MATH 669 ( COMBINATORICS AND COMPLEXITY OF PARTITION FUNCTIONS ) NOTES. Alexander Barvinok. April 3, 2017

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1 MATH 669 COMBINATORICS AND COMPLEXITY OF PARTITION FUNCTIONS ) NOTES Alexander Barvinok April 3, 2017 Abstract. These are condensed notes for the course, updated as the course progresses. While the main content of the lectures is reflected here, some details and, occasionally, topics might be missing. 1. Matching polynomial 1.1) Weighted graphs, matchings and matching polynomials. We consider a finite, undirected, without multiple edges or loops) weighted graph G = V,E;a) with set V of vertices, set E of edges and non-negative real weights a : E R + on the edges. A matching in G is a set e 1,...,e k of pairwise disjoint edges of weight ae 1 ) ae k ). We consider the empty set a matching of weight 1. The matching polynomial is the univariate polynomial Mat G λ) = 1+ e 1,...,e k matching λ k ae 1 ) ae k ). Together with Mat G λ) it is convenient to consider the matching defect polynomial q G λ) = λ V Mat G 1 λ 2 ). It is easy to see that q G λ) is indeed a monic polynomial of degree V. One gets a lot of milage from the following simple recurrence: 1.1.1) Mat G λ) = Mat G v λ)+λ u V: {v,u} E a vu Mat G v u λ), where v V is a vertex, G v is the graph obtained from G by deleting v and all incident edges, G v u is the graph obtained from G by deleting vertices v and u 1 Typeset by AMS-TEX

2 and all incident edges and a uv is the weight on the edge {v,u}. Consequently, for the matching defect polynomial, we get 1.1.2) q G λ) = λq G v λ) u V: {v,u} V a vu q G v u λ). Our first main result is the Heilmann - Lieb Theorem 1972). 1.2) Theorem. The roots of Mat G λ) are negative real. The proof uses the concept of interlacing polynomials. 1.3) Interlacing polynomials. Let f be a polynomial of degree n and n distinct real roots α 1 < α 2 <... < α n and let g be a polynomial of degree n 1 and n 1 distinct real roots β 1 < β 2 <... < β n 1. We say that g interlaces f if α i < β i < α i+1 for i = 1,...,n 1. A canonical example is supplied by the Rolle s Theorem: f is a polynomial with distinct real roots and g = f. Here is a useful lemma. 1.4) Lemma. 1) Let f and g 1,...,g k be real polynomials such that each g k interlaces f and the highest degree terms of g 1,...,g k have the same sign. Let a 1,...,a k be non-negative real, not all equal to 0, and let g = a 1 g a k g k. Then g interlaces f. 2) Suppose that g interlaces f and the highest terms of f and g have the same sign. Then for any real a, the polynomial f interlaces the polynomial hλ) = λ a)fλ) gλ). Sketch of Proof. Let α 1 < α 2 <... < α n be the roots of f. To prove Part 1), we note that each g k changes its sign once on each interval between consecutive roots of f and all g k change the sign in the same way. Hence g also changes its sign once on each interval between consecutive roots of f and g interlaces f. To prove Part 2) we note that h changes its sign once on each interval between consecutive roots of f and also once on the interval,α 1 ) and once on the interval α n,+ ). Now we are ready to prove the Heilmann - Lieb Theorem. 1.5) Sketch of proof of Theorem 1.2. It suffices to prove that the roots of q G λ) are real. Since the roots of a non-zero polynomial depend continuously on the polynomial, it suffices to prove that the roots of q G λ) are real, assuming that G is the complete graph on n vertices and that a uv > 0 for all {u,v} E. We use 1.1.2) to prove that by induction on n that the roots of q G λ) are real and, moreover, for every v V, the polynomial q G v λ) interlaces q G λ). This is checked immediately if V = 2. Suppose that V > 2 and let v V be a vertex. By the induction hypothesis, for any u V such that {v,u} E the polynomial q G v u λ) interlaces q G v λ). Using 1.1.2) and Lemma 1.4, we conclude that q G v λ) interlaces q G λ). 2

3 1.6) Examples Problems). In what follows, ae) = 1 for all e E ) Interval. Let V = {1,...,n} and let E consist of all pairs {i,i+1} for i = 1,...,n 1. Then q G λ) = U n λ/2), where U n is the Chebyshev polynomial of the second kind, defined by U n x) = sinn+1)θ sinθ where cosθ = x ) Circle. Let V = {1,...,n} and let E consist of all pairs {i,i + 1} for i = 1,...,n 1 and {n,1}. Then q G λ) = 2T n λ/2), where T n is the Chebyshev polynomial of the first kind defined by T n x) = cosnθ where cosθ = x ) Complete graph. Let V = {1,...,n} and let E consist of all pairs {i,j} for 1 i < j n. Then q G λ) = H n λ), where H n is the Hermite polynomial defined by H n x) = 1) n e x2 /2 dn 2 /2. dx ne x 1.6.4) Complete bipartite graph. Let V = {1,...,2n} and let E consist of all pairs {i,j} where 1 i n and n +1 j 2n. Then q G λ) = 1) n n!l n λ 2 ), where L n is the Laguerre polynomial defined by L n λ) = ex d n e x n! dx n x n). 1.7) Bounds on the roots Problems). Given non-negative weights a : E R + on the edges of G, let us define β G = max v V u V {v,u} E If q G λ) = 0 then λ 2 β G and if Mat G λ) = 0 then λ 1. The bounds 4β G are pretty useful, though not optimal. Suppose that a e = 1 for all e E and that G) is the largest degree of a vertex of G, so that β G = G). If q G λ) = 0 then λ 2 G) 1 and if Mat G λ) = 0 1 then λ. These bounds are asymptotically optimal. 4 G) 1) Here is a useful property log-concavity) of the coefficients of a real polynomial with all roots real. 3 a vu.

4 1.8) Theorem. Suppose that the roots of the real polynomial n px) = a k x k are also real. Let Then k=0 b k = a k n for k = 0,...,n. k) b 2 k b k+1 b k 1 for k = 1,...,n 1. If a k 0 for k = 0,...,n then the above inequalities imply that a 2 k a k+1a k 1 for k = 1,...,n 1. Sketch of Proof. By the repeated application of the Rolle s Theorem, the roots of the polynomial qx) = dk 1 dx k 1px) of degq n k +1 are real. Then the roots of the polynomial ) 1 rx) = x n k+1 q x of degr n k +2 are real. Then the roots of the quadratic polynomial sx) = dn k 1 dx n k 1rx) are real. Hence the discriminant of sx) is non-positive, which implies the desired inequality. 2. Correlation decay 2.1) The probability space of matchings. Let G = V, E) be a graph for now, we let ae) = 1 for all edges) and let λ 0 be a parameter. We consider the set of all matchings in G as a finite) probability space, with probability of a matching of k edges equal to λ k /Mat G λ) for k = 0,1,... We are interested in the following question: given a vertex v of G, what is the probability that a random matching contains v? We introduce a metric on G, where distu,v) is the smallest number of edges of G in a path connecting u and v we allow distu,v) = + if u and v lie in different components of the graph). Let > 1 be an upper bound on the degrees of vertices of G. Our main goal is to establish the correlation decay result of Bayati, Gamarnik, Katz, Nair and Tetali 2007) that says that within an additive error 0 < ǫ < 1, the probability that a random matching contains v depends only on the structure of G in an m-neighborhood of v for some m = mǫ,λ, ). One can choose, roughly, m = O λ ln 1 ǫ 4 ).

5 2.2) Probabilities and recursions. For a set S V and v V \S, let p S,v = p S,v λ) denote the probability that a random matching does not contain v provided it contains no vertices from S. Denoting by G S the graph obtained from G by deleting all vertices in S together with incident edges, from 1.1.1) we get Mat G S λ) = Mat G S v λ)+λ u V\S: {v,u} E Mat G S v u λ), which we rewrite as Mat G S v λ) Mat G S λ) = 1+λ u V\S: {v,u} E Mat G S v u λ) Mat G S v λ) 1 and further interpret as 2.2.1) p S,v = 1+λ u V: {v,u} E p S v,u 2.3) Theorem. Let us consider the set of all non-negative vectors x = x S,v ) with coordinates parameterized by pairs S V and v V \ S and let T be a transformation of this set defined by 1) If then y = Tx) where y S,v = 1+λ 1 u V\S {v,u} E. x S v,u 1 1+λ x S,v 1 for all S,v 1 1+λ y S,v 1 for all S,v; 2) Suppose that y = Tx ) and y = Tx ). Then 1. max lny S,v lny S,v S,v λ 1+λ max lnx S,v lnx S,v. S,v 5

6 Sketch of Proof. Part 1) is obvious and Part 2) follows since if we write T in the logarithmic coordinates, that is, if so that then x S,v = e ξ S,v η S,v = ln 1+λ u V\S {v,u} E η S,v ξ S v,u and y S,v = e η S,v u V\S {v,u} E e ξ S v,u λ 1+λ. Theorem 2.3 asserts that T is a contraction and while 2.2.1) implies that p S,v ) is the necessarily unique) fixed point of T. Let us define x by x S,v = 1 for all S V and v V \S and let y = T m x). Then, by Theorem 2.3, we have ) m ln1+λ ). In particular, if we choose then max S,v lny S,v lnp S,v λ 1+λ m 1+λ )ln 1 ǫ +lnln1+λ ), lny S,v lnp S,v ǫ for all S V and v V \S. Next, we note that computing y,v, we need to access only the coordinates S,u), wherethetheverticesofs andulieinthem-neighborhoodofv. Henceweconclude: 2.4) Corollary. Let p v be the probabilitythat a random matching does not contain vertex v. For any m 1+λ )ln 1 ǫ +lnln1+λ ), up to an additive error of 0 < ǫ < 1, the value of lnp v is determined by the m- neighborhood of v in G. 2.5) Remarks. In fact, Bayati, Gamarnik, Katz, Nair and Tetali 2007) established a stronger bound, by proving that T 2 isacontractionwith a factor of roughly λ We can incorporate non-negative weights a e on the edges of G, roughly by replacing by throughout. max v V u V: {v,u} E 6 a vu

7 3. Graphs of large girth In what follows, we consider the set of all matchings in a graph G = V,E) as a probability space, with probability of a matching with k edges equal λ k /Mat G λ) for some fixed λ 0. We are going to exploit the correlation decay phenomenon. 3.1) Tree T k n. Let k 2. We define the rooted k-tree Tk n with n levels as follows: the root at level 0) is connected to k 1 vertices descendants) at level 1, each vertex at level 1 is connected to the root and k 1 vertices at level 2, etc., with each vertex at level m connected to one vertex at level m 1 ancestor) and k 1 vertices at level m+1 descendants). Finally, each vertex at level n leaf) is connected only to its ancestor at level n ) Lemma. Let p n be the probability that the root of T k n is not covered by a random matching. Then 1+4λk 1) 1 lim p n =. n 2λk 1) Sketch of Proof. That the limit, say p, exists, follows from Corollary 2.4, since the m-neighborhoods of the root of T k n look the same for n m. The recursion 2.2.1) becomes p n = 1+λk 1)p n 1 ) 1, which gives the quadratic equation for p: solving which we get the formula. p = 1+λk 1)p) 1, 3.3) Definitions. A graph G = V,E) is called k-regular if every vertex of G has k neighbors. The girth of G is the smallest integer m 3 such that G has an m cycle {v 1,...,v m }, where {v k,v k+1 } for k = 1,...,m 1 and {v m,v 1 } are edges of G. If G has no cycles, we say that grg = ) Lemma. Let G n be a sequence of k-regular graphs such that grg n +. Let us pick a vertex v n of G n and let p n be the probability that v n is not contained in a random matching. Then lim p n = n 2k 2 k 1+4λk 1)+k 2. Moreover, for any λ 0 > 0, the convergence is uniform over all 0 λ λ 0 and vertices v n of G n. Sketch of Proof. That the limit, say q, exists follows from Corollary 2.4, since for any m, the m-neighborhood of v n looks identical for all sufficiently large n. By 2.2.1), we get the equation q = 1+λkp) 1, where p is the limit in Lemma

8 3.5) Lemma. Let G = V,E) be a graph. For a vertex v V, let p v be the probability that v is not covered by a random matching. Then λ d dλ lnmat Gλ) = 1 1 p v ). 2 Sketch of Proof. Since 1 p v is the probability that a random matching contains v, the right hand side of the formula is half of the expected number of vertices covered by a random matching, that is, the expected number of edges in a random matching. On the other hand, Mat G λ) = the number of k-matchings in G)λ k, so that and k=0 v V λmat G λ) = kthe number of k-matchings in G)λ k k=0 λ d dλ lnmat Gλ) = λmat G λ) Mat G λ) = λ k kthe number of k-matchings in G) Mat G λ), k=0 is also the expected number of k-matchings in G. 3.6) Theorem. Let G n = V n,e n ) be a sequence of k-regular graphs such that grg n +. Then 1 lim n V n lnmat G n λ) = k ) λk 4λ 2 ln 2 k 2 2 k 1+4λk 4λ+k 2 ln 2k 2 Sketch of Proof. Combining Lemma 3.4 and Lemma 3.5, we conclude that 3.6.1) lim n 1 V n d dλ lnmat G n λ) = 1 2λ 1 λ k 1 ). k 1+4λk 1)+k 2 and that the convergence is uniform on any interval λ 0 < λ < λ 1 for any 0 < λ 0 < λ 1. Moreover, the right hand side of 3.6.1) is regular at λ = 0, it is k 2k 1)kλ +Oλ 2 ) for λ 0, 2 2 so we can integrate and obtain ) 1 λ lim n V n lnmat 1 G n λ) = 2τ 1 k 1 τ k dτ, 1+4λk 1)+k 2 and the proof follows. 0 8

9 4. Lifts and bipartite graphs Our next is goal is to prove a theorem of Csikvári 2014): 4.1) Theorem. Let G = V,E) be a k-regular bipartite graph. Then for λ 0. lnmat G λ) V k ) λk 4λ 2 ln 2 k 2 ) k 1+4λk 4λ+k 2 ln 2 2k 2 The proof of Theorem 4.1 hinges on the construction of a 2-lift of a graph. 4.2) 2-lift of a graph. Let G = V,E) be a graph. For every vertex v V, we introduce a pair v + and v of vertices, and for every edge {u,v} E, we introduce either the pair {u +,v + } and {u,v } of edges, or the pair {u +,v } and {u,v + } of edges we have a choice). The resulting graph H is called a 2-lift of G. If G is a bipartite k-regular graph then H is a bipartite k-regular graph with twice as many vertices. Similarly, we define a 2-lift of a weighted graph, by copying the weight of each edge of G on its liftings. 4.3) Lemma. If H is a 2-lift of a bipartite graph G then Mat H λ) Mat 2 G λ) for all λ 0. The same inequality holds for weighted graphs with non-negative weights on the edges). Sketch of Proof. Let Ĝ be the trivial 2-lift of G, consisting of two disconnected copies of G one with vertices v + and the other with vertices v for v V). Then MatĜλ) = Mat 2 G λ), so we need to prove that Mat Hλ) MatĜλ). Let e 1,...,e k beamatchinginh andletw bethemultisetofedgesofgobtainedfrome 1,...,e k by the natural projection v +,v v as we can obtain the same edge more than once, W is a multiset). It suffices to prove that given such a W, the contribution sum of monomials) to MatĜλ) of all matchings in Ĝ that project onto W is at least as large as the contribution of all matchings in H that project onto W. First, we note that it suffices to check the last statement when W is connected. Second, each vertex belongs to at most two edges of W. Therefore, assuming that W is connected, it is of the following 4 types: a) W is an edge, b) W is a double edge, c) W is a path, and d) W is a cycle, in which case W has to be an even cycle since G is bipartite. In a), there are two matchings of one edge each that project onto W, so the contributions to Mat H λ) and MatĜλ) are equal. In b), there is one matching of two edges projected onto W, so the contributions to Mat H λ) and MatĜλ) are 9

10 equal. In c), the inverse image of W consists of two vertex-disjoint paths, so there are two matchings that project onto W and hence the contributions to Mat H λ) and MatĜλ) are equal. In d) the inverse image of W consists of either two cycles of the same length W, in which case there are exactly two matchings projecting onto W recall that W is even) or a cycle of length 2 W, in which case there are no matchings projecting onto W recall that W is even). Hence we conclude that in all cases the contribution to MatĜλ) is at least as big as the contribution to Mat H λ). The following lemma is due to Linial. 4.4) Lemma. Let G be a graph. Then there exists a sequence G n of graphs such that G 0 = G, G n+1 is a 2-lift of G n for n 0 and grg n + as n. Sketch of Proof. Clearly, if H is a 2-lift of G then grh grg. It suffices to prove that if grg = k and G contains g cycles of length k then there is a 2-lift of H of G which contains fewer than g cycles of length k. Let us consider a random 2-lift: independently for each edge {u,v} of G, we pick the pair {u +,v + } and {u,v } with probability 1/2 or the pair {u +,v } and {u,v + } with probability 1/2 as edges of H. We claim that the expected number of k-cycles in H is g. Indeed, since the inverse image of every path in G consists of two vertex-disjoint paths in H, with probability 1/2, we have two k-cycles in H projecting onto a given k-cycle in G and with probability 1/2, we have one 2k-cycle in H projecting onto a given cycle in G. Since with positive probability we obtain a trivial lift Ĝ which has 2g cycles of length k, there ought to be a 2-lift that contains fewer than g cycles of length k. 4.5) Sketch of Proof of Theorem 4.1. Using Lemma 4.4, we construct a sequence of graphs G n = V n,e n ), where G 0 = G, G n+1 is a 2-lift of G n and grg n +. We observe that each G n is bipartite, so by Lemma 4.3, for any λ 0, the sequence lnmat Gn λ) V n is non-increasing. Besides, each G n is k-regular and the proof follows by Theorem ) Perfect matchings. A matching in G = V,E) is called perfect if it covers all vertices of V. We obtain the following corollary of Theorem 4.1: 4.7) Corollary. Let G = V, E) be a k-regular bipartite graph with V = 2n vertices. Then there are at least ) k 1)n k 1 k n k perfect matchings in G. 10

11 Sketch of Proof. Let N be the number of perfect matchings in G. Assuming that N > 0, we get Mat G λ) = Nλ n 1+o1)) as λ +. Therefore, lnn = lim λ + lnmat Gλ) nlnλ, where the limit is if N = 0. The bound now follows from Theorem H-stable polynomials and capacity 5.1) Definitions. A complex polynomial pz 1,...,z n ) is called H-stable if pz 1,...,z n ) 0 whenever Iz 1 > 0,...,Iz n > 0. Given a polynomial px 1,...,x n ) with non-negative real coefficients, we define its capacity by px 1,...,x n ) capp = inf. x 1 >0,...,x n >0 x 1 x n Our goal is to prove the following theorem due to Gurvits 2008). 5.2) Theorem. Let px 1,...,x n ) be an H-stable polynomial with non-negative real coefficients. For k = n,...,0, let us define polynomials p k x 1,...,x k ) by p n = p and p k x 1,...,x k ) = p k+1 x 1,...,x k+1 ) x k+1 xk+1 =0 for k = n 1,...,0. In particular, p 0 is the coefficient of x 1 x n in p. Suppose that the degree of p k in x k does not exceed d k for some integer d k 0. Then p 0 n ) ) dk 1 dk 1 capp, d k k=1 where we agree that dk 1 d k ) dk 1 = 1 if d k = 1 or d k = 0. The proof of Theorem 5.2 hinges on the following three lemmas. 5.3) Lemma [Gauss-Lucas Theorem]. If pz) is a non-constant complex polynomial with roots z 1,...,z d C, d 1, then every root w of p z) can be written as w = α 1 z α d z d for some real α 1,...,α d 0 such that α α d = 1. 11

12 Sketch of Proof. Without loss of generality, we assume that w z i for i = 1,...,d and that f is monic, so that Then fz) = d z z j ). j=1 d 0 = f w) = w z k ) and hence j=1 k j d w z k ) = 0. j=1 k j Multiplying both sides of the last identity by fw), we obtain d j j=1w z ) w z k 2, k j from which α j = k j w z k 2 d k i w z k 2 for j = 1,...,d. 5.4) Lemma. 1) Let f m : C n C, m = 1,2,... be a sequence of H-stable polynomials and let f : C n C be yet another polynomial such that f m f uniformly on compact subsets of C n. Then f is either H-stable or identically 0. 2) Let f z 1,...,z n ) be an H-stable polynomial and let gz 1,...,z n 1 ) = f z 1,...,z n 1,0). Then g is either H-stable or identically 0. 3) Let f z 1,...,z n ) be an H-stable polynomial and let gz 1,...,z n ) = z n f z 1,...,z n ). Then g is either H-stable or identically 0. Sketch of Proof. The proof of Part 1) follows from a theorem of Hurwitz, which claims that if Ω C n is a connected open set and f m is a sequence of functions analytic in Ω such that f m z) 0 for all z Ω and all m and f m f uniformly on compact subsets of Ω then either fz) 0 for all z Ω or f 0 on Ω. The proof of Hurwitz theorem reduces to that for n = 1 consider a section of Ω through 12

13 a given point by a complex line identified with C), while for n = 1 it follows from the Rouche Theorem. To prove Part 2), consider a sequence g m z 1,...,z n 1 ) = f z 1,...,z n 1,im 1). Then g m are H-stable and g m g uniformly on compact sets in C n 1, so the proof follows from Part 1). To prove Part 3), let d be the degree of f in z n. If d = 0 then g 0. Hence we assume that d > 0, in which case we write f z 1,...,z n ) = d zn k h kz 1,...,z n 1 ), k=0 where h d z 1,...,z n 1 ) 0. Let us consider a sequence f m of polynomials, Then f m are H-stable and f m z 1,...,z n ) = m d f z 1,...,z n 1,mz n ). f m z 1,...,z n ) z d n h dz 1,...,z n 1 ) uniformly on compact subsets of C n. It follows from Part 1) that the polynomial z d n h dz 1,...,z n 1 ) is H-stable and hence h d z 1,...,z n 1 ) 0 whenever Iz 1 > 0,...,Iz n 1 > 0. Let us fix arbitrary z 1,...,z n 1 such that Iz 1 > 0,...,Iz n 1 > 0 and let us consider a univariate polynomial pz) = f z 1,...,z n 1,z). Hence pz) is a polynomial of degree d > 0. Since f is H-stable, we have Iz 0 for every root z of p. By Lemma 5.3, for every root w of p we have Iw 0 so that p z) = gz 1,...,z n 1,z) 0 whenever Iz > 0, which completes the proof of Part 3). 5.5) Lemma. Suppose that Rt) is a polynomial with non-negative real coefficients and real roots such that degr d. Then R 0) d 1 d 13 ) d 1 inf t>0 Rt), t

14 where we agree that ) d 1 d 1 = 1 if d = 1 or d = 1. d Sketch of Proof. It is not hard to see that the function ) x 1 x 1 x for x 1 x is decreasing, so without loss of generality, we may assume that degr = d. If degrt) 1 then Rt) = a+bt for some a,b 0 and R 0) = b = inf t>0 Rt), t so the desired inequality holds. Hence we assume that d 2. If R0) = 0 then Rt) inf = R 0), t>0 t the infimum is obtained as t 0+) and the desired inequality holds as well. Hence we assume that R0) > 0. Scaling, if necessary, we assume that R0) = 1. Then we can write d ) Rt) = 1 tαi, where α 1,...,α d < 0 are the roots of Rt). Denoting a i = α 1 i, we obtain We have Rt) = d 1+a i t) for some a 1,...,a d > 0. R 0) = and using the arithmetic-geometric mean inequality, we obtain Rt) 1+ a d a d t) = d d a i ) d 1+ R 0) d t for t > 0. Therefore, Rt) 5.5.1) inf t>0 t ) d inf t>0 t 1 1+ R 0) d t. 14

15 Since R 0) > 0 and d 2, the infimum in the right hand side of 5.5.1) is attained at a critical point of t. It is not hard to show that t 0 = is the unique critical point and hence d d 1)R 0) Rt) inf t>0 t t 1 0 ) d ) d 1 1+ R 0) d d t 0 = R 0) d 1 and the proof follows. 5.6) Sketch of proof of Theorem 5.2. It suffices to prove that 5.6.1) p k 1 x 1,...,x k ) dk 1 d k ) dk 1 for all x 1,...,x k 1 > 0, p k x 1,...,x k 1,x k ) inf x k >0 x k since combining 5.6.1) for k = 1,2,...,n, we get the desired inequality. By Parts 2) and 3) of Lemma 5.4, each polynomial p k is either H-stable or identically 0. If p k 0 then p k 1 0 and 5.6.1) follows. Hence we assume that p k is H-stable. Given x 1,...,x k 1 > 0, let us define a univariate polynomial Rt) by Rt) = p k x 1,...,x k 1,t). Hence Rt) is a polynomial with non-negative real coefficients with degr d k and we claim that all roots of Rt) are real. Indeed, suppose that α±βi is a pair of complex conjugate roots of Rt) for some β > 0. By continuity of roots, we conclude that for a sufficiently small ǫ > 0, the univariate polynomial z p k x 1 +ǫi,...,x k +ǫi,z) has a root with Iz > 0, which contradicts the H-stability of p k. Now, by Lemma 5.5, we have R 0) dk 1 d k ) dk 1 inf t>0 Rt), t which is 5.6.1). 15

16 6. Corollaries for the permanent 6.1) Permanent. Let A = a ij ) be an n n matrix real, complex, or over an arbitrary field). The permanent of A is pera = σ S n n a iσi), where S n is the symmetric group of the permutations of {1,...,n}. If a ij {0,1} for all i and j then pera is the number of perfect matchings in the bipartite graph on n+n vertices with biadjacency matrix A: we have a ij = 1 if and only if vertex i on one left) side of the graph is connected to vertex j on the other side right) of the graph by an edge. OurgoalistoprovethefollowingresultknownasthevanderWaerdenconjecture originally proved by Falikman and Egorychev in 1981). The proof below is due to Gurvits 2008). 6.2) Theorem. Let A = a ij ) be an n n doubly stochastic matrix, that is, non-negative real matrix with all row and column sums equal to 1. Then pera n! n n. Sketch of Proof. Given an n n matrix A = a ij ), we define a polynomial p A x 1,...,x n ) = n n a ij x j. It is not hard to see that the coefficient p 0 of x 1 x n in p A is pera. Moreover, if A is a non-negative real matrix with no zero rows then p A is H-stable: if Iz 1 > 0,...,Iz n > 0 then n I a ij z j = j=1 j=1 n a ij Iz j > 0 and hence p A z 1,...,z n ) 0. Next, we claim that if A is a doubly stochastic matrix then capp A = 1. Indeed, p A 1,...,1) = j=1 n n 16 j=1 a ij = 1,

17 since the row sums of A are 1 and hence capp A 1. On the other hand, for any x 1,...,x n > 0, applying the arithmetic-geometric mean inequality, we have n n n n n p A x 1,...,x n ) = a ij x j = x j, j=1 since the column sums of A are 1. Hence capp A 1. Starting with p n = p A, let us define polynomials p k as in Theorem 5.2. Then degp k k, so we can choose d k = k. Applying Theorem 5.2, we obtain pera n ) ) k 1 k 1 k=1 k j=1 x a ij j capp A = n! n n. In fact, if A = a ij ) is an n n doubly stochastic matrix and pera = n!/n n then a ij = 1/n for all i,j exercise). As Gurvits 2008) noticed, the bound can be sharpened if A has few non-zero entries. 6.3) Theorem. Let A = a ij ) be an n n doubly stochastic matrix with at most m k 1 non-zero entries in the k-th column for k = 1,...,n. Then n ) min{mk,k} 1 min{mk,k} 1 pera min{m k,k} k=1 with the usual agreement that the k-th factor is 1 if min{m k,k} = 1. Sketch of Proof. We construct the polynomial p A and polynomials p k as in the proof of Theorem 6.2. Then the degree of p k in x k does not exceed min{m k,k} and the proof follows from Theorem ) Perfect matchings in regular bipartite graphs. If G = V,E) is a k- regular bipartite graph with n + n vertices then the number of perfect matchings in G is pera = k n perk 1 A), where A is the biadjacency matrix of G. We note that k 1 A is a doubly stochastic matrix and hence by Theorem 6.3 the number of perfect matchings in G is at least ) k 1)n k+1) k 1 k 1 ) j 1 j ) k n, k j which is a stronger bound than the one of Corollary 4.7. In particular, if k = 2 then 6.4.1) transforms into 2 n 1 = 2, 2n 1 which is of course sharp, and if k = 3 then 6.4.1) transforms into ) n 3 4 6, 3 which was established first by Voorhoeve 1979) by a different method. 17 j=1 j=1

18 7. Ramifications We prove yet another theorem of Gurvits 2015). 7.1) Theorem. Let px 1,...,x n ) be an H-stable polynomial with non-negative real coefficients. Suppose that the degree of p in x k does not exceed d k for k = 1,...,n, where d k are positive integers. Let r 1,...,r n be positive integers such that r k d k for k = 1,...,n. Then the coefficient of x r 1 1 xr n n in p is at least as big as n k=1 r r k k d k r k ) d k r k d k! r k!d k r k )!d k d k ) inf x 1 >0,...,x n >0 px 1,...,x n ) x r. 1 1 xr n n We note that the coefficient of x r 1 1 xr n n in p trivially does not exceed px 1,...,x n ) inf x 1 >0,...,x n >0 x r. 1 1 xr n n Sketch of Proof of Theorem 7.1. We deduce the result from Theorem 5.2. Let us introduce a polynomial q in variables by y 11,...,y 1r1,...,y 21,...,y 2r2,...,y n1,...,y nrn q...,y k1,...,y krk,...) = p..., y ) k y krk,..., r k that is, q is obtained from p by the substitution 7.1.1) x k = y k y krk r k for k = 1,...,n. Clearly, the coefficients of q are non-negative real and q is H-stable. Moreover, the coefficient of x r 1 1 xr n n in p is the coefficient of y k1 y krk in q multiplied n r r k k by r k!. Next, we observe that the degree of q in variables {y k1,...,y krk } does k=1 not exceed d k and that the degree of j y k1 y kj q in {y k1,...,y kr } does not exceed d k j. Finally, we observe that q...,y k1,...,y krk,...) 7.1.2) inf...y k1 >0,...y krk >0... y k1 y krk 18 px 1,...,x n ) inf x 1 >0,...,x n >0 x r. 1 1 xr n n

19 Indeed, given...y k1 > 0,...,y krk > 0..., we define x k by 7.1.1) and notice that by the arithmetic-geometric mean inequality, we have k=1 x r k k y k1 y krk, from which 7.1.2) follows. Combining all the above observations, we deduce from Theorem 5.2 that the coefficient of x r 1 1 xr n n in p is at least as big as n ) r r k n r k ) dk j k dk j px 1,...,x n ) inf r k! d k j +1 x 1 >0,...,x n >0 x r, 1 1 xr n n as required. k=1j=1 7.3) Counting subgraphs with prescribed degrees. Let R = r 1,...,r m ) and C = c 1,...,c n ) be positive integer vectors and let ΣR,C) be the set of all m n matrices D = d ij ) with row sums R, column sums C and 0-1 entries. Gale- Ryser Theorem 1957) states a convenient necessary and sufficient condition for ΣR, C) to be non-empty: assuming that we must have and m c 1 c 2... c n > 0 and n r i > 0 for i = 1,...,m, m min{r i,k} k c j for k = 1,...,n j=1 m r i = n c j. Given a non-negative real m n matrix W = w ij ), we define a polynomial FlR,C;W) = w d ij ij, j=1 D ΣR,C) D=d ij ) where we agree that 0 0 = 1, so that Fl is indeed a polynomial in w ij and remains a continuous function as w ij 0+). If w ij {0,1} then FlR,C;W) is the number of subgraphs with degrees r 1,...,r m ;c 1,...,c n of vertices in the bipartite graph on m+n vertices with biadjacency matrix W, or, equivalently, the number of 0-1 flows in the bipartite network with biadjacency matrix W, demands r 1,...,r m and supplies c 1,...,c n. Let us define a polynomial p W x 1,...,x m ;y 1,...,y n ) = x i +w ij y j ). 19 i,j 1 i m 1 j n

20 Then FlR,C;W) is the coefficient of x n r 1 1 x n r m m y c 1...y c m m in p W and hence FlR,C;W) inf x 1,...,x m >0 y 1,...,y n >0 p W x 1,...,x m ;y 1,...,y n ) x n r 1 1 x n r m m y c. 1 1 yc n n If Ix i,iy j > 0 then Ix i +w ij y j ) > 0, from which it follows that p W is H-stable. Applying Theorem 7.1, we get m ) r r i i n r i ) n r i n! n c c j FlR,C;W) j m c j) m c j m! r i!n r i )!n n c j!m c j )!m m inf x 1,...,x m >0 y 1,...,y n >0 j=1 p W x 1,...,x m ;y 1,...,y n ) x n r 1 1 x n r m m y c. 1 1 yc n n 8. Capacity, convexity, log) concavity 8.1) Capacity. Given a polynomial px 1,...,x n ) with non-negative real coefficients and a multi-index R = r 1,...,r n ) of non-negative integers, we define cap R p) = px 1,...,x n ) inf x 1 >0,...,x n >0 x r. 1 1 xr n n This extends our Definition 5.1 of capacity. Our first observation is that the capacity can be found by minimizing a convex function on R n, which is easy. Making the substitution x i = e t i, we write 8.1.1) lncap R p) = inf lnp ) n e t 1,...,e t n t 1,...,t n 8.2) Lemma. The function t 1,...,t n ) lnp ) e t 1,...,e t n is convex. Sketch of Proof. It suffices to prove that the restriction of the function onto any affine line t i = a i t+b i is convex, that is, that the function t lngt), where r i t i. gt) = m α i e λ it where α 1,...,α m > 0 is convex. Indeed, lngt)) = g t) gt) and lngt)) = g t)gt) g t)g t) g 2, t) 20

21 where g t) = m α i λ i e λ it and g t) = m α i λ 2 i eλ it, so that g t)gt) g t)g t) = and the proof follows. m i,j=1 = {i,j} i j α i α j λ 2 i eλ i+λ j )t m i,j=1 λ i λ j ) 2 α i α j e λ i+λ j )t 0 α i α j λ i λ j e λ i+λ j )t Our next observation is that the function R lncap R p) is concave. 8.3) Lemma. Let R 1,...,R k ;R be multi-indices of n non-negative integers such that k R = α i R i for some α 1,...,α k 0 such that k α i = 1. Then cap R p) k capri p) ) α i. Sketch of Proof. By 8.1.1), the function R lncap R p) is the infimum of a family indexed by t 1,...,t n ) of affine functions in R. Hence R lncap R p) is concave. For R = r 1,...,r n ), let a R be the coefficient of x r 1 1 xr n n in the polynomial px 1,...,x n ). Clearly, a R cap R p). As follows from Lemma 8.3, if cap R p) approximates a R reasonably well for all possible multi-indices R, then the function R a R is approximately log-concave that is, the function R lna R is approximately concave). The following lemma shows that the converse is approximately true. 8.4) Lemma. Let px 1,...,x n ) be a polynomial with non-negative coefficients. For a multi-index M = m 1,...,m n ) of non-negative integers, let a M be the coefficient of x m 1 1 x m n n in p. Suppose that the function M lna M is concave, 21

22 that is, whenever M = α 1 M α k M k for some α 1,...,α k 0 such that α α k = 1, we have k a M a α i M i. Then for every multi-index R = r 1,...,r n ) of non-negative integers, one can find x 1,...,x n > 0 such that a R x r 1 1 xr n n a M x m 1 1 x m n n for all M = m 1,...,m n ). In particular, a R cap R p) number of monomials in p. Sketch of proof. Let us consider the set S R n+1 of points M,lna M ), where M ranges over all multi-indices of monomials in p. Let us choose any γ > lna R and consider a ray {R,β) : β γ} in R n+1. We claim that the ray and the convex hull the set of all convex combinations of points from S) are disjoint. Indeed, if R = k α i M i where α 1,...,α k 0 and k α i = 1, then β γ > lna R k α i lna Mi and the point R,β) is not a convex combination of points from S. Since the ray and the convex hull of S do not intersect, we can separate them by an affine hyperplane, that is, we can find real t 1,...,t n ;t n+1 such that t n+1 β + k t i r i t n+1 lna M + k t i m i for R = r 1,...,r n ), all β γ > lna R and all M = m 1,...,m n ). Taking the separating hyperplanesufficiently generic, wecanassumethatt n+1 0, from which it follows that t n+1 > 0, from which we can further scale it to t n+1 = 1. Then we have k k lna R + t i r i lna M + t i m i for R = r 1,...,r n ) and all M = m 1,...,m n ). Letting x i = e t i, we complete the proof. 22

23 9. Capacity and matrix scaling The following theorem was proved by Sinkhorn 1964). 9.1) Theorem. Let A = a ij ) be an n n matrix with positive entries. Then there exists an n n doubly stochasticmatrix B = b ij ) and positiveλ 1,...,λ n ;µ 1,...,µ n such that a ij = λ i µ j b ij for all i,j. Given A, the matrix B is unique and numbers λ 1,...,λ n ;µ 1,...,µ n are unique up to a rescaling λ i λ i τ, µ j µ j τ 1 for some τ > 0. Sketch of proof. Let us consider a function gx) = n i,j=1 x ij ln x ij a ij on the set of all n n non-negative matrices X = x ij ) and let P n be the set of all n n doubly stochastic matrices X. Then g attains its minimum on P n at some doubly stochastic matrix B = b ij ) since g is strictly convex, the minimum B is unique). We observe that g X) = 1+ln x ij, x ij a ij and if x ij = 0 then the derivative is understood as the right derivative and its value is. It follows then that we must have b ij > 0 for all i,j since otherwise, letting Bt) = b ij t)), where b ij = 1 t)b ij +t/n, we obtain gb t ) < gb) for all sufficiently small t > 0. Since b ij > 0 for all i,j, we conclude that B is a critical point of g on the affine subspace of all n n matrices X = x ij ) satisfying n x ij = 1 for i = 1,...,n and j=1 n x ij = 1 for j = 1,...,n. Therefore, there exist Lagrange multipliers α 1,...,α n and β 1,...,β n such that ln b ij a ij = α i +β j for all i,j. We let λ i = e α i and µ j = e β j for all i,j. The original Sinkhorn s proof was based on iterated scaling of the matrix first to row sums 1, then to column sums 1, then to row sums 1 again, etc., and proving that the process converges. As follows from Theorem 9.1, the function fa) = λ 1 λ n µ 1 µ n is a welldefined function on positive n n matrices. 23

24 9.2) Connections to capacity. Given an n n positive matrix A = a ij ), we define the polynomial n n p A x 1,...,x n ) = a ij x j, as in the proof of Theorem 6.2. Let B = b ij ) be an n n doubly stochastic matrix, and let λ 1,...,λ n,µ 1,...,µ n be positive reals such that a ij = λ i µ j b ij. Then j=1 p A x 1,...,x n ) capp A = inf x 1,...,x n >0 x 1 x n p B µ 1 x 1,...,µ n x n ) =λ 1 λ n µ 1...µ n inf x 1,...,x n >0 µ 1 x 1 µ n x n =λ 1 λ n µ 1 µ n capp B = λ 1 λ n µ 1 µ n, see the proof of Theorem 6.2. Let us indeed define a function f on n n positive matrices A by fa) = λ 1 λ n µ 1 µ n. Here are some exercises: Prove that the function f is homogeneous of degree n, that is, fta) = t n fa) for all t > 0, that f 1/n is concave and that f is monotone: fc) fa) provided 0 < c ij a ij for all i,j. 9.3) Bounds on the permanents of doubly stochastic matrix. Suppose that A is obtained from B by scaling, a ij = λ i µ j b ij for all i,j. Then pera = λ 1 λ n µ 1 µ n perb. In view on Theorem 9.1, it would be nice to get some bounds on the permanent of a doubly stochastic matrix B. It is known that 1 b ij ) 1 b ij perb 2 n 1 b ij ) 1 b ij. i,j i,j The lower bound is due to Schrijver 1998). Lelarge 2015) proved the lower bound using the 2-lift construction, see Section 4. The upper bound is due to Gurvits and Samorodnitsky 2014), who also conjectured that 2 n can be replaced by 2 n/2, after which the bound becomes sharp on block-diagonal matrices with diagonal 2 2 blocks 1/2 1/2 1/2 1/2 ). 10. Entropy 10.1) The entropy function. Given x 1,...,x n 0 such that x x n = 1, we define n Hx 1,...,x n ) = x i ln 1, x i 24

25 where we agree that the i-th term is 0 if x i = 0. Since H is a strictly) concave function, it attains its minimum on the simplex x x n = 1, x i 0 for i = 1,...,n, at an extreme point vertex), where one x i = 1 and all other are 0. Therefore, H x 1,...,x n ) 0. On the other hand, since lnx is a strictly) concave function, we have n x i ln 1 n ) ln x i 1 = lnn x i x i and hence H x 1,...,x n ) lnn, with equality attained if and only if x 1 =... = x n = 1/n. 10.2) The entropy of a partition. Let Ω be a probability space and let F = {F 1,...,F n } be a partition of Ω into pairwise disjoint events: Ω = n F i and F i F j = for i j. We define the entropy of the partition by HF) = H p 1,...,p n ) = n p i ln 1 p i where p i = PrF i ) for i = 1,...,n. 10.3) Conditional entropy. Let Ω be a probability space and let F and G be partitions of Ω into finitely many disjoint events. We say that G refines F, if every event F in F is a union of events in G, in which case we write F G. For such partitions we define the conditional entropy ) HG F) = F F PrF) G G G F PrG) PrF) ln PrF) PrG). If PrF) = 0 then the corresponding term in ) is 0. In words: we consider each event F in F henceforth called block) as a probability space with conditional probability measure, compute the entropy of the partition of F by events of G and average over all blocks F of F. It follows that HG F) 0 and that HG) = HF)+HG F). 25

26 Telescoping, we get n ) HF n ) = HF 0 )+ HF k F k 1 ) k=1 provided F 0 F 1... F n. Suppose that Ω is finite and that every ω Ω is an event. Let F G be a pair of partitions of Ω. For ω Ω, let Fω) be the block of F containing ω. Assuming that PrFω)) 0, weconsider Fω)asaprobabilityspacewithconditionalprobability measure. Let Fω) be the partition of Fω) induced by events of G. Using that PrFω)) = ω F Prω), we can rewrite ) as ) HG F) = ω ΩPrω)H Fω) ). If PrFω)) = 0, we just assume that the corresponding term in ) is 0. Similarly, we can rewrite ) as ) HF n ) = H F 0 )+ ω ΩPrω) n H F k 1 ω) ), where F k 1 ω) is the partition of the block of F k 1 containing ω by events of F k. k=1 11. The Bregman-Minc inequality Our goal is to prove the following result. 11.1) Theorem. Let A = a ij ) be an n n matrix with 0-1 entries and let r i > 0 be the number of 1s in the i-th row of A for i = 1,...,n. Then pera n r i!) 1/r i. The inequality of Theorem 11.1 was conjectured by Minc and proved by Bregman 1973). We follow the entropy-based approach of Radhakrishnan 1997). The proof hinges on the following combinatorial lemma. 26

27 11.2) Lemma. Let A be a matrix as in Theorem 11.1, let us fix an integer 1 i n and a permutation ω S n such that a iωi) = 1. Let us choose a permutation τ S n uniformly at random, find 1 k n such that τk) = i and cross out from A the columns indexed by ωτ1)),...,ωτk 1)). Let X be the number of 1s in the i-th row that remain after the columns are crossed, so X = Xτ) is a random variable on S n. Then PrX = x) = 1 r i for x = 1,...,r i. Sketch of Proof. Let J be the set of indices j of columns such that a ij = 1. Hence J = r i and ωi) J. Let I = ω 1 J), so that I = r i and i I. The number of 1s that remain in the i-th row is the number of indices in τ 1 I) that are greater than or equal to τ 1 i). Since τ is chosen uniformly at random, τ 1 i) is equally likely to be the largest, the second largest, etc. element of τ 1 I). 11.3) Sketch of Proof of Theorem Let Ω = { ω S n : a iωi) = 1 for i = 1,...,n }, so that pera = Ω. Without loss of generality, we assume that Ω and consider Ω as a finite probability space with uniform measure. For a permutation τ S n, we consider a family of partitions F τ,0 F τ,1... F τ,n of Ω, where a block of F τ,k consists of permutations ω Ω with prescribed values of ωτ1)),...,ωτk)). In particular, F τ,0 consists of a single block Ω, while F τ,n is the partition of Ω into singletons. Applying ), we obtain ln Ω = ω ΩPrω) and averaging over τ S n, we get n H F τ,k 1 ω) ) k= ) ln Ω = ω ΩPrω) 1 n! τ S n k=1 n H F τ,k 1 ω) )). We fix an ω Ω, make a substitution i = τ 1 k) and consider the sum ) 1 n! τ S n k=1 n H F τ,k 1 ω)) = 1 n! 27 τ S n n H F τ,τ 1 i) 1ω) ).

28 Now, the space of F τ,τ 1 i) 1 consists of the permutations σ Ω such that στ1)) = ωτ1)),...,στk 1)) = ωτk 1)) further subdivided into blocks depending on the value of στk)) for k = τi). Hence the number of blocks in F τ,τ 1 i) 1ω) does not exceed the number X of 1s in the i-th row of A, after the columns numbered ωτ1)),...,ωτk 1)) have been crossed out. Since from Lemma 11.2, we get 1 n! τ S n H H F τ,τ 1 i) 1ω) ) lnx, Fτ,τ 1 i) 1ω) ) 1 r i lnx = 1 lnr i!). r i r i Summing over i = 1,...,n, we conclude from ) and ) that x=1 ln Ω n 1 r i lnr i!), which is the desired bound. Here are some applications of Theorem ) The number of perfect matchings in k-regular bipartite graphs. The number of perfect matchings in k-regular bipartite graph with n + n vertices is the permanent of n n matrix of 0s and 1s with exactly k of 1s in every row and column. Hence by Theorem 11.1, this number does not exceed ) k!) n/k On the other hand, by Corollary 4.7 see also Section 6.4), this number is at least ) k 1)n k ) k n. k Curiously, when k is large enough, both ) and ) are roughly k n e n in the logarithmic order, say). 11.5) The number of Latin squares or 3-dimensional permutations. Let us consider n n n arrays A = a ijk ) of 0s and 1s, such that each of the 3n 2 skewers, obtained by fixing some two coordinates and letting the remaining coordinate vary, contains exactly one 1. Such 3-dimensional permutations are in one-to-one correspondence with n n Latin squares, consisting of n n matrices such that each row and each column contains a permutation of 1,...,n). If we are to fill A layer by layer, we can fill the 1st layer by choosing a permutation matrix, that is, a perfect matching in the complete bipartite graph on n+n vertices. After 28

29 the first layer is filled, we fill the 2nd layer by choosing a perfect matching in the bipartite graph on n + n vertices obtained from the complete bipartite graph by deleting the perfect matching chosen at the first layer. In any case, at the 2nd layer we choose a perfect matching in an n 1)-regular bipartite graph on n+n vertices. Continuing in this way, to fill the k-th layer, we need to choose a perfect matching in the bipartite graph on n+n vertices obtained from the complete bipartite graph by deleting k 1 pairwise edge-disjoint perfect matchings chosen at the previous k 1 layers. Hence at the k-th layer we are choosing a perfect matching in an n k)-regular bipartite graph on n+n vertices. From Section 11.4, the number of 3-dimensional permutation is between n k=1 ) k 1)n k 1 k n and k n k!) n/k, k=1 which can be written as 1+o1)) n e 2 ) n 2 as n. 11.6) The number of d-dimensional permutations. Similarly, one can define a d-dimensional permutation as a d-dimensional cubical array n... n filled by 0s and 1s such that each of the dn d 1 skewers contains exactly one 1. Using the entropy method, Luria and Linial 2011) proved an upper bound ) n ) n 1+o1)) e d 1 d 1 as n for the number of such d-dimensional permutations. For d > 3, no lower bound matching ) is known. Here is a heuristic argument purported to show that11.6.1) is a plausible asymptotic formula. With the entries of the array we associate n d independent Bernoulli random variables, each taking value 1 with probability 1/n and value 0 with probability n 1)/n. Then the sum of the variables in a skewer is roughly a Poisson random variable with parameter 1, so the probability that a skewer contains exactly one 1 is roughly 1/e. Assuming that the sums in skewers are independent they are not!), we conclude that the probability that we get a d-dimensional permutation is roughly e dnd 1. Since the probability to get any particular permutation is ) n d n n) nd nd 1 e nd 1 n nd, we obtain ). 29

30 11.7) Permanents of doubly stochastic matrices. Let A = a ij ) be an n n stochastic i.e. non-negative, with row sums 1) matrix and suppose that and some positive integer r i. Then ) per A a ij 1 r i for j = 1,...,n n r i!) 1/r i r i. Indeed, pera is a linear function in each row and hence the maximum value of pera as a function of the i-th row a i = a i1,...,a in ) on the polytope n a ij = 1 and 0 a ij 1 for j = 1,...,n r i j=1 is attained at an extreme point, where a ij {0,1/r i } for all j and then ) follows from Theorem This argument is due to Samorodnitsky 2000). Assume now that A is a doubly stochastic matrix with all entries not exceeding α/n for some constant α 1. Combining Theorem 6.2 and ), we conclude that pera = e n n Oα), so the permanent of doubly stochastic matrices with small entries is strongly concentrated. A positive matrix A is called α-balanced if the ratio of any two entries in every row or column does not exceed α. Exercise: prove that if B is a doubly stochastic matrix obtained from an α-balanced matrix by scaling then B is α 2 -balanced and deduce that n n pera = e n n Oα2) capp A where p A x 1,...,x n ) = a ij x j and A is α-balanced. 12.1) D-stable polynomials. Let 12. D-stable polynomials D = {z C : z 1} be the closed unit disc in the complex plain. A polynomial pz 1,...,z n ) is called D-stable provided pz 1,...,z n ) 0 whenever z 1,...,z n D. Our maingoalistoprovethe followingresult of Hinkannen1997)who wasbuilding on results of Ruelle 1971) who was building on results of Asano 1970). 30 j=1

31 12.2) Theorem. Suppose that f z 1,...,z n ) = S {1,...,n} a S z i and gz 1,...,z n ) = i S are D-stable. Then the polynomial hz 1,...,z n ) = S b S S {1,...,n}a ) i S is also D-stable we agree that for S = we have i S z i = 1). S {1,...,n} b S The polynomial h is called the Hadamard product or the Schur product of polynomials f and g and denoted h = f g. The proof is based on the following lemma due to Asano 1970), known as the Asano contraction lemma. 12.3) Lemma. Suppose that the bivariate polynomial z i i S z i fz 1,z 2 ) = a+bz 1 +cz 2 +dz 1 z 2 is D-stable. Then the univariate polynomial is also D-stable. gz) = a+dz Sketch of Proof. Since f is D-stable, we have a 0. Suppose that g is not D-stable. Then d a. Without loss of generality, we assume that b c. Let us find a z 2 such that z 2 = 1 and b+dz 2 = b + d. Note that a+cz 2 a + c b + d = b+dz 2. Solving the equation fz 1,z 2 ) = 0 for z 1, we obtain z 1 = a+cz 2 b+dz 2, so that z 1 = a+cz 2 b+dz 2 1, which contradicts the D-stability of f. 31

32 12.4) Sketch of Proof of Theorem We proceed by induction on n. If n = 1 then fz) = a 0 +a 1 z and gz) = b 0 +b 1 z where a 0 > a 1 and b 0 > b 1. Hence hz) = a 0 b 0 )+a 1 b 1 )z and a 0 b 0 > a 1 b 1. Therefore h is D-stable. Suppose that n > 1. We write f z 1,...,z n ) = gz 1,...,z n ) = S {1,...,n 1} S {1,...,n 1} as +a S {n} z n ) i S z i ) bs +b S {n} z n z i. i S and Therefore, for any w 1,w 2 D, the n 1)-variate polynomials z 1,...,z n 1 ) z 1,...,z n 1 ) S {1,...,n 1} as +a S {n} w 1 ) i S as +a S {n} w 2 ) S {1,...,n 1} i S z i z i and are D-stable and hence by the induction hypothesis the n 1)-variate polynomial z 1,...,z n 1 ) as +a S {n} w 1 ) bs +b S {n} w 2 ) S {1,...,n 1} i S is D-stable. Therefore, for any z 1,...,z n 1 D, the bivariate polynomial w 1,w 2 ) as +a S {n} w 1 ) bs +b S {n} w 2 ) S {1,...,n 1} i S is D-stable. Hence by Lemma 12.3, for any z 1,...,z n 1 D, the univariate polynomial ) z as b S +za S {n} b S {n} S {1,...,n 1} i S is D-stable. Therefore, h is D-stable. 12.5) Corollary. Let α > 0 and β > 0 be real and let f z 1,...,z n ) = a S z i and gz 1,...,z n ) = S {1,...,n} i S S {1,...,n} i S be polynomials such that f z 1,...,z n ) 0 whenever z 1 α,..., z n α z i z i z i b S z i and gz 1,...,z n ) 0 whenever z 1 β,..., z n β. 32

33 Then for we have hz 1,...,z n ) = a S b S S {1,...,n} i S hz 1,...,z n ) 0 whenever z 1 αβ,..., z n αβ. z i Sketch of Proof. The polynomials z 1,...,z n ) f αz 1,...,αz n ) and z 1,...,z n ) gβz 1,...,βz n ) are D-stable and hence by Theorem 12.2 the polynomial is D-stable. z 1,...,z n ) hαβz 1,...,αβz n ) 12.6) Ramifications. For a multi-index α = α 1,...,α n ) of non-negative integers we denote z α = z α 1 1 zα n n, where we agree that zα i i = 1 if α i = 0. For γ = γ 1,...,γ n ) we say that α γ if α i γ i for i = 1,...,n, in which case we denote ) γ n ) γi n γ i! = = α α i!γ i α i )!. α i Borcea and Brändén 2009) proved the following extension of Theorem Suppose that the polynomials f z 1,...,z n ) = ) γ a α z α and gz 1,...,z n ) = ) γ b α z α α α α γ α γ are D-stable. Then the polynomial hz 1,...,z n ) = ) γ a α b α α α γ z α is also D-stable. Ruelle 1971) proved the following extension of Lemma Let A,B C be closed sets such that 0 / A and 0 / B. Let be a polynomial such that Then for the polynomial we have pz 1,z 2 ) = a+bz 1 +cz 2 +dz 1 z 2 pz 1,z 2 ) = 0 = z 1 A or z 2 B. qz) = a+dz qz) = 0 = z = z 1 z 2 for some z 1 A and z 2 B. 33

34 13. The Lee-Yang Circle Theorem The following remarkable result was obtained by Lee and Yang 1952). 13.1) Theorem. Let A = a ij ) be an n n Hermitian matrix such that a ij 1 for all i,j. Then the roots of the univariate polynomial Cut A z) = S {1,...,n} z S i S,j/ S lie on the circle z = 1. As usual, we assume that the constant term for S = ) and the coefficient of z n for S = {1,...,n}) are equal to 1. The proof is based on Theorem 12.2 and the following lemma. 13.2) Lemma. Suppose that a 1. Then 1+az 1 +az 2 +z 1 z 2 0 whenever z 1 < 1 and z 2 < 1. a ij Sketch of Proof. Solving the equation for z 2, say, we obtain We claim that for the map we have φz) 1 for all z < 1. Indeed and z 2 = 1+az 1 a+z 1. φz) = 1+az a+z φz) 2 = 1+az)1+az) a+z)a+z) 1+az)1+az) a+z)a+z) = 1+ a 2 z 2 a 2 z 2 = 1 a 2) 1 z 2) ) Sketch of Proof of Theorem Let us define an n-variate polynomial p A z 1,...,z n ) = S {1,...,n} i S,j/ S a ij z i. For an unordered pair {i,j} where 1 i j n, let us define p ij z 1,...,z n ) = 1+a ij z i +a ji z j +z i z j ) 34 i S k i,j 1+z k ).