Asymptotic Combinatorics and Algebraic Analysis
ANATOLY M. VERSHIK*
Steklov Mathematical Institute
St. Petersburg Branch
Fontanka 27, St. Petersburg 191011, Russia
1 Asymptotic problems in combinatorics and their algebraic equivalents
A large number of asymptotic questions in mathematics can be stated as combinatorial problems. I can give examples from algebra, analysis, ergodic theory, and
so on. Therefore the study of asymptotic problems in combinatorics is stimulated
enormously by taking into account the various approaches from different branches
of mathematics. Recently we found many new aspects of this development of combinatorics. The main question in this context is: What kind of limit behavior can
have a combinatorial object when it "grows" ?
One of the recent examples we can find in a very old area, namely the theory
of symmetric and other classical groups and their representations. Let me quote the
remarkable words of Weyl from his book Philosophy of mathematics and natural
science (1949): Perhaps the simplest combinatorial entity is the group of permutations of rc objects. This group has a different constitution for each individual
number rc. The question is whether there are nevertheless some asymptotic uniformities prevailing for large rc or for some distinctive class of large rc. He continued:
Mathematics has still little to tell about such a problem.
In the meantime, a lot of progress has been made in this direction. We should
mention the names of some persons who have made important contributions to
this area, namely P. Erdös, V. Goncharov, P. Turan, A. Khinchin. W. Feller, and
others. In the more general context of what is called nowadays the asymptotic
theory of representations, I want to mention the names of H. Weyl and J. von
Neumann.
2 Typical objects in asymptotic combinatorial theory
Besides the symmetric groups there arc other classical objects in mathematics and
in combinatorics, namely partitions of natural numbers. They provide another
source of extremely important asymptotic problems that are also closely related
to analysis, algebra, number theory, measure theory, and statistical physics.
The third class of objects, which plays the role of a link between combinatorics
on one side and algebra and analysis on the other side, is a special kind of graphs,
email: vershik@pdmi.ras.ru
Proceedings of the International Congress
of Mathematicians, Zürich, Switzerland 1994
© Birkhäuser Verlag, Basel, Switzerland 1995
Asymptotic Combinatorics and Algebraic Analysis
1385
the so-called Bratteli diagrams, i.e. Z+-graded locally finite graphs. These are the
combinatorial analogues of locally semisimple algebras. This important class of
algebras arises in asymptotic theory of finite and locally finite groups, and can be
considered as an algebraic equivalent of asymptotic theory in analysis.
We now have described some of the objects that used to be basic in the theory
of asymptotic combinatorial problems.
3 Problems
Next we will formulate the typical problems for these objects. We will start with the
problems related to symmetric groups. It is important to emphasize that the same
problems can also be stated for any other series of classical groups like Coxeter
groups, GL(rc,Fp), and so on.
In all the considerations we use some probability measure. For example it is
natural to provide the symmetric group S7l (rc G N) with the uniform distribution
(Haar measure).
PROBLEM. Describe the asymptotic behavior (on n) of conjugacy classes; more
precisely, find the common limit distribution of the numerical invariants of the
classes.
Now let us consider linear representations of those groups or their dual objects
Sn provided with the Plancherel measure. (Then the measure of a representation
is the normalized square of its dimension; this is the right analogue of the Haar
measure for the dual space. The deep connection between these two measures is
given by the RSK-Robinson-Shensted-Knuth-correspondence.)
Describe the asymptotic behavior, i.e. find the common limit distribution of a complete system of invariants of the representations.
PROBLEM.
For the symmetric groups there are natural parameters both for conjugacy
classes, namely the lengths of cycles, and for representations, namely Young diagrams. So we have to study asymptotic combinatorial problems about random
partitions or random Young diagrams. Both of these problems were posed by the
author in the early 1970s and were solved in the 1970s in joint papers of the author with Kerov (see [KV1]) and Schmidt (see [SV]), and also partially in papers
of Shepp and Logan (see [LS]).
Now let us consider partitions of natural numbers V(n). As we mentioned before, the previous problems can be reduced to problems about partitions. Roughly
speaking, all the questions concern the following problem: suppose we have some
statistics on the space of partitions V(n) for all rc, say pn; how do we scale the
space V(n) in order to obtain the true nontrivial limit distribution of the measures
pn? The same question can be asked for Young diagrams, graphs, configurations,
and higher-dimensional objects of such a type.
A possible kind of answer can be a limit-shape theorem, which asserts that the
limit distribution is a ó-measure concentrated at one configuration, called the limit
shape of the random partition diagram, configuration, etc. In the problems that we
discuss below, examples are Plancherel statistics, uniform statistics, and convex
Anatoly M. Vershik
1386
problems. Other examples for the same situation are the so-called RichardsonEden model in the probability theory of a many-particles system, one-dimensional
hydrodynamics, Maxwell (Bell) statistics on partitions, etc. We obtain rich information about asymptotics from the properties of the limit shape.
In the opposite (nonergodic) case the limit distribution is a nondegenerate
distribution. Examples for this case are conjugacy classes in symmetric groups
(see above) and other series of classical groups, and harmonic measures on Young
diagrams.
In all these examples we have a completely different scaling in comparison
with the first case. The dichotomy of the two cases can be compared with the
dichotomy of trivial and nontrivial Poisson boundary in probability theory —
there is a deep analogy. A systematic theory as well as general criteria for the two
cases are still unknown.
4 Results
Here we will list the main results that have been obtained in this direction. Let us
first define some of the important statistics on partitions, some of which we have
mentioned shortly above. We use the bijection between Young diagrams with rc
cells and the set of partitions V(n) of rc (see Figure 1).
K
i = #{i:
x(t) = ^2 *ii
x
j = *};
i>t
\{t) = — V " Ki;
Vn i>tlpn
ipniljn = rc.
Figure 1
(a) Haar statistics (for conjugacy classes in symmetric groups): Let A G V(n) and
let ki,..., kn be the multiplicities of the summands 1,2,..., rc respectively. Then
"KW =m =ftl o ^ : •
Asymptotic Combinatorics and Algebraic Analysis
1387
(b) Uniform statistics:
/£(*) = p(n)
where p(n) is the Euler-Hardy-Ramanujan function.
(c) Maxwell or Bell statistics:
rfw-nensnK
This is the image on V(n) of the uniform distribution on partitions of n distinct
objects.
2
\
'
1
\ v
\
i
/
15
\
i
. . . .
. . .
y
0.5
. \/ . . . .
i
.
x _ f - (s arcsin s + \ / l — s 2 )
îî(s ;
IN
.
.
.
i
kl >i-
Figure 2
(d) Plancherel statistics on Young diagrams:
tfW=n\/([lhaf
where Q is a cell of the Young diagram, ha is the hooklength of the cell a, and the
product here is taken over all cells of the diagram. This is the probability of the
diagram (or partition) as a representation of the symmetric group: the probability
is proportional to the square of the dimension of the representation.
(e) Uniform statistics on Young diagrams which sit inside a given rectangle.
1388
Anatoly M. Vershik
(f) Fermi statistics on Young diagrams (no rows with equal length).
(g) Uniform statistics on convex diagrams: This is one of the first two-dimensional
problems. We consider the set of all diagrams inside a given square of a lattice
whose border is convex (or concave). This set can be considered as the set of vector
partitions of the vector (rc, rc). The correspondence with the previous description
is established by considering the slopes of the edges (see Figure 3), etc.
y/1 - \x\ + y/1 - \y\ = 1
Figure 3
All the measures are defined for all natural numbers rc. In order to be able to
speak about convergence we have to normalize or rescale the axes of the diagrams
(partitions), dividing by appropriate sequences of numbers <j>n and xßn which depend on the cases; the choice of those numbers is unique (and they exist). Suppose
the rescaling is done and wc can consider the measures pn for all rc in the same
limit topological space of normalized diagrams. Let us say that we have the ergodic case if the weak limit of the measures pn is a ^-measure at some point of the
space — this is usually some curve — "continuous" diagram. If the limit of the
measures pn does exist but is a nondegenerate measure we will say that the case
is nonergodic.
1. The case (a) is nonergodic, the cases (b)-(e) are ergodic. The normalizations of the axes are the following:
(a) For A G V(n), X = ( A i , . . . , A n ): A* —> A?:/rc fori = l , . . . , r c , there is no
normalization along the second axis.
(b), (d), (e), (f), (g) The normalization along both axes is 1/y/n.
(c) For X G V(n), the normalization of the values of X — s is 1/lnrc and the
normalization of the indices is rc/lnrc.
THEOREM
Now we will give the precise answer to the questions about limit measures or
limit shapes.
Asymptotic Combinatorics and Algebraic Analysis
e v^ + e
VG2'
1389
= 1
Figure 4
2. The following curves are the limit shapes:
Case (b) exp[— (n/\/6)x] +exp[-(7r/\/6)y] = 1 (see Figure 4),
Case (c) y(x) = 1.
Case (d) Let w = (x +1/)/2, s = (x — y)/2, then
THEOREM
2
n(s)
/(2/7T )(sarcsins + i / ( l — s ))
\s\
for |a;| < 1
for bl > 1
(Kerov and Vershik [KVÎ], Logan and Shepp [LS]), see Figure 2.
Case (e) (1 — exp[-cA]) exp[—cy] + (1 — exp[—cp]) exp[—cx] = 1 — exp[-c(A + p)].
X,p are the size of the rectangle, c = c(X,p). (See Figure 5 for the case
X = p = 2).
Case (f) exp[-ir/y/l2y] - cxpt-Tr/y 7 !^] = 1.
Case(g) y / l ^ \ + y/ï^]y~\ = 1
(Barany [B], Sinai [S], Vershik [V4]), see Figure 3.
This means that in each of those cases the following is true: for any e > 0
there exists an N such that if rc > N then the measure pn on V(n) has the property
pn{X : the normalized A G V^T)} > 1 — e, where T is the limit shape curve, Ve is
the e-neighborhood of the curve T in the uniform topology, and the normalization
of the diagram for the cases is as above.
A completely different situation occurs in case (a), i.e. the limit distributions
of normalized lengths of cycles. The complete answer for this case was given in
a joint paper with Schmidt [SV]. In this case the limit measure is concentrated
in the space of positive series with sum 1. This remarkable measure also appears
in the context of number theory (the distribution of logarithms of prime divisors
of natural numbers) as was recently described by the author [VI] (see also P.
Billingsley, D. Knuth, and Trab-Pardo).
Anatoly M. Vershik
1390
e-cx
+
e-cy
=
1 +
e-cX.
X = p = 2 => c « 0.853138
Figure 5
Let me give some examples of the applications of Theorem 2.
For A G V(n) let dim A = nl/J\hQ (hook formula).
PROBLEM.
Find maxjdimA : A G V).
This functional is rather complicated and, as H. Weyl suspected, the optimal
diagram has a different feature for each rc, hardly depending on the arithmetic
of rc. But it happens that asymptotically there is a prevailing form of diagram: this
is again the limit shape Q that I mentioned above. We obtain:
THEOREM
3 (Kerov and Vershik [KV3]). There are constants ci and c^ such that
1 _ ,
dim A.
•= lnfmax —•==-) < C2 < oc .
v^
Vn!
It is an important observation that the average diagram with respect to the
Plancherel measure asymptotically coincides with the diagram of maximal dimension. Another important fact is that the limit shape Q arises in many different
contexts such as the asymptotic of the spectrum of random matrices, zeros of the
orthogonal polynomials, etc.
For the case of Maxwell-Bell statistics the limit shape is not interesting —
then the generic block of the partition has size r where r is a solution of the
equation x exp x = rc. But it is possible to refine this answer in the spirit of CLT:
Let ip(i) = (b(i) — r)/y/r, where b(i) is the length of the block containing i.
0 < ci <
THEOREM 4.
41 {;
Ump'bliXeV(n)
for all a G M and all e > 0.
- # { « : <p(i) <a}-
a) <4 =
Erf(
Asymptotic Combinatorics and Algebraic Analysis
1391
This theorem together with case (c) of Theorem 2, proven by my students
Yu. Yakubovich and D. Alexandrovsky, describes the limit structure of generic
finite partitions (see [Ya]).
5 Techniques
Now we will discuss some technical aspects that are important by themselves. We
emphasize four tools:
(i) generating functions and the Hardy method, the saddle-point method;
(ii) the variational principle in combinatorics;
(iii) functional equations and the ergodic approach;
(iv) methods from statistical physics: big canonical set and the local limit theorem
of probability theory.
The classical method for studying enumeration problems uses generating
functions. For our goals we also can use them, but with some modification: we
need to consider the generating function for the number of combinatorial objects
with some special properties. For example, instead of the Euler function for partitions F(z) = {IlfcliCl ~~ 2 f c )} - 1 we have to use the generating function for the
number of partitions with a given number of blocks whose lengths are less than
a constant. In a different context such a method has been applied by Turan and
Szalay.
For the higher-dimensional case (g) of convex diagrams or convex lattice
polygons we introduce a new kind of generating function of two variables:
F(t,s)=
n
(i-tv-)- 1 .
(fc,r) coprirne
Let p(n,m) = Coeff(tnsm:F);
then this is the number of convex lattice
polygons in the rectangle (0, rc) x (0,ra) which meet the points (0,0) and (rc, rrc).
The formula
ln(p(n,n))/n 2 / 3 = 3^C(3)/<(2)(1 + o(l))
LEMMA.
holds. This formula also gives us the number of vector partitions without collinear
summands.
The two-dimensional saddle point method applied to this function is the main
ingredient to obtain the limit-shape theorem for this case. But in addition we need
considerations on generating functions (see Barany [B] and Vershik [V4]). We can
combine this approach with an approach from statistical physics. For case (g) this
was done by Sinai [S]. In the early 1950s A. Khinchin [Kh] used this method in
statistical physics.* Here wc present a general context that covers all these papers.
The main idea is the following. Instead of studying the asymptotics of the
coefficients of the generating function with the help of methods from the theory of
*)
We want to emphasize that the difference between the saddle point method (Darwin and
Fauler's method) and the local limit theorem (Khinchin's approach) is not so big: technically to find a saddle point is the same as to find the value of a parameter which realizes
a needed mathematical expectation. (See [V5]).
1392
Anatoly M. Vershik
complex variables we can introduce one-parametric families of measures for which
the natural coordinates, say the number of rows of given length, are independent.
After this we can easily find the distribution of the functionals and then (the
hardest part) prove that the distributions are the same as in the initial problem.
This is completely analogous to the method of equivalence of great and small
canonical sets.
The general definition: let F(t) = Y\fi(t) = 1 + bxz + b2z2 + • • • and let
fi(t) = 1 + ant + a^t2 H
be series converging in a circle and with nonnegative
coefficients a^, i = 1,2,
Now we introduce two sets of measures on the set
of partitions: the first set consists of the measures pn on the sets V(n), defined
by p(X) = CnHaijft, where a partition A = (j(l),..., j(n)) G V(n) has j(s)
summands equal to s, and cn is a constant; the second set is a one-parametric set
of measures vt (t G (0,1)) on the big canonical set \jV(n), rc G N, and defined as
follows:
vt(X) = (F(t))-lbnin,
where A G V(n).
The simple observation is that the number of summands of a given size is independent of vt and consequently we can use powerful methods of probability theory
such as large deviations and so on. The main fact is contained in the following
theorem.
5. There exists a sequence tn such that the main terms of the asymptotics of the expected smooth functionals on V(n) with respect to the measures pn
and i/tn coincide.
THEOREM
But we cannot claim that the expected distributions of the functionals also
coincide. For this we need some additional assumptions. In particular, it is true in
the cases (b)-(e) above. This assertion is essentially the above-mentioned equivalence between two sets. The technique is slightly simpler but parallel to the saddle
point method.
Variational principles in these problems are very useful — we will give some
examples. From the combinatorial point of view the variational principle defines a
functional on the configuration (or diagram, partitions) which gives the main term
in the asymptotics (like energy or entropy).
Suppose the continuous diagram T is fixed (see Figure 3) and T is the graph
of a differentiable function 7(-). We want to find the asymptotic of the number of
convex diagrams Tn that are close to V in the uniform metric.
THEOREM 6.
[V4]
rc"? inT r 3
•°= {M(L*"°d'+°w)
where K is a curvature ofT.
This means that in case (g) the integral takes its maximal value on the class
of all monotone differentiable curves in the limit shape curve.
Asymptotic Combinatorics and Algebraic Analysis
1393
6 Combinatorics of infinite objects
Our approach can be extended by considering certain limit objects preserving the
combinatorial structure. That gives us an interpretation of some limit distributions
that previously appeared as pure limit objects.
The best example of such an extension is the notion of virtual permutations.
This theory is developed in a recent paper by Kerov, Ol'shansky, and myself [KOV].
The main definition is the following.
There is a unique projection pn: Sn —• S n - i that commutes with the
two-sided action of Sn-i.
LEMMA.
The inverse limit X = \im(Sn,pn) is called the space of virtual permutations.
On this space X a two-sided action of the infinite symmetric group is well defined.
The space of virtual permutations does not form a group, but it is possible to
define the notion of cycles, and their normalized length, Haar measure, and other
group-like notions. We will not give the precise definitions for those notions here
(see [KOV]).
7. The common distribution of normalized lengths of virtual permutations coincides with the above-defined limit distribution for normalized lengths of
cycles of ordinary permutations.
THEOREM
The main application of the space of virtual permutations consists in the
construction of some new types of representations of the infinite symmetric group
like the regular representation based on the analogue of the Haar measure and its
one-parametric deformation.
One of the main problems in this area is the problem of limit shapes for multidimensional configurations, Young diagrams etc. Perhaps variational principles
with ideas coming from statistical physics will help to solve them (see [V5]).
References
[B]
I. Barany, The limit shape theorem of convex lattice polygons, Discrete Comput.
Geom. 13 (1995), 279-295.
[KI]
S. Kerov, Gaussian limit for the Plancherel measure of the symmetric groups, C.
R. Acad. Sci. 316 (1993), 303-308.
[K2]
S. Kerov, Transition probabilities of continuous Young diagrams and the Markov
moment problem, Functional Anal. Appi. 27 (3) (1993), 32-49.
[KOV] S. Kerov, G. Ol'shansky, and A. Vershik, Harmonic analysis on the infinite symmetric group, C. R. Acad. Sci. 316 (1993), 773-778.
[KV1] S. Kerov and A. Vershik, Asymptotics of the Plancherel measure of the symmetric
group and limit shapes of Young diagrams, Soviet Math. Dokl. 18 (1977), 527531.
[KV2] S. Kerov and A. Vershik, Characters and factor representations of the infinite
symmetric group, Soviet Math. Dokl. 23 (5) (1981), 1037-1040.
[KV3] S. Kerov and A. Vershik, The asymptotic of maximal and typical dimension of
irreducible representations of the symmetric group, Functional Anal. Appi. 19
(1) (1985), 21-31.
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[Kh]
[LS]
[SV]
[S]
[VI]
[V2]
[V3]
[V4]
[V5]
[Ya]
Anatoly M. Vershik
A. Khinchin, Mathematical Foundations of Quantum Statistics, Moscow, 1951,
(in Russian).
B. Logan and L. Shepp, A variational problem for random Young tableaux, Adv.
in Math. 26 (1977), 206-292.
A. Schmidt and A. Vershik, Limit measures that arise in the asymptotic theory
of symmetric groups I, Teor. Veroyatnast. i. Primenen 22 (1) (1977), 72-88; II,
Teor. Veroyatnast. i. Primenen. 23 (1) (1978), 42-54.
Ya. Sinai, Probabilistic approach to analysis of statistics of convex polygons,
Functional Anal. Appi. 28 (2) (1994), 41-48.
A. Vershik, Asymptotic distribution of decompositions of natural numbers into
prime divisors, Soviet Math. Dokl. 289 (2) (1986), 269-272.
A. Vershik, Statistical sum associated with Young diagrams, J. Soviet Math. 47
(1987), 2379-2386.
A. Vershik, Asymptotic theory of the representation of the symmetric group,
Selecta Math. Soviet. 19 (2) (1992), 281-301.
A. Vershik, Limit shape of convex lattice polygons and related questions, Functional Anal. Appi. 28 (1) (1994), 16-25.
A. Vershik, Statistical physics of combinatorial partitions and its limit configuration, Fune. Anal. 30 (1) (1996).
Yu. Yakubovich, Asymptotics of a random partitions of a set, Zap. nauch. sem.
POMI, Representation theory, dynamical systems, combinatorial and algorithmic methods I 223 (1995), 227-249 (in Russian).