Asymptotic Combinatorics and Algebraic Analysis

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1 Asymptotic Combinatorics and Algebraic Analysis ANATOLY M. VERSHIK* Steklov Mathematical Institute St. Petersburg Branch Fontanka 27, St. Petersburg , 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, vershik@pdmi.ras.ru Proceedings of the International Congress of Mathematicians, Zürich, Switzerland 1994 Birkhäuser Verlag, Basel, Switzerland 1995

2 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 S 7l (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 S n 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.) PROBLEM. Describe the asymptotic behavior, i.e. find the common limit distribution of a complete system of invariants of the representations. 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 p n ; how do we scale the space V(n) in order to obtain the true nontrivial limit distribution of the measures p n? 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

3 1386 Anatoly M. Vershik problems. Other examples for the same situation are the so-called Richardson- Eden 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). x(t) = ^2 *ii i>t \{t) = V" Ki; Vn i>tlpn Figure 1 K i = #{i : x j = *}; ip n ilj n = rc. (a) Haar statistics (for conjugacy classes in symmetric groups): Let A G V(n) and let ki,..., k n be the multiplicities of the summands 1,2,..., rc respectively. Then "KW = ft o^: m=l

4 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 \ \ v 1 ' 5 1 y / \ 0.5 i.... i.... \ /.... i.... i îî(s x _ f - (s arcsin s + \/l s2 ) ; IN Figure 2 kl >i- (d) Plancherel statistics on Young diagrams: tfw=n\/([lh a f where Q is a cell of the Young diagram, h a 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.

5 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 xß n 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 p n 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 p n is a ^-measure at some point of the space this is usually some curve "continuous" diagram. If the limit of the measures p n does exist but is a nondegenerate measure we will say that the case is nonergodic. THEOREM 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 = (Ai,..., A n ): A* > A?: /rc fori = l,...,rc, 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. Now we will give the precise answer to the questions about limit measures or limit shapes.

6 Asymptotic Combinatorics and Algebraic Analysis 1389 e v^ + e VG 2 ' = 1 Figure 4 THEOREM 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 n(s) /(2/7T)(sarcsins + i/(l s 2 )) for a; < 1 \s\ 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 p n on V(n) has the property p n {X : the normalized A G V^T)} > 1 e, where T is the limit shape curve, V e 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).

7 1390 Anatoly M. Vershik e-cx + e-cy = 1 + e-cx. X = p = 2 => c« Figure 5 Let me give some examples of the applications of Theorem 2. For A G V(n) let dim A = nl/j\h Q (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. 0 < ci < = 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. THEOREM 4. Ump' b l ixev(n) for all a G M and all e > {; -#{«: <p(i) <a}- Erf( a) <4 =

8 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 fc )} -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 LEMMA. Let p(n,m) = Coeff(t n s m :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)) 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]).

9 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 + b x z + b 2 z 2 + and let fi(t) = 1 + ant + a^t 2 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 p n 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 c n is a constant; the second set is a one-parametric set of measures v t (t G (0,1)) on the big canonical set \jv(n), rc G N, and defined as follows: v t (X) = (F(t))- l b n i n, where A G V(n). The simple observation is that the number of summands of a given size is independent of v t 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. THEOREM 5. There exists a sequence t n such that the main terms of the asymptotics of the expected smooth functionals on V(n) with respect to the measures p n and i/ tn coincide. 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 T n that are close to V in the uniform metric. THEOREM 6. [V4] where K is a curvature oft. rc"? int r = 3 {M(L*" d ' + w ) 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.

10 Asymptotic Combinatorics and Algebraic Analysis 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. LEMMA. There is a unique projection p n : S n S n -i that commutes with the two-sided action of S n -i. The inverse limit X = \im(s n,p n ) 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]). THEOREM 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. 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), [KI] S. Kerov, Gaussian limit for the Plancherel measure of the symmetric groups, C. R. Acad. Sci. 316 (1993), [K2] S. Kerov, Transition probabilities of continuous Young diagrams and the Markov moment problem, Functional Anal. Appi. 27 (3) (1993), [KOV] S. Kerov, G. Ol'shansky, and A. Vershik, Harmonic analysis on the infinite symmetric group, C. R. Acad. Sci. 316 (1993), [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), [KV2] S. Kerov and A. Vershik, Characters and factor representations of the infinite symmetric group, Soviet Math. Dokl. 23 (5) (1981), [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),

11 1394 Anatoly M. Vershik [V5] [Kh] A. Khinchin, Mathematical Foundations of Quantum Statistics, Moscow, 1951, (in Russian). [LS] B. Logan and L. Shepp, A variational problem for random Young tableaux, Adv. in Math. 26 (1977), [SV] 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), [S] Ya. Sinai, Probabilistic approach to analysis of statistics of convex polygons, Functional Anal. Appi. 28 (2) (1994), [VI] A. Vershik, Asymptotic distribution of decompositions of natural numbers into prime divisors, Soviet Math. Dokl. 289 (2) (1986), [V2] A. Vershik, Statistical sum associated with Young diagrams, J. Soviet Math. 47 (1987), [V3] A. Vershik, Asymptotic theory of the representation of the symmetric group, Selecta Math. Soviet. 19 (2) (1992), [V4] A. Vershik, Limit shape of convex lattice polygons and related questions, Functional Anal. Appi. 28 (1) (1994), A. Vershik, Statistical physics of combinatorial partitions and its limit configuration, Fune. Anal. 30 (1) (1996). [Ya] 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), (in Russian).