The Geometry of Polynomials and Applications Julius Borcea (Stockholm) based on joint work with Petter Brändén (KTH) and Thomas M. Liggett (UCLA)
Rota s philosophy G.-C. Rota: The one contribution of mine that I hope will be remembered has consisted in just pointing out that all sorts of problems of combinatorics can be viewed as problems on location of the zeros of certain polynomials and in giving these zeros a combinatorial interpretation. Phase transitions, Lee-Yang type theorems (Random) Matrix theory, representation theory, total positivity, number theory Interacting particle systems, probability theory, negative dependence 2
Polynomials galore Stable Polynomial f C[z 1,..., z n ] is stable if Im(z j ) > 0 j = f(z 1,..., z n ) 0 Real stable if f R[z 1,..., z n ]. Related notions: Half-Plane Property (HPP) Y. Choe, J. Oxley, A. Sokal, D. Wagner (Gårding) Hyperbolic Polynomial (PDE theory, control theory, optimization) f R[z 1,..., z n ] of degree d is real stable iff its homogenization f H (z 1,..., z n, z n+1 ) = zn+1 d f(z 1zn+1 1,..., z nzn+1 1 ) is hyperbolic w.r.t. every v R n + {0}, i.e., t f H (u + tv) has all real zeros if u R n+1, v R n + {0}. 3
Rayleigh polynomial (h-nlc+) Multi-affine polynomial f(z 1,..., z n ) with (nonnegative) real coefficients satisfying f f z i z j f z i z j f 0 for 1 i, j n and z R n +. (Electrical networks: Kirchhoff-Rayleigh monotonicity for relative effective conductances; cf. D. Wagner) Strongly Rayleigh polynomial Multi-affine polynomial f(z 1,..., z n ) with (nonnegative) real coefficients satisfying f f z i z j f z i z j f 0 for 1 i, j n and z R n. Theorem [P. Brändén]. A multi-affine polynomial f(z 1,..., z n ) is strongly Rayleigh iff it is stable. 4
Examples. (i) f R[t] is stable iff all its zeros are real; (ii) Spanning tree polynomials are stable. Notation P n is the set of multi-affine polynomials f(z) = S [n] a Sz S with a S 0 S [n] = {1,..., n} and f(1,..., 1) = 1, where z = (z 1,..., z n ) and z S = i S z i. P n is the set of probability measures on 2 [n] = {S : S [n]} {0, 1} [n]. The i-th coordinate function on 2 [n] is the binary r.v. X i (S) = 1 if i S and 0 otherwise. 1-1 correspondence P n P n : if µ P n its generating polynomial g µ (z) = z S dµ = S [n] µ(s)zs P n. µ P n is (strongly) Rayleigh if g µ is (strongly) Rayleigh. 5
Positive dependence Positive Association (PA) µ P n is PA if F Gdµ F dµ Gdµ for all increasing F, G : 2 [n] R. Positive Lattice Condition (PLC) µ P n satisfies PLC if for all S, T [n] µ(s T ) µ(s T ) µ(s) µ(t ). Negative dependence FKG Theorem PLC = PA. Negative Association (NA) µ P n is NA if F Gdµ F dµ Gdµ for all increasing F, G : 2 [n] R depending on disjoint sets of variables. R. Pemantle & others: We need a theory of Negative Dependence. Under what conditions do we have NA? 6
Closure properties and CNA+ Strongly Rayleigh polynomials are closed under (1). taking limits (2). partial differentiation (conditioning on X j = 1) (3). scaling of variables with positive numbers (external fields) (4). setting variables equal to nonnegative numbers (projections and conditioning on X j = 0) CNA+ := NA + (2) + (3) + (4) Theorem [J.B., P. Brändén, T. Liggett]. Strongly Rayleigh measures are CNA+. The strongly Rayleigh class includes e.g. uniform random spanning tree measures, product meaures, ballsand-bins measures, determinantal measures. 7
Example: determinantal measures Let A be an n n matrix with A(S) 0, S [n], A(S) := principal minor of A indexed by S = [n]\s. Determinantal measure µ = µ A P n defined by µ(s) = A(S)/ det(i + A), S [n]. Occurences: Number theory: Montgomery, Conrey,... Rep. theory and random permutations: Johansson, Borodin-Okounkov-Olshanski-Reshetikhin,... Probability theory: Lyons-Steif, Peres,... Mathematical physics: Daley, Vere-Jones,... The generating polynomial of µ A is g µa (z) = det(i + A) 1 det(z + A), where Z = diag(z 1,..., z n ). 8
Hadamard-Fischer-Kotelyansky inequalities Let A = (a ij ) be positive semidefinite (A 0). Hadamard: det(a) a 11 a nn Fischer: det(a) A(S) A(S ) Kotelyansky: A(S T ) A(S T ) A(S) A(T ) Theorem [R. Lyons]. If A 0 then µ A is CNA+. Theorem [J.B., P. Brändén, T. Liggett]. Strongly Rayleigh measures are CNA+. + Theorem [J.B., P. Brändén]. If B is Hermitian and A 1,..., A n 0 (all of the same order) then f(z 1,..., z n ) = det(z 1 A 1 +...+z n A n +B) is stable. In particular, if A 0 then g µa (z) is stable (= strongly Rayleigh). 9
Example: graphs, Laplacians, spanning trees Let G = (V, E), V = [n], be a graph, and e an edge connecting i < j. Let A e be the n n matrix with nonzero entries (A e ) ii = (A e ) jj = 1, (A e ) ij = (A e ) ji = 1 ( A e 0). The Laplacian of G is L(G) = w e A e. e E Let f G (z, w) = det(l(g)+z), where w = (w e ) e E, z = (z 1,..., z n ) and Z = diag(z 1,..., z n ). Kirchhoff s Matrix-Tree Theorem: If G is connected then det(l(g) ii ) = f G (z, w) z i = z=0 T where the sum is over all spanning trees. All Minors Matrix-Tree Theorem: w T f G (z, w) = F z roots(f) w edges(f) where the sum is over all rooted spanning forests. 10
A positive answer to a conjecture of Wagner For each spanning forest F let ρ(f) = # of ways to root F and define µ P E by µ(s) = ρ(f)/r if S is a spanning forest, 0 otherwise, where R = F ρ(f). Conjecture [D. Wagner]. µ is Rayleigh. Proof. The generating polynomial of µ is g µ (w) = µ(s)w S = R 1 det(l(g) + I) S E = R 1 f G (z, w) z1 = =z n =1 so g µ is in fact stable/strongly Rayleigh. 11
Example: the symmetric exclusion process Let τ = (ij) be a transposition and for S [n] let τ(s) = {τ(s) : s S}. Given µ P n and p [0, 1] define µ τ,p (S) := pµ(s) + (1 p)µ(τ(s)). Theorem [J.B., P. Brändén, T. Liggett]. µ is strongly Rayleigh = µ τ,p is strongly Rayleigh. In particular, this proves the following Conjecture [R. Pemantle, T. Liggett]. If the initial configuration of a symmetric exclusion process is deterministic then the distribution at time t is NA t 0. 12
Strongly Rayleigh CNA+: main ideas 1. Symmetric homogenization: let µ P n and define µ H P 2n by ( ) µ S [n] µ H (S) = ( n S [n] ) if S = n, 0 otherwise, for S [2n]. The g.p. g µh = g µh (z 1,..., z 2n ) is g µh = S [n] µ(s) ) z S e n S (z n+1,..., z 2n ), ( n S where z = (z 1,..., z n ) and e j (w 1,..., w m ) = 1 i 1 < <i j m w i1 w ij is the j-th elementary symmetric polynomial. Theorem [J.B., P. Brändén, T. Liggett]. µ strongly Rayleigh µ H strongly Rayleigh. 13
Sketch of proof: S [n] µ(s)zs is stable = {Using Gårding s results on hyperbolicity cones} S [n] µ(s)zs y n S is stable = {Grace-Walsh-Szegö Coincidence Theorem} ) 1z S e n S (z n+1,..., z 2n ) S [n] µ(s)( n S is stable. 2. Feder-Mihail Theorem: Let S be a class of probability measures such that S is closed under conditioning, µ(x i X j ) µ(x i ) µ(x j ) for all i j, µ S, any µ S has a homogeneous gen. polynomial. Then every µ S is NA. 3. Facts: (I) (Strongly) Rayleigh measures with homogeneous gen. polynomials meet the requirements. (II) NA is closed under projections. 14
More (positive) results on negative dependence Proof of Pemantle s conjecture on stochastic domination for truncations of strongly Rayleigh measures and counterexamples for other classes of CNA+ measures (J.B., P. Brändén, T. Liggett). Extensions of Lyons results on NA, the Löwner order and stochastic domination for determinantal measures (J.B., P. Brändén, T. Liggett). Conjecture [Pemantle, Wagner]. Rayleigh = CNA+. True for e.g. exchangeable measures (Pemantle). Proof for the class of almost exchangeable measures (J.B., P. Brändén, T. Liggett). Distributional limits for the symmetric exclusion process (T. Liggett), the HPP for certain matroids studied by A. Sokal, D. Wagner (J.B.). 15
Negative results on negative dependence Let (a k ) n k=0 be a sequence of nonnegative numbers. Log-Concave (LC) a 2 k a k 1a k+1 Ultra-Log-Concave (ULC) ( n k a 2 k ( n ) ( a k+1 ) n k 1 k+1 ) 2 a k 1 i < j < k, a i a k 0 = a j 0 The rank sequence of µ P n is (a k ) n k=0 : a k = µ n i=1 X i = k, i.e., g µ (t,..., t) = n k=0 where g µ (z 1,..., z n ) = S [n] µ(s)zs is the generating polynomial of µ. µ is ULC if its rank sequence is ULC. a k t k, 16
Fact g(z 1,..., z n ) is stable g(t,..., t) is real-rooted. + Newton s Inequalities If m k=0 a k t k is real-rooted then (a k ) m k=0 is ULC. If µ P n is strongly Rayleigh then µ is ULC. Big Conjecture [R. Pemantle, Y. Peres, D. Wagner ]. If µ P n is Rayleigh then µ is ULC. (would Mason s conjecture for Rayleigh matroids) Conjecture [R. Pemantle, Y. Peres ]. If µ P n is CNA+ then µ is ULC. Counterexamples [J. B., P. Brändén, T. Liggett ]. For any n 20 almost exchangeable µ P n for which both the Big Conjecture and CNA+ ULC fail. 17