Determinantal Probability Measures. by Russell Lyons (Indiana University)

Transcription

1 Determinantal Probability Measures by Russell Lyons (Indiana University) 1

2 Determinantal Measures If E is finite and H l 2 (E) is a subspace, it defines the determinantal measure T E with T = dim H P H (T ) := det[p H ] T,T, where the subscript T, T indicates the submatrix whose rows and columns belong to T. This representation has a useful extension, namely, D E P H [D T ] = det[p H ] D,D. In case E is infinite and H is a closed subspace of l 2 (E), the determinantal probability measure P H is defined via the requirement that this equation hold for all finite D E. 2

3 Matroids Let E be a finite set, called the ground set, and let B be a nonempty collection of subsets of E. We call the pair M := (E, B) a matroid with bases B if the following exchange property is satisfied: B, B B e B \ B e B \ B (B \ {e}) {e } B. All bases have the same cardinality, called the rank of the matroid. Example: If E is the set of edges of a finite connected graph and B is the set of spanning trees of the graph, this is called a graphical matroid. Proof. Example: If E is a finite subset of a vector space and B is the set of maximal linearly independent subsets of E, this is called a vectorial matroid. Represent E by columns of a matrix. Example: graphical. Use the incidence matrix. 3

4 A spanning tree of a graph with the edges of the tree in red. 4

5 e x y h k f w g z e f g h k x y z w

6 A representable matroid is one that is isomorphic to a vectorial matroid. A regular matroid is one that is representable over every field. For example, graphical matroids are regular. The dual of a matroid M = (E, B) is the matroid M := (E, B ), where B := {E \ B ; B B}. 6

7 Determinantal Probability Measures (Random Matrix Theory 1950s present, Macchi , Daley, Vere-Jones 1988, many papers since late 1990s; L. 2005) For representable matroids only. The measure depends on the representation. The usual way of representing a vectorial matroid M over R (or over C) of rank r on a ground set E is by an (s E)-matrix M whose columns are the vectors in R s representing M. The column space of M is r-dimensional, so the rank of M is r, and the row space H R E of M is r-dimensional. Suppose that the first r rows, say, of M span H. For an r-subset B E, let M B denote the (r r)-matrix determined by the first r rows of M and the columns of M indexed by those e belonging to B. Let M (r) denote the matrix formed by the first r rows of M. Define P H [B] := det M B 2 / det(m (r) M T (r)), where the superscript T denotes (conjugate) transpose. This depends only on H. row ops and scale 7

8 Simpler formula: Identify e E with 1 {e} l 2 (E). Let P H be the orthogonal projection onto H. Then P H [B] = det[(p H e, e )] e,e B = det[(p H e, P H e )] e,e B. Thus, for r-element subsets B E, we have B B iff P H B is a basis for H. One also obtains A E P H [A B] = det[(p H e, e )] e,e A. ( ) Example: For a graphical matroid, M is the vertex-edge incidence matrix (each edge has a fixed arbitrary orientation). The row space is the space spanned by the stars or cuts. The measure P is uniform measure on spanning trees. Equation ( ) is called the Transfer Current Theorem of Burton and Pemantle (1993). Remark. For any given matroid M, there exists some real representation with a row space H such that P H is uniform on B iff M is regular. 8

9 Why is this a probability measure? Suppose first that H is 1-dimensional (r = 1). Choose a unit vector v H. Then P H [{e}] = (v, e) 2. The general case arises from multivectors. Recall that (u 1 u k, v 1 v k ) = det [ (u i, v j ) ] i,j [1,k]. Also, vectors u 1,..., u k l 2 (E) are linearly independent iff u 1 u k 0. If dim H = r, then r H is a 1-dimensional subspace of Ext ( l 2 (E) ) ; denote by ξ H a unit multivector in this subspace. 9

10 Review of Exterior Algebra E k := choice of ordered k-subsets of E ( k {e1 } ) l 2 (E) := l 2 e k ; e 1,..., e k E k =: multivectors of rank k. k k e σ(i) = sgn(σ) e i for any permutation σ of {1, 2,..., k} i=1 i=1 k a i (e)e = k a j (e j ) k e i for any scalars a i (e) (i [1, k], e E). i=1 e E e 1,...,e k E j=1 i=1 Ext ( l 2 (E) ) := E k=1 k l 2 (E), orthogonal summands For H l 2 (E), we identify Ext(H) with its inclusion in Ext ( l 2 (E) ), that is, k H is the linear span of {v 1 v k ; v 1,..., v k H}. 10

11 Why P H is a probability measure: P H [{e 1,..., e r }] = ( ξ H, r ) e i 2. i=1 To prove this, we use: Lemma. For any subspace H l 2 (E), any k 1, and any u 1,..., u k l 2 (E), P k H(u 1 u k ) = (P H u 1 ) (P H u k ). Proof. Write u 1 u k = (P H u 1 + P H u 1 ) (P H u k + P H u k ) and expand the product. All terms but P H u 1 P H u k have a factor of P H u i in them, making them orthogonal to k H. 11

12 Proof that We have ( ξ H, P H [{e 1,..., e r }] = ( ξ H, r ) e i 2 = P rh( i=1 i ( e i ) 2 r ) e i 2 : i=1 = P r H( e i ), P r H( i i ( = P r H( e i ), ) e i i i ( = P H e i, ) e i i i = det[(p H e i, e j )]. e i ) ) 12

13 Completion of Calculation Let the ith row of M be m i. For some constant c, we thus have r ξ H = c m i, i=1 whence P H [B] = (ξ H, e) 2 e B = c 2 det [(mi, e)] i r, e B 2 = c 2 det M B 2. Now we calculate c 2 : r 1 = ξ H 2 = c 2 m i 2 i=1 = c 2 det [(m i, m j )] i,j r = c 2 det(m (r) M T (r)). 13

14 Matrix-Tree Theorem The Matrix-Tree Theorem. the number of spanning trees of G equals Let G be a finite connected graph and o V. Then det [ ( x, y ) ] x o,y o. Proof. In other words, we want to show that if u is the wedge product (in some order) of the stars at all the vertices other than o, then (u, u) = u 2 is the number of spanning trees. Any set of all the stars but one is a basis for. Thus, u is a multiple of ξ. Since represents the graphic matroid, the only non-zero coefficients of u are those in which choosing one edge in each x for x o yields a spanning tree; moreover, each spanning tree occurs exactly once since there is exactly one way to choose an edge incident to each x o to get a given spanning tree. This means that its coefficient is ±1. This proof also shows that P is uniform. 14

15 Additional Probabilities: Recall that P H [A B] = det[(p H e, e )] e,e A = ( ) P Ext(H) θ A, θ A, where, for a finite subset A = {e 1,... e k } E, we write k θ A := e i. This is proved by proving an extension: For any A 1, A 2 E, P H [A 1 B, A 2 B = ] = ( ) P Ext(H) θ A1 P Ext(H )θ A2, θ A1 θ A2. (First prove when A 1 A 2 = E, then sum over partitions.) Therefore P H [B] = P H [E \ B]. Orthogonal subspaces thus correspond to dual matroids. 15 i=1

16 Additional Property: Extend P H from B to the collection 2 E of all subsets of E. An event A is called increasing if whenever A A and e E, we have also A {e} A. Given two probability measures P 1, P 2 on 2 E, we say that P 1 is stochastically dominated by P 2 and write P 1 P 2 if P 1 [A] P 2 [A] for all increasing A. Theorem (L.). If H H l 2 (E), then P H P H. 16

17 A monotone coupling of two probability measures P 1, P 2 on 2 E is a probability measure µ on 2 E 2 E whose coordinate projections give P 1, P 2 and which is concentrated on the set {(A 1, A 2 ) ; A 1 A 2 }. That is, A 1 E µ(a 1, A 2 ) = P 1 [A 1 ], A 2 E A 2 E A 1 E µ(a 1, A 2 ) = P 2 [A 2 ], A 1, A 2 E µ(a 1, A 2 ) 0 = A 1 A 2. Strassen s theorem (proved, say, by Max Flow-Min Cut Theorem) says that stochastic domination is equivalent to existence of a monotone coupling. Open Question: Find an explicit monotone coupling of P H and P H when H H. 17

18 Extension to Infinite E Let E = {e 1, e 2,...}. If H l 2 (E) is finite-dimensional, then write H k for the image of the orthogonal projection of H onto the span of {e 1, e 2,..., e k }. Then the matrix entries of P Hk limit of P H k. converge to those of P H, whence we may define P H to be the weak* If H l 2 (E) is closed and infinite-dimensional, then let H k be finite-dimensional subspaces of H that are increasing with union dense in H. Again, the matrix entries of P Hk converge to those of P H, whence we may define P H to be the weak* limit of P H k. Theorem (L.). Let E be finite or infinite and let H H be closed subspaces of l 2 (E). Then P H P H, with equality iff H = H. This means that there is a probability measure on the set { (B, B ) ; B B } that projects in the first coordinate to P H and in the second to P H. 18

19 Trees, Forests, and Determinants Let G = (V, E) be a finite graph. Choose one orientation for each edge e E. Let = B 1 (G) denote the subspace in l 2 (E) spanned by the stars (coboundaries) and let = Z 1 (G) denote the subspace spanned by the cycles. Then l 2 (E) =. For an infinite graph, let := B c 1 (G) be the closure in l 2 (E) of the span of the stars. For an infinite graph, Benjamini, Lyons, Peres, and Schramm (2001) showed that WUSF is the determinantal measure corresponding to orthogonal projection on, while FUSF is the determinantal measure corresponding to. Thus, WUSF FUSF, with equality iff =. 19

20 Open Questions: Orthogonal Decomposition Suppose that H = H 1 H 2. Is there a disjoint coupling of P H 1 with P H 2 whose union marginal is P H? I.e., is there a probability measure µ on 2 E 2 E such that A 1 E µ(a 1, A 2 ) = P H 1 [A 1 ], A 2 E A 2 E A 1 E µ(a 1, A 2 ) = P H 2 [A 2 ], A 1, A 2 E µ(a 1, A 2 ) 0 = A 1 A 2 =, A E µ{(a 1, A 2 )} = P H [A]? A 1 A 2 =A E.g., if H = l 2 (E), then yes since then P H 1 and P H 2 correspond to dual matroids and complementary subsets. In general, there is some computer evidence. 20

21 Open Questions: Group Representations We can ask for even more. Suppose that E is a group. Then l 2 (E) is the group algebra. Invariant subspaces H give subrepresentations of the regular representation and give invariant probability measures P H. There is a canonical decomposition l 2 (E) = s H i, i=1 where each H i is an invariant subspace containing all isomorphic copies of a given irreducible subrepresentation. Can we disjointly couple all measures P H i so that every partial union has marginal equal to P H for H the corresponding partial sum? P Hi given by characters Consider the case E = Z n. All irreducible representations are 1-dimensional and there are n of them: for each k Z n, we have the representation j e 2πikj/n. Thus, a coupling as above would be a random permutation of Z n with special properties. comp evid to n = 7. 21

22 REFERENCES Benjamini, I., Lyons, R., Peres, Y., and Schramm, O. (2001). Uniform spanning forests. Ann. Probab. 29, Brooks, R.L., Smith, C.A.B., Stone, A.H., and Tutte, W.T. (1940). The dissection of rectangles into squares. Duke Math. J. 7, Burton, R.M. and Pemantle, R. (1993). Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer-impedances. Ann. Probab. 21, Kirchhoff, G. (1847). Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird. Ann. Phys. und Chem. 72, Lyons, R. (2003). Determinantal probability measures. Publ. Math. Inst. Hautes Études Sci. 98,