minimum β such that γ2 (β) > log(2d).
2
RSK correspondence and related probabilistic models
This section deals with the classical RSK correspondence, so called because it dates
back to Robinson’s, Schensted’s and Knuth’s mathematical works. Subsection 2.1 is devoted to provide the general combinatorial framework. In the next subsections, we will
explain the fundamental role that RSK plays in understanding some probabilistic models
closely related to random polymers: we will talk about longest increasing subsequences
in random permutations and maximal directed paths on Poisson points (subsection 2.2),
directed last passage percolation (subsection 2.3) and polynuclear growth model (subsection 2.4).
2.1
Robinson-Schensted-Knuth correspondence
The Robinson-Schensted-Knuth correspondence (RSK) is a combinatorial algorithm
providing a bijection between matrices of non-negative integers and semistandard Young
tableaux of the same shape. In this section we explain it in detail and show its main
properties.
A partition of n of length l is a weakly decreasing sequence λ = (λ1 , . . . , λl ) of l positive
integers that sum up to n: we write λ ` n and l(λ) = l. The (only) partition of length 0 is
λ = €. A graphical representation of a partition λ is a Young diagram, i.e. a collection of
l(λ) left-justified rows of boxes such that i-th row contains λi boxes; row lengths turn out
9
to be weakly decreasing. The following is an example of partition of 8 of length 3, with its
corresponding Young diagram:
λ = (4, 3, 1)
←→
A Young tableau P is obtained by filling the boxes of a Young diagram with positive
integer numbers; the shape of P , denoted by sh(P ), is the partition which corresponds to
the Young diagram; the size of P is its total number of boxes; the type of P is the vector
(P1 , P2 , . . . ) such that Pj is the number of j’s in P (since the size is finite, Pj = 0 for j large
enough). A Young tableau is called semistandard if its rows are weakly increasing and
its columns are strictly increasing. Here, we show an example of semistandard Young
tableaux of shape λ = (4, 3, 1), therefore of size 8, and type (2, 2, 1, 3):
1 1 2 3
2 4 4
4
We now define the insertion of a positive integer k into a semistandard Young tableau
P = (pi,j ). In row i, starting from i = 1, we search for the smaller j such that pi,j > k. If
such a j does not exist, we simply add a box filled with k at the end of row i; if such a j
does exist, we fill box (i, j) with k and bump the old entry pi,j to the next row i + 1, where
we will try to insert it in the same way. Clearly, the procedure must stop in a finite number
of steps, producing a new semistandard Young tableau whose size is increased by one. For
example, inserting 2 in the tableau above, we obtain:
1 1 2 2
2 3 4
4 4
For any m × n matrix W = (wi,j ) with non-negative integer entries, we consider two
words of the same length composed of weakly increasing words in the alphabet of nonnegative integer numbers:
(i) the word w := w1 . . . wm such that wi := 1wi,1 . . . nwi,n ;
(ii) the word w0 := w10 . . . wn0 such that wj0 := 1w1,j . . . mwm,j .
The RSK correspondence is defined as the map that associates a matrix W to the pair
of semistandard Young tableaux (P , Q) such that P is obtained by inserting all numbers
appearing in w successively (starting from the empty tableau) and Q is obtained in the
same way using w0 instead.
10
Example 2.1.


1 2 1 1


W = 0 1 1 0


3 0 0 1
0
w = 12234 23 1114
| {z } |{z} |{z}
w1
w2
w = 1333 112 12 13
|{z} |{z} |{z} |{z}
w10
w3
w20
?
w30
w40
?
1 1 1 1 3 4
P= 2 2 2
3 4
1 1 1 1 1 3
Q= 2 2 3
3 3
♦
It is no accident that P and Q have the same shape in the latter example. Indeed, Q
can also be constructed at the same time as P this way: when inserting a number of wi
into P , the diagram increases by one box; a box must be added to Q in the same position
and filled with i. It is clear from this alternative construction that P and Q have finally
(actually, at every insertion step) the same shape. Moreover, the denominations insertion
tableau and recording tableau for P and Q respectively are now natural.
We also observe that the j-th column of W sum up to the number of j’s in P , i.e. Pj .
Similarly, the i-th row of W sum up to the number of i’s in Q, i.e. Qi . We then have the
following fundamental theorem, whose proof can be found for example in [22, §7.11].
Theorem 2.2. The RSK correspondence is a bijection between matrices W = (wi,j ) with
non-negative integer entries and pairs (P , Q) of semistandard Young tableaux with the
same shape. Under this bijection, the type of P and Q is determined by:
X
X
Pj =
wi,j ∀j ,
Qi =
wi,j ∀i .
(2.1)
i
j
Transposing matrix W , the roles of P and Q are interchanged:
RSK
RSK
Theorem 2.3. If W −−−−→ (P , Q), then W T −−−−→ (Q, P ).
Let us now consider the special case of permutation matrices of order n, i.e. n × n
matrices such that every row and every column has one entry equal to 1 and all the others
equal to 0. In this case, the corresponding words w and w0 are permutations of {1, . . . , n},
inverse of each other. The resulting tableaux under RSK are standard Young tableaux of
size n: namely, they are composed of n boxes occupied by the numbers 1, . . . , n, and both
rows and columns are strictly increasing.
11
Example 2.4.

0

0
W = 
0

1
1
0
0
0
0
0
1
0

0

1

0

0
-
w0 = 4132
w = 2431
?
?
1 3
P= 2
4
1 2
Q= 3
4
♦
The restriction of RSK to permutations is the so-called called Robinson-Schensted (RS)
correspondence, and was actually studied first:
Theorem 2.5. The RS correspondence is a bijection between permutations of n objects
and pairs of standard Young tableaux of size n with the same shape.
We finally introduce another combinatorial object which is in a bijective correspondence with semistandard Young tableaux. Let P be a semistandard Young tableau such
that maxi,j pi,j = n and let us define zi,j as the total number of 1’s, 2’s, ..., i’s in row j of P .
By the column strict rule, below row i there cannot be numbers ≤ i, so that zi,j = 0 for all
j > i; for the same reason, zi,j = 0 for all i > n. We can thus arrange all (possibly) nonzero
zi,j , j ≤ i ≤ n, in a triangular array:
z1,1
z2,1
z3,1
Z=
.
..
zn,1
z2,2
z3,2
···
z3,2
..
···
zn,2
···
.
···
zn,n
The i-th row (zi,1 , . . . , zi,i ) of Z is the shape of the semistandard Young tableau obtained by
removing all numbers > i from the original tableau P ; it is hence clear that the last row
(zn,1 , . . . , zn,n ) is the shape of P , and it is called shape of Z by analogy. On the other hand,
by definition the type of P can be recovered from Z through the following equations:
Pj =
j
X
i=1
zj,i −
j−1
X
zj−1,i
∀j = 1, . . . , n ,
(2.2)
i=1
where the empty sum is set to 0. The latter numbers define the type of Z by analogy.
Another important property of Z is the interlacing condition:
zi+1,j ≥ zi,j ≥ zi+1,j+1
12
∀1 ≤ j ≤ i < n ,
(2.3)
which follows from the column strict rule for P as well. Triangular arrays satisfying
condition (2.3) are called Gelfand-Tsetlin patterns. It is thus easy to see that there is
a bijective correspondence between semistandard Young tableaux and Gelfand-Tsetlin
patterns. Consequently, Theorem 2.2 can be reformulated this way:
Theorem 2.6. The RSK correspondence induces a bijection between matrices W = (wi,j )
with non-negative integer entries and pairs (Z, Z 0 ) of Gelfand-Tsetlin patterns with the
same shape. Under this bijection, the type of Z and Z 0 is determined by:
j
X
i=1
zj,i −
j−1
X
zj−1,i =
X
i=1
wi,j
i
X
∀j ,
i
0
zi,j
−
j=1
i−1
X
0
zi−1,j
=
X
j=1
wi,j
∀i .
j
Example 2.7. The Gelfand-Tsetlin patterns associated to tableaux P and Q of Example 2.1
are respectively:
4
4
Z=
5
6
3
3
5
3
0
2
5
Z =
1
6
0
2
3
2
♦
2.2
Longest increasing subsequence of random permutations and directed
polymers on Poisson points
Let Sn be the symmetric group of order n. For each permutation σ ∈ Sn , we say that
a subsequence (n1 , . . . , nk ) of the sequence (σ (1), . . . , σ (n)) is an increasing subsequence of
length k if n1 < · · · < nk . Let Ln (σ ) be the length of the longest increasing subsequence for
the permutation σ . If we think of Sn as a probability space with the uniform distribution,
we can investigate the asymptotic law of the random variable Ln for large n: this is a
classical problem, proposed by Ulam [23] in 1961.
A graphical formulation of the problem is the following. Let us define a directed
path in R2 from (x0 , y0 ) to (x, y) as any piecewise linear path between these endpoints
that is increasing in both coordinates; it is determined by a finite sequence of points
(x1 , y1 ), . . . , (xl , yl ) (endpoints of the linear pieces) such that x0 < x1 < · · · < xl < x and
y0 < y1 < · · · < yl < y; l is called length of the path. Let us now fix a square, say [0, 1]2 , and
n points (x1 , y1 ), . . . , (xn , yn ) in the square with distinct abscissas and ordinates. Assuming
that x1 < x2 < · · · < xn , let σ ∈ Sn be the permutation such that yσ (1) < yσ (2) < · · · < yσ (n) .
Then increasing subsequences of σ are clearly in bijective correspondence with directed
paths from (0, 0) to (1, 1) on the points (x1 , y1 ), . . . , (xn , yn ) (i.e., consisting of a subset of these
points). Therefore, if we consider the n points as independent random variables uniformly
distributed on the square, the length of the longest directed path on these points has the
same law as Ln .
The key in analysing Ulam’s problem is the Young tableau representation of permutations, i.e. the RS correspondence introduced in subsection 2.1. In this respect, the
following theorem is fundamental, see [12]:
13
RS
Theorem 2.8. Let σ ∈ Sn such that σ −−→ (P , Q), and let λ be the common shape of
(k)
tableaux P and Q. For k ≤ n, let Ln (σ ) be the length of the longest subsequence of σ
consisting of k disjoint increasing subsequences. Then
(k)
Ln (σ ) = λ1 + · · · + λk .
(1)
In particular, the length Ln (σ ) = Ln (σ ) of the longest increasing subsequence of σ is
the length of the first row of the Young tableaux associated to σ via RS. The asymptotic
law of Ln can therefore be studied by analyzing the behavior of λ1 , with the law inherited
by the uniform distribution on Sn , which is called Plancherel measure (see Theorem 2.5):
Pl(λ) = P
dλ2
2
µ`n dµ
∀λ ` n ,
where dµ is the number of semistandard Young tableaux of shape µ, for any µ ` n.
Using this representation, Vershik and Kerov [24] carried out the first important step
in attacking the problem, proving that
E[L ]
lim √ n = 2 .
n→∞
n
Only in 1999, Baik, Deft and Johansson [1] finally solved Ulam’s problem, proving
that the fluctuations of Ln are of order n1/6 and its asymptotic law is F2 , the Tracy-Widom
distribution for the Gaussian Unitary Ensemble (GUE):
√
Ln − 2 n
lim P
≤
s
= F2 (s),
s ∈ R.
(2.4)
n→∞
n1/6
They proved this result using a closely related problem, called Poissonized version.
Instead of fixing the length n, they considered it as a Poisson random variable with mean
λ, and studied the asymptotic law of Ln as λ gets large; afterwards, they used the obtained
result to solve the original problem.
As Ulam’s problem has a graphical interpretation in terms of directed paths, also
its Poissonized version can be visualized as a point-to-point problem of directed paths on
Poisson points. Let ω be a configuration of points in the square [0, t]2 , corresponding to
the realization of a Poisson process of intensity 1 on the plane. Let us define L(t) as the
length of the longest direct path from (0, 0) to (t, t) on these Poisson points. Since t 2 is the
2
area of the square, the probability of having k points in the square is just e−t (t 2 )k /k!: this
shows the equivalence to the Poissonized version of Ulam’s problem, setting λ := t 2 . In
this setting, the fluctuations are of order t 1/3 and the convergence in law stated in (2.4)
becomes:
L(t) − 2t
lim P
≤
s
= F2 (s),
s ∈ R.
t→∞
t 1/3
Finally, we mention a modification of the latter model, called point-to-line problem of
directed paths on Poisson points. On point configurations arising from a Poisson process
of intensity 1 on the plane, we now consider the length L∗ (t) of the longest directed
path starting from (0, 0) and ending at any point (x, y) of the first quadrant on the line
14
(m, n)
(m + n − 2, −m + n)
(0, 0)
(1, 1)
Figure 3. Two graphical representations of directed paths on the (1 + 1)-dim.
lattice.
{x + y = 2t}. Fluctuations are still of order t 1/3 , but the asymptotic law is now F1 , the
Tracy-Widom distribution for the Gaussian Orthogonal Ensemble (GOE), see [2]:
lim P
t→∞
2.3
L∗ (t) − 2t
≤
s
= F1 (s),
t 1/3
s ∈ R.
Directed last passage percolation
Let us consider the following discrete model on the lattice N2 . Let us define Πm,n as
the set of up-right paths from (1, 1) to (m, n): namely, every path π ∈ Πm,n is a sequence
((i1 , j1 ), (i2 , j2 ), . . . , (im+n , jm+n )) such that (i1 , j1 ) = (1, 1), (im+n , jm+n ) = (m, n) and (ik+1 , jk+1 ) −
(ik , jk ) is either (1, 0) or (0, 1). Assigning to every point (i, j) ∈ N2 a non-negative weight
wi,j , we define:
X
Tm,n := max
π∈Πm,n
wi,j
∀m, n ≥ 1 .
(2.5)
(i,j)∈π
If the weights wi,j are interpreted as waiting times, Tm,n is the maximal passage time
for directed paths from (1, 1) to (m, n), and turns out to be the point-to-point version of the
directed last passage percolation defined in (1.3). To see this, let us identify in the obvious
way a path π ∈ Πm,n with a path x = (k, xk )m+n−2
∈ (N × Z)m+n−1 such that x0 = 0, xk−1 ∼ xk
k=0
for 1 ≤ i ≤ m + n − 2 and xm+n−2 = −m + n: namely, up steps (resp., right steps) in the first
path corresponds to up-right steps (resp., down-right steps) in the latter, see also Figure 3.
Also, set V (k, xk ) := wi,j if k = i + j − 2 and xk = −i + j. Then,
Tm,n = max
x
m+n−2
X
V (k, xk ) ,
k=0
where the maximum is over all paths x defined above. The latter equation is clearly a
point-to-point version of (1.3).
We note that Tm,n can also be defined by the recursive formula:
Tm,n = max(Tm−1,n , Tm,n−1 ) + wm,n
∀m, n ≥ 1 ,
(2.6)
setting Tm,n := 0 whenever m or n are zero.
We now explain why directed last passage percolation is strongly related to RSK. For
this, we need a generalization of Theorem 2.8 from the RS setting to the RSK setting (see
for example [11]):
15
Theorem 2.9. Let W = (wi,j ) be an m × n matrix of non-negative integers such that
RSK
W −−−−→ (P , Q), and let λ be the common shape of tableaux P and Q. For k ≤ n, let us
define
X
(k)
Tm,n := max
wi,j ,
π(1) ,...,π(k)
(i,j)∈π(1) ∪···∪π(k)
where the maximum is over all k-tuples of non-intersecting directed lattice paths π(i)
from (1, i) to (m, n − k + i), for i = 1, . . . , k. Then
(k)
Tm,n = λ1 + · · · + λk .
(1)
In particular, taking k = 1, Tm,n = Tm,n turns out to be equal to the length λ1 of the first
row of the tableaux P and Q obtained from the weight matrix (wi,j ) via RSK.
Let us now consider independent weights Wi,j geometrically distributed with parameter pj qi , where (pj )j≥1 and (qi )i≥1 are two sequences of numbers in (0, 1):
P(Wi,j = wi,j ) = (1 − pj qi )(pj qi )wi,j
∀wi,j ≥ 0 .
We are interested in calculating the law of Tm,n . Let us fix an m × n matrix (wi,j ) and the
corresponding tableaux (P , Q) via RSK. Denoting by (P1 , . . . , Pn ) and (Q1 , . . . , Qm ) the type
of P and Q respectively, by independence of the weights and Theorem 2.2:
h
i
RSK
P (Wi,j )i≤m,j≤n −−−−→ (P , Q) = P(Wi,j = wi,j ∀i ≤ m, j ≤ n)
=
n Y
m
Y
(1 − pj qi )(pj qi )wi,j
j=1 i=1
=
"Y
n Y
m
# "Y
# "Y
#
n
m
Pj
Qi
(1 − pj qi ) ·
pj ·
qi .
j=1 i=1
|
j=1
{z
i=1
}
=:cm,n
If sh is the (random) shape of the tableaux obtained from (Wi,j )i≤m,j≤n via RSK, then for a
fixed partition λ of length ≤ min(m, n)
P(sh = λ) =
X
h
i
RSK
P (Wi,j )i≤m,j≤n −−−−→ (P , Q)
(P ,Q):
sh(P )=sh(Q)=λ

 

n
m
 X Y
 X Y

Pj 
Q

= cm,n · 
pj  · 
qi i 
sh(P )=λ j=1
sh(Q)=λ i=1
= cm,n · sλ (p1 , . . . , pn ) · sλ (q1 , . . . , qm ) .
Here sλ is a Schur polynomial, which is a well-known symmetric polynomial in n variables
p1 , . . . , pn , indexed by a partition λ of length ≤ n and defined by
n
X Y
sλ (p1 , . . . , pn ) =
sh(P )=λ j=1
16
Pj
pj ,
where the sum is over all semistandard Young tableaux of shape λ in the alphabet {1, . . . , n}.
Since Tm,n = λ1 by Theorem 2.9, we can now express the distribution function of Tm,n in
terms of Schur polynomials:
Theorem 2.10. If the weights Wi,j are independent and geometrically distributed with
parameter pj qi , then for all m, n ≥ 1 and t ∈ R
X
P(Tn,m ≤ t) = cm,n
sλ (p1 , . . . , pn ) · sλ (q1 , . . . , qm ) ,
λ: λ1 ≤t
where the sum is over all partitions λ of length ≤ min(m, n), and
n Y
m
Y
cn,m :=
(1 − pj qi ) .
j=1 i=1
This model with geometric weights is therefore exactly solvable: the law of Tn,m
can be explicitly computed in terms of Schur polynomials, which also have an explicit
determinantal expression (see [22, § 7.15]). In the special case of i.i.d. geometric weights,
Johansson [14] proved that Tn,n , appropriately scaled, converges in law to the Tracy-Widom
distribution for the GUE.
The case of independent exponentially distributed weights can be also proven to be
exactly solvable; furthermore, it is of particular interest because, if all the weights Wi,j
have exponential distribution with mean 1, Tm,n also describes passage times of particles
in the well-known totally asymmetric exclusion process (TASEP). This is a continuous
time Markov process (ηt )t≥0 on the state space {0, 1}Z , interpreted as an interacting particle
system: any η ∈ {0, 1}Z is a configuration of particles and holes (1’s and 0’s respectively)
on the one dimensional integer lattice. A particle at site k jumps with exponential rate
one to k + 1, provided that this site is vacant, otherwise nothing happens. Considering the
initial configuration η such that η(k) = 1 if and only if k ≤ 0 (i.e. all non-negative sites are
occupied and all positive sites are empty), let us call Wi,j the time that the particle starting
at site −i has to wait to perform its j-th jump once that site −j + i is empty. Then Tm,n is
the time that the particle at site −m needs to wait to go beyond site −m + n; in particular,
Tn,n is the time that the particle at site −n needs to wait to go beyond site 0. Indeed, it is
easily seen that Tm,n defined in such a way satisfies the recursive formula (2.6).
2.4
Polynuclear growth model
In statistical mechanics, surface growth models simulate the behavior of atoms or
molecules that, once ejected onto a surface, attach to each other and form growing islands.
Every growth model is given by a random function h(x, t), which represents the height of
the surface and depends on a d-dimensional spatial coordinate x and a time coordinate t.
KPZ universality class is the most studied class of random growth models characterized by
both local evolution and local randomness: for such models, the height function is solution
of a (d + 1)-dimensional nonlinear stochastic equation introduced by Kardar, Parisi and
Zhang [15].
The polynuclear growth (PNG) model is a surface growth model in one spatial dimension belonging to the KPZ universality class, and can be thought of as a further
graphical representation of RSK correspondence, see [10] and [9]. We now wish to give
17
a discrete time version of the PNG model. Starting from an m × n matrix (wi,j ) of independent non-negative random weights (to simplify notation we assume m = n), we will
construct a growing surface of height h(x, t); in fact, for a complete characterization of
matrix (wi,j ), we will describe an ensemble of height functions {hl }l=0,−1,−2,... such that
h0 ≡ h and hl (x, t) ≥ hl−1 (x, t) for all level l and for all (x, t). For all t and l, hl (·, t) will be
constant on any spatial interval [x − 1/2, x + 1/2), x ∈ Z; therefore, we can see each hl (·, t)
as a profile of columns of unit width centered on the integers. This way, it suffices to
define the evolution on all integers x. We start from the flat initial condition: at time 0,
hl (x, 0) = 0 for all x ∈ Z, l ≤ 0. For t ≥ 1, this is the outline of the evolution:
• in the time interval from t − 1 to t, at any level l every column grows one unit to the
left and one unit to the right, if it is higher than its neighbors;
• if two columns meet, they merge and the highest column “wins”, creating an overlap;
• each overlap created at level l falls down to level l − 1, growing the corresponding
column;
• at level 0, weights wi,j “fall down” to grow columns further on: precisely, column at
site x also grows by wi,j if i − j = x and i + j − 1 = t;
• the evolution stops after instant 2n − 1.
We now describe this evolution formally, constructing from matrix (wi,j ) a new n × n
matrix (ti,j ) = T ((wi,j )) which will contain all the information about the height functions.
This map T : (R≥0 )n×n → (R≥0 )n×n will be expressed as a composition of simple maps li,j
called local moves, as explained in [19, §8]. For all i, j = 1, . . . , n, let us define li,j : (R≥0 )n×n →
(R≥0 )n×n as the map that takes as input an n × n matrix (wi,j ) and replaces the submatrix
"
#
wi−1,j−1 wi−1,j
wi,j−1
wi,j
with its image under the map
"
# "
#
min(b, c) − a
b
a b
→
.
c d
c
max(b, c) + d
(2.7)
To give a meaning to li,j also for i = 1 or j = 1, we simply set wi,j = 0 whenever i = 0 or
j = 0; in other words:
• l1,1 is the identity map;
• for j = 2, . . . , n, l1,j replaces w1,j with w1,j−1 + w1,j ;
• for i = 2, . . . , n, li,1 replaces wi,1 with wi−1,1 + wi,1 .
We note that li,j is bijective. For i, j = 1, . . . , n, set:



l1,j−i+1 ◦ · · · ◦ li−1,j−1 ◦ li,j
%ji := 

li−j+1,1 ◦ · · · ◦ li−1,j−1 ◦ li,j
i≤j
i ≥j.
(2.8)
For convenience, we set %ji to be the identity map if (i, j) < {1, . . . , n}2 . For t = 1, . . . , 2n − 1,
we also set
2
Dt := %1t ◦ %2t−1 ◦ · · · ◦ %t−1
◦ %t1 .
(2.9)
18
The desired map T : (R≥0 )n×n → (R≥0 )n×n is then defined by
T := D2n−1 ◦ · · · ◦ D1 .
(2.10)
The height profiles at time 2n − 1 are now determined by
ti,j = hmax(i,j) (i − j, 2n − 1)
∀(i, j) ∈ {1, . . . , n}2 ,
and hl (x, 2n − 1) = 0 for any other x ∈ Z and l ≤ 0.
Since the local moves are bijective, T is also bijective, and it turns out to be a version
of RSK extended to all matrices of non-negative real numbers (not necessarily integers).
More precisely, let us change notation of (ti,j ) this way:

 t1,1 t1,2
t1,n


 t2,1



tn−1,n


tn,n−1 tn,n
 tn,1
 
0
0
 
λn
zn−1,n−1
z1,1
 
 
  zn−1,n−1
 = 
 
0
 
zn−1,1
 
 
 
z1,1
zn−1,1 λ1






 .





The main diagonal of T in reverse order
0
0
)
(tn,n , . . . , t1,1 ) = (λ1 , . . . , λn ) = (zn,1 , . . . , zn,n ) = (zn,1
, . . . , zn,n
turns out to be the common shape of two Gelfand-Tsetlin patterns Z and Z 0 , corresponding
to the entries of T below and above the diagonal respectively. If wi,j are integers, W →
(Z, Z 0 ) coincides with the version of RSK stated in Theorem 2.6. This new representation
of RSK correspondence in terms of local moves will be especially useful in subsection 3.1,
where we will talk about geometric RSK.
Geometric RSK and polymers in 1 + 1 dimensions
3
This section deals with the geometric lifting of the RSK correspondence introduced
by Kirillov in [16], and applications to some exactly solvable random polymer models.
In subsection 3.1, we describe the geometric RSK, following [19] (the construction is
equivalent to the one given in [18]). In subsection 3.2, we show the exact solvability of the
random polymer model with inverse gamma weights [19]. In subsection 3.3, we sketch
more polymer models with inverse gamma weights characterized by symmetries or other
constraints, and we also set out some open problems.
3.1
Geometric RSK
Consider the classical RSK correspondence for square matrices as a map (R≥0 )n×n →
(R≥0 )n×n , as explained in subsection 2.4: the output matrix (ti,j ) is obtained from the input
matrix (wi,j ) by applying a sequence of local moves li,j defined by (2.7). By construction,
such local moves only involve sums, subtractions, max and min operations: all of them
can be defined in the (max, +) semiring, i.e. R ∪ {−∞} with the algebraic structure defined
by the binary operations max (“addition”) and + (“multiplication”)† .
† Note that min(b, c) = − max(−b, −c).
19
In brief, the geometric RSK is the mapping T : (R>0 )n×n → (R>0 )n×n consisting in the
composition of the analogous sequence of maps li,j , where the operations in the (max, +)
semiring (combinatorial setting) are formally replaced with the corresponding ones in the
usual (+, ·) ring (geometric setting):
a+b → a·b,
a−b →
a
,
b
max(a, b) → a + b .
In other words, the new local move li,j , for all i, j = 1, . . . , n, is defined as the map
(R>0 )n×n → (R>0 )n×n that takes as input an n × n matrix (wi,j ) and replaces the submatrix
"
wi−1,j−1 wi−1,j
wi,j−1
wi,j
#
with its image under the map
"
# "
#
a b
bc/[a(b + c)]
b
→
.
c d
c
(b + c)d
To give a meaning to li,j also for i = 1 or j = 1, the convention is that wi,j = 0 whenever
i = 0 or j = 0, but w0,1 + w1,0 = 1. In other words:
• l1,1 is the identity map;
• for j = 2, . . . , n, l1,j replaces w1,j with w1,j−1 · w1,j ;
• for i = 2, . . . , n, li,1 replaces wi,1 with wi−1,1 · wi,1 .
With this new definition of local moves, we set T to be the same sequence of local moves
defined in (2.8), (2.9) and (2.10). Since many local moves commute with each other
(li,j ◦ li 0 ,j 0 = li 0 ,j 0 ◦ li,j whenever |i − i 0 | + |j − j 0 | > 2), there are other equivalent ways to define
the order of local moves in the sequence, see [19, § 3]. Since the local moves li,j are clearly
birational, i.e. bijective rational mappings, T is also birational.
Similarly to the RSK case, we will also adopt the notation:

 t1,1 t1,2
t1,n


 t2,1



tn−1,n


tn,n−1 tn,n
 tn,1
 
0
0
 
λn
zn−1,n−1
z1,1
 
 
  zn−1,n−1
 = 
 
0
zn−1,1
 
 
 
z1,1
zn−1,1 λ1







 .





This way, we can see the output matrix T as a pair (Z, Z 0 ) of triangular patterns of height n
0
z1,1
z1,1
..
Z=
zn,1
.
..
···
Z0 =
,
.
..
0
zn,1
zn,n
.
..
···
where
0
0
(zn,1 , . . . , zn,n ) = (zn,1
, . . . , zn,n
) = (λ1 , . . . , λn ) = (tn,n , . . . , t1,1 )
20
,
.
0
zn,n
will be referred to as the (common) shape of Z and Z 0 , as in the case of Gelfand-Tsetlin
gRSK
gRSK
patterns. We will write either W −−−−−→ T or W −−−−−→ (Z, Z 0 ). We define the type (P1 , . . . , Pn )
of a pattern Z of height n by
Qj
i=1 zj,i
Pj := Qj−1
i=1 zj−1,i
∀j = 1, . . . , n ,
(3.1)
which is the analog of (2.2) in the geometric setting (here the empty product is equal to 1
by convention).
We now state some properties of geometric RSK, most of which have a straightforward
equivalent in the combinatorial setting, see [18] and [19].
gRSK
Theorem 3.1. Let W ∈ (R>0 )n×n and W −−−−−→ T = (Z, Z 0 ). Let (P1 , . . . , Pn ) and (Q1 , . . . , Qn )
denote the types of Z and Z 0 respectively, and (λ1 , . . . , λn ) their common shape. Then:
Pj =
n
Y
wi,j
and
∀j
Qi =
n
Y
i=1
wi,j
∀i ;
(3.2)
j=1
gRSK
W T −−−−−→ T T = (Z 0 , Z) ;
X
Y
λ1 · · · λk =
(3.3)
∀k ≤ n ;
wi,j
(3.4)
π(1) ,...,π(k) (i,j)∈π(1) ∪···∪π(k)
X 1
1
=
+
wi,j t1,1
i,j
X
ti−1,j + ti,j−1
(i,j),(1,1)
ti,j
.
(3.5)
In (3.4), the sum is over all k-tuples of non-intersecting directed lattice paths π(i) from
(1, i) to (n, n − k + i), for i = 1, . . . , k. In (3.5), the convention is that ti−1,j = 0 if i = 1 and
ti,j−1 = 0 if j = 1.
Formulas (3.2) are the equivalent of (2.1) in the geometric setting. Formulas (3.3)
and (3.4) are the analogs of Theorem 2.3 and Theorem 2.9 respectively in the geometric
setting. Formula (3.5) is the cornerstone of [19] and can be proved by induction on n.
3.2
Inverse gamma polymers
From (3.4) for k = 1, we see that
tn,n = λ1 =
X Y
wi,j
∀n ≥ 1 ,
(3.6)
π∈Πn,n (i,j)∈π
where Πn,n is the set of directed paths from (1, 1) to (n, n) in N2 . Similarly to what we
noticed in subsection 2.3 for last passage percolation, (3.6) is a point-to-point version of
the (1 + 1)-dimensional directed polymer partition sum defined in (1.2). Again, to see
2n−1 such that x = 0,
this we identify a path π ∈ Πn,n with a path x = (k, xk )2n−2
0
k=0 ∈ (N × Z)
xk−1 ∼ xk for 1 ≤ i ≤ 2n − 2 and x2n−2 = 0. If we set
wi,j = exp[βV (k, xk )]
21
for k = i + j − 2 and xk = −i + j, then
tn,n =
X
" 2n−2
#
X
exp β
V (k, xk ) ,
x
k=0
where the maximum is over all paths x defined above. The latter equation is clearly a
point-to-point version of (1.2).
We now wish to study the distribution of the polymer partition sum tn,n in the special
case of inverse gamma distributed weights. We recall that X −1 ∼ Γ (α, β) if on R>0
!
β α −α−1
β
dx .
P(X ∈ dx) =
x
exp −
Γ (α)
x
−1
If (wi,j )1≤i,j≤n are independent and wi,j
∼ Γ (αj + α̂i , 1) for all i, j, then the joint law of
matrix W is given by
Y
Y −α −α̂
X 1 ! Y dwi,j
1
j
i
να,α̂ (dw) =
wi,j
.
exp −
Γ (αj + α̂i )
wi,j
wi,j
i,j
i,j
i,j
i,j
We now need the following lemma:
Lemma 3.2. The gRSK mapping in logarithmic variables
log(wi,j )1≤i,j≤n → log(ti,j )1≤i,j≤n
has Jacobian ±1.
The proof is based on the local move decomposition of the gRSK mapping: it is easy
to see that all maps li,j have Jacobian ±1. This lemma permits to make the change of
variables from (wi,j ) to (ti,j ) in the measure να,α̂ :
Theorem 3.3. The push-forward of να,α̂ under the gRSK map T is given by
να,α̂ ◦ T
−1
(dt) =
Y
i,j
−α

 1
1
−α −α̂
P Q exp  −
−
Γ (αj + α̂i )
t1,1
−α
X
(i,j),(1,1)

ti−1,j + ti,j−1  Y dti,j

.

t
t
i,j
i,j
i,j
Here, P −α = P1 1 · · · Pn n , where (P1 , . . . , Pn ) is the type of Z as usual, and the same for
Q−α̂ . This theorem follows from (3.2), (3.5) and Lemma 3.2.
We are now able to give an expression for the Laplace transform of the polymer
0
partition sum, using the push forward of να,α̂ and writing (ti,j ) in terms of (zi,j ), (zi,j
) and
22
λ: for θ ∈ R,


i
 Y
 h

Γ (αj + α̂i )E e−θtn,n

i,j
Z
=
e
−θtn,n −α
P
Q
(R>0 )n×n
Z

 1
−
exp  −
t
1,1
e−θλ1
=
−α̂
(R>0 )n×n
n
Z
zj,1 . . . zj,j
j=1
zj−1,1 . . . zj−1,j−1
=
(R>0
)n
(i,j)
i,j
e−θλ1 −1/λn ψα (λ)ψα̂ (λ)
i,j
i,j
0
0
zj,1
. . . zj,j
i,j
!−α̂j
...
0
0
zj−1,1
. . . zj−1,j−1
j=1

0
0
0
n
zi−1,j + zi+1,j+1
 Y dzi,j Y dzi,j Y
dλi

+


0
0
zi,j
λi
zi,j
zi,j
1≤j≤i<n
n
Y
dλ
i=1
(i,j),(1,1)

ti−1,j + ti,j−1  Y dti,j


t
t
!−αj Y
n
n
Y

 1 X zi−1,j + zi+1,j+1
−
. . . exp  −
λ
z
X
λi
i
1≤j≤i<n
i=1
.
The convention is that zi,j = 0 whenever (i, j) does not satisfy 1 ≤ j ≤ i ≤ n. In the latter
formula, ψα and ψα̂ (where α = (α1 , . . . , αn ) and α̂ = (α̂1 , . . . , α̂n )) are GL(n, R)-Whittaker
functions, which play an analogous role of Schur polynomials in the geometric setting:


!−αj
Z Y
n
zj,1 . . . zj,j
 X zi−1,j + zi+1,j+1  Y dzi,j

ψα (λ) =
exp  −
,

zj−1,1 . . . zj−1,j−1
zi,j
zi,j
j=1
(i,j)
1≤j≤i<n
where the integral is over all triangular patterns (zi,j ) of height n and fixed shape λ =
(zn,1 , . . . , zn,n ).
3.3
Variations and future developments
In [19], O’Connell, Seppäläinen and Zygouras also analyze the case of symmetric input
matrix with inverse gamma distributed weights: in this case, by (3.3), the output matrix is
also symmetric. They prove that an analog of Lemma 3.2 holds, i.e. the geometric RSK
map in logarithmic variables has Jacobian ±1: again, this leads to a successful change of
variable in the polymer measure, and to an expression of the Laplace transform for the
polymer partition sum in terms of Whittaker functions. In the same spirit, one might try
to figure out what happens with antisymmetric matrices, for example finding a sequence
of local moves that somehow connects the antisymmetric case to the symmetric one, and
then computing the Jacobian of such a transformation.
More variations on the same theme are possible, imposing other constraints on the
geometry of the input matrices, and thus of the polymers. In [19], the case of triangular
input arrays is also analyzed: the corresponding polymers are forced to stay “below a
hard wall”. A further interesting open problem consists in analyzing the behavior of the
geometric RSK and the corresponding polymers for paths not allowed to go outside a
diagonal strip.
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