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Solution to a problem arising from Mayers theory of cluster integrals Olivier Bernardi, C.R.M. Barcelona October 2006, 57 th Seminaire Lotharingien de Combinatoire CRM, Barcelona Olivier Bernardi p.1/37 Content of the talk Mayers

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Solution to a problem arising from Mayer’s theory of cluster integrals

Olivier Bernardi, C.R.M. Barcelona October 2006, 57th Seminaire Lotharingien de Combinatoire

CRM, Barcelona Olivier Bernardi – p.1/37

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Content of the talk

Mayer’s theory of cluster integrals Pressure = Generating function of weighted connected graphs.

CRM, Barcelona

▽Olivier Bernardi – p.2/37

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Content of the talk

Mayer’s theory of cluster integrals Pressure = Generating function of weighted connected graphs. Hard-core continuum gas Pressure = Generating function of Cayley trees.

CRM, Barcelona

▽Olivier Bernardi – p.2/37

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Content of the talk

Mayer’s theory of cluster integrals Pressure = Generating function of weighted connected graphs. Hard-core continuum gas Pressure = Generating function of Cayley trees. Why ? [Labelle, Leroux, Ducharme : SLC 54]

CRM, Barcelona

▽Olivier Bernardi – p.2/37

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Content of the talk

Mayer’s theory of cluster integrals Pressure = Generating function of weighted connected graphs. Hard-core continuum gas Pressure = Generating function of Cayley trees. Why ? [Labelle, Leroux, Ducharme : SLC 54] Combinatorial explaination

CRM, Barcelona Olivier Bernardi – p.2/37

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Mayer’s theory of cluster integrals

CRM, Barcelona Olivier Bernardi – p.3/37

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Statistical physics

Gas of n particules in a box Ω. Ω x1 x2 x3

CRM, Barcelona

▽Olivier Bernardi – p.4/37

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Statistical physics

Gas of n particules in a box Ω. Ω x1 x2 x3 The energy of a configuration x1, . . . , xn is ǫ(x1, . . . , xn) =

  • i

µ(xi) +

  • i<j

φ(xi, xj).

CRM, Barcelona

▽Olivier Bernardi – p.4/37

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Statistical physics

Gas of n particules in a box Ω. Ω x1 x2 x3 The energy of a configuration x1, . . . , xn is ǫ(x1, . . . , xn) =

  • i

µ(xi) +

  • i<j

φ(xi, xj). No external field : µ(xi) = µ.

CRM, Barcelona Olivier Bernardi – p.4/37

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Statistical physics

Ω x1 x2 x3 Energy : ǫ(x1, . . . , xn) = nµ +

i<j φ(xi, xj).

CRM, Barcelona

▽Olivier Bernardi – p.5/37

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Statistical physics

Ω x1 x2 x3 Energy : ǫ(x1, . . . , xn) = nµ +

i<j φ(xi, xj).

The partition function is Z(Ω, T, n) = 1 n!

  • Ωn exp
  • −ǫ(x1, . . . , xn)

kT

  • dx1..dxn

.

CRM, Barcelona

▽Olivier Bernardi – p.5/37

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Statistical physics

Ω x1 x2 x3 Energy : ǫ(x1, . . . , xn) = nµ +

i<j φ(xi, xj).

The partition function is Z(Ω, T, n) = 1 n!

  • Ωn exp
  • −ǫ(x1, . . . , xn)

kT

  • dx1..dxn

= 1 λnn!

  • Ωn
  • i<j

exp

  • −φ(xi, xj)

kT

  • dx1..dxn.

CRM, Barcelona Olivier Bernardi – p.5/37

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Example

Hard particules in Ω = {1, . . . , q}. λ = 1 and φ(x, y) = +∞

if x = y

  • therwise.

CRM, Barcelona

▽Olivier Bernardi – p.6/37

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Example

Hard particules in Ω = {1, . . . , q}. λ = 1 and φ(x, y) = +∞

if x = y

  • therwise.

The partition function : Z(Ω, T) ≡ 1 n!

  • Ωn
  • i<j

exp

  • −φ(xi, xj)

kT

  • CRM, Barcelona

▽Olivier Bernardi – p.6/37

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Example

Hard particules in Ω = {1, . . . , q}. λ = 1 and φ(x, y) = +∞

if x = y

  • therwise.

The partition function : Z(Ω, T) ≡ 1 n!

  • Ωn
  • i<j

exp

  • −φ(xi, xj)

kT

  • =

q n

  • CRM, Barcelona

Olivier Bernardi – p.6/37

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Mayer’s Idea (1940)

exp

  • −φ(xi, xj)

kT

  • = 1 + f(xi, xj).

CRM, Barcelona

▽Olivier Bernardi – p.7/37

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Mayer’s Idea (1940)

  • i<j

exp

  • −φ(xi, xj)

kT

  • =
  • i<j

1+f(xi, xj) =

  • G⊆Kn
  • (i,j)∈G

f(xi, xj).

CRM, Barcelona

▽Olivier Bernardi – p.7/37

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Mayer’s Idea (1940)

  • i<j

exp

  • −φ(xi, xj)

kT

  • =
  • i<j

1+f(xi, xj) =

  • G⊆Kn
  • (i,j)∈G

f(xi, xj). ⇒ Partition function can be written as a sum over graphs : Z(Ω, T, n) ≡ 1 λnn!

  • Ωn
  • i<j

exp

  • −φ(xi, xj)

kT

  • dx1..dxn

= 1 λnn!

  • G⊆Kn

W(G), where W(G) =

  • Ωn
  • (i,j)∈G

f(xi, xj)dx1..dxn is the Mayer’s weight of G.

CRM, Barcelona Olivier Bernardi – p.7/37

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For those familiar with the Tutte Polynomial

Mayer’s tranformation is the analogue (for general partition function) of the correspondence Partition function of the Potts model ⇐ ⇒ Tutte polynomial (coloring expansion) (subgraph expansion) [Fortuin & Kasteleyn 72]

CRM, Barcelona Olivier Bernardi – p.8/37

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Example

Hard particules in Ω = {1, . . . , q}. φ(x, y) = +∞

if x = y

f(x, y) = −1

if x = y

  • therwise.
  • therwise.

CRM, Barcelona

▽Olivier Bernardi – p.9/37

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Example

Hard particules in Ω = {1, . . . , q}. φ(x, y) = +∞

if x = y

f(x, y) = −1

if x = y

  • therwise.
  • therwise.

Mayer’s weight of G : W(G) =

  • Ωn
  • (i,j)∈G

f(xi, xj) = (−1)e(G)qc(G).

CRM, Barcelona

▽Olivier Bernardi – p.9/37

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Example

Hard particules in Ω = {1, . . . , q}. φ(x, y) = +∞

if x = y

f(x, y) = −1

if x = y

  • therwise.
  • therwise.

Mayer’s weight of G : W(G) =

  • Ωn
  • (i,j)∈G

f(xi, xj) = (−1)e(G)qc(G). Mayer’s correspondence

  • G⊆Kn

W(G) = n!Z(Ω, n) shows :

  • G⊆Kn

(−1)e(G)qc(G) = n! q n

  • = q(q − 1) . . . (q − n + 1).

CRM, Barcelona Olivier Bernardi – p.9/37

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Allowing any number of particules

The grand canonical partition function is Zgr(Ω, T, z) =

  • n

Z(Ω, T, n)λnzn. In terms of Mayer’s weights : Zgr(Ω, T, z) =

  • n
  • 1

λnn!

  • G⊆Kn

W(G)

  • λnzn =
  • G

W(G)z|G| |G|! .

CRM, Barcelona Olivier Bernardi – p.10/37

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Pressure

The pressure of the system is given by P(Ω, T, z) = kT |Ω| log (Zgr(Ω, T, z)) .

CRM, Barcelona

▽Olivier Bernardi – p.11/37

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Pressure

The pressure of the system is given by P(Ω, T, z) = kT |Ω| log (Zgr(Ω, T, z)) . Since Mayers weights are multiplicative P(Ω, T, z) = kT |Ω| log (Zgr(Ω, T, z)) = kT |Ω|

  • G connected

W(G)z|G| |G|! .

CRM, Barcelona Olivier Bernardi – p.11/37

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Example

Hard particules in Ω = {1, . . . , q}. Grand canonical partition function : Zgr(Ω, T, z) =

  • n

Z(Ω, T, n)zn =

  • n

q n

  • zn = (1 + z)q.

CRM, Barcelona

▽Olivier Bernardi – p.12/37

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Example

Hard particules in Ω = {1, . . . , q}. Grand canonical partition function : Zgr(Ω, T, z) =

  • n

Z(Ω, T, n)zn =

  • n

q n

  • zn = (1 + z)q.

Pressure : P(Ω, T, z) = kT |Ω| log (Zgr(Ω, T, z)) = kT log(1 + z).

CRM, Barcelona Olivier Bernardi – p.12/37

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Example

Mayer’s weights : W(G) = (−1)e(G)qc(G). Pressure : P(Ω, T, z) = kT |Ω|

  • G connected

W(G)z|G| |G|! = kT

  • G connected

(−1)e(G)z|G| |G|! .

CRM, Barcelona

▽Olivier Bernardi – p.13/37

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Example

Mayer’s weights : W(G) = (−1)e(G)qc(G). Pressure : P(Ω, T, z) = kT |Ω|

  • G connected

W(G)z|G| |G|! = kT

  • G connected

(−1)e(G)z|G| |G|! . Comparing the two expressions of the pressure yields :

  • G connected

(−1)e(G)z|G| |G|! = log(1 + z).

CRM, Barcelona

▽Olivier Bernardi – p.13/37

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Example

Mayer’s weights : W(G) = (−1)e(G)qc(G). Pressure : P(Ω, T, z) = kT |Ω|

  • G connected

W(G)z|G| |G|! = kT

  • G connected

(−1)e(G)z|G| |G|! . Comparing the two expressions of the pressure yields :

  • G connected

(−1)e(G)z|G| |G|! = log(1 + z). In other words :

  • G⊆Kn connected

(−1)e(G) = (−1)n−1(n − 1)!.

CRM, Barcelona Olivier Bernardi – p.13/37

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How did we get there ?

Mayer

Z(Ω, T, z) log A B log =

  • G

W(G)z|G| |G|!

CRM, Barcelona Olivier Bernardi – p.14/37

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A killing involution

  • G⊆Kn connected

(−1)e(G) = (−1)n−1(n − 1)!

CRM, Barcelona

▽Olivier Bernardi – p.15/37

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A killing involution

  • G⊆Kn connected

(−1)e(G) = (−1)n−1(n − 1)! We define an involution Φ on the set of connected graphs :

  • Order the edges of Kn lexicographicaly.
  • Define E∗(G) = {e = (i, j) / i and j are connected by G>e},

Φ(G) = G

if E∗(G) = ∅

G ⊕ min(E∗(G))

  • therwise.

CRM, Barcelona

▽Olivier Bernardi – p.15/37

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A killing involution

  • G⊆Kn connected

(−1)e(G) = (−1)n−1(n − 1)! We define an involution Φ on the set of connected graphs :

  • Order the edges of Kn lexicographicaly.
  • Define E∗(G) = {e = (i, j) / i and j are connected by G>e},

Φ(G) = G

if E∗(G) = ∅

G ⊕ min(E∗(G))

  • therwise.

Prop [B.] : The only remaining graphs are the increasing spanning trees. (Known to be in bijection with the permutations of {1, .., n − 1}.)

CRM, Barcelona Olivier Bernardi – p.15/37

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Increasing trees

4 3 2 1 4 3 2 1 4 3 2 1 4 3 2 1 4 3 2 1 4 3 2 1

CRM, Barcelona Olivier Bernardi – p.16/37

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For those familiar with the Tutte Polynomial

The sum of the Mayer’s weight correspond to the evaluations of TKn(1, 0). This is the number of internal spanning trees. Subgraph expansion ⇐ ⇒ Spanning tree expansion

CRM, Barcelona Olivier Bernardi – p.17/37

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Hard-core continuum gas

CRM, Barcelona Olivier Bernardi – p.18/37

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Hard-core continuum gas

Hard particules in Ω = [0, q].

Ω

x3 x1 x2 φ(x, y) = +∞

if |x − y| < 1

f(x, y) = −1

if |x − y| < 1

  • therwise.
  • therwise.

CRM, Barcelona

▽Olivier Bernardi – p.19/37

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Hard-core continuum gas

Hard particules in Ω = [0, q].

Ω

x3 x1 x2 φ(x, y) = +∞

if |x − y| < 1

f(x, y) = −1

if |x − y| < 1

  • therwise.
  • therwise.

W(G) =

  • Ωn
  • (i,j)∈G

f(xi, xj)dx1..dxn. P(Ω, T, z) = kT |Ω|

  • G connected

W(G)z|G| |G|! .

CRM, Barcelona Olivier Bernardi – p.19/37

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Thermodynamical limit (|Ω| → ∞)

x3 x1 x2 P(T, z) ≡ lim

|Ω|→∞ P(Ω, T, z) = kT

  • G connected

W∞(G)z|G| |G|! where, W∞(G) ≡ lim

|Ω|→∞

WΩ(G) |Ω| =

  • n−1; x1=0
  • (i,j)∈G

f(xi, xj)dx2..dxn.

CRM, Barcelona Olivier Bernardi – p.20/37

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Mayer’s weight for the hard-core gas

f(x, y) = −1

if |x − y| < 1

  • therwise.

W(G) =

  • n−1; x1=0
  • (i,j)∈G

f(xi, xj)dx2..dxn.

CRM, Barcelona

▽Olivier Bernardi – p.21/37

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Mayer’s weight for the hard-core gas

f(x, y) = −1

if |x − y| < 1

  • therwise.

W(G) =

  • n−1; x1=0
  • (i,j)∈G

f(xi, xj)dx2..dxn. P(T, z) = kT

  • G connected

W(G)z|G| |G|!

CRM, Barcelona Olivier Bernardi – p.21/37

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Mayer’s diagram for the hard-core gas

Mayer

log log =

  • G

W(G)z|G| |G|! Z(T, z)

  • G⊆Kn W∞(G)

(−1)n−1nn−1

Cayley trees ! ? [Labelle, Leroux, Ducharme : SLC 54]

CRM, Barcelona Olivier Bernardi – p.22/37

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Slicing W(G) [Lass]

W(G) =

  • n−1; x1=0
  • (i,j)∈G

f(xi, xj)dx2..dxn, = (−1)e(G) × Volume(ΠG), where ΠG ⊂ Rn−1 is the polytope

  • (i,j)∈G

|xi − xj| ≤ 1.

CRM, Barcelona

▽Olivier Bernardi – p.23/37

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Slicing W(G) [Lass]

W(G) =

  • n−1; x1=0
  • (i,j)∈G

f(xi, xj)dx2..dxn, = (−1)e(G) × Volume(ΠG), where ΠG ⊂ Rn−1 is the polytope

  • (i,j)∈G

|xi − xj| ≤ 1. Example :

G : ΠG :

x2 x1 x3 x3 x2

CRM, Barcelona

▽Olivier Bernardi – p.23/37

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Slicing W(G) [Lass]

W(G) =

  • n−1; x1=0
  • (i,j)∈G

f(xi, xj)dx2..dxn, = (−1)e(G) × Volume(ΠG), where ΠG ⊂ Rn−1 is the polytope

  • (i,j)∈G

|xi − xj| ≤ 1. Example :

G : ΠG :

x2 x1 x3 x3 x2

CRM, Barcelona Olivier Bernardi – p.23/37

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Slicing W(G) [Lass]

Fractional representation xi = h(xi) + ǫ(xi). 2 3 ǫ = 1 xi −1 ǫ(xi) ǫ = 0 h(xi) = 1

CRM, Barcelona

▽Olivier Bernardi – p.24/37

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Slicing W(G) [Lass]

Fractional representation xi = h(xi) + ǫ(xi). |xi − xj| < 1 ⇐ ⇒ 2 3 −1 xi xj 1 xi xj

CRM, Barcelona

▽Olivier Bernardi – p.24/37

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Slicing W(G) [Lass]

Fractional representation xi = h(xi) + ǫ(xi). Prop [Lass] : (x2, .., xn) ∈ ΠG ? only depends on the integer parts h(x2), .., h(xn) and the order of the fractional parts ǫ(x2), .., ǫ(xn).

CRM, Barcelona

▽Olivier Bernardi – p.24/37

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Slicing W(G) [Lass]

Fractional representation xi = h(xi) + ǫ(xi). Prop [Lass] : (x2, .., xn) ∈ ΠG ? only depends on the integer parts h(x2), .., h(xn) and the order of the fractional parts ǫ(x2), .., ǫ(xn). x2 x1 x3 x4 x5 x6 h6 = 0 h5 = −1 h4 = 2 h3 = 0 h2 = 1 h1 = 0 3 2 1 −1 0=ǫ1 <ǫ4 <ǫ6<ǫ2 <ǫ5 <ǫ3

CRM, Barcelona

▽Olivier Bernardi – p.24/37

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Slicing W(G) [Lass]

Fractional representation xi = h(xi) + ǫ(xi). Prop [Lass] : (x2, .., xn) ∈ ΠG ? only depends on the integer parts h(x2), .., h(xn) and the order of the fractional parts ǫ(x2), .., ǫ(xn). x2 x6 x3 x1 x5 x4 2 3 −1 x2 1 x1 x3 x4 x5 x6

CRM, Barcelona

▽Olivier Bernardi – p.24/37

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Slicing W(G) [Lass]

Fractional representation xi = h(xi) + ǫ(xi). Prop [Lass] : (x2, .., xn) ∈ ΠG ? only depends on the integer parts h(x2), .., h(xn) and the order of the fractional parts ǫ(x2), .., ǫ(xn). Each subpolytope ∆ defined by h2, .., hn and an order on ǫ(x2), .., ǫ(xn) has volume 1 (n − 1)!.

CRM, Barcelona Olivier Bernardi – p.24/37

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Counting labelled schemes

G : ΠG :

x2 x1 x3 x2 x3 x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x1 x3 x1 x2 x1 x2 x3 Each labelled scheme has weight (−1)e(G) (n − 1)! .

CRM, Barcelona Olivier Bernardi – p.25/37

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Rearanging the sum

  • G⊆Kn

connected

W(G) =

  • G⊆Kn

connected

(−1)e(G) (n − 1)! #{S labelled scheme containing G}

CRM, Barcelona

▽Olivier Bernardi – p.26/37

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SLIDE 55

Rearanging the sum

  • G⊆Kn

connected

W(G) =

  • G⊆Kn

connected

(−1)e(G) (n − 1)! #{S labelled scheme containing G} =

  • S labelled scheme

1 (n − 1)!

  • G contained in S

(−1)e(G)

CRM, Barcelona

▽Olivier Bernardi – p.26/37

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SLIDE 56

Rearanging the sum

  • G⊆Kn

connected

W(G) =

  • G⊆Kn

connected

(−1)e(G) (n − 1)! #{S labelled scheme containing G} =

  • S labelled scheme

1 (n − 1)!

  • G contained in S

(−1)e(G) =

  • S scheme
  • G contained in S

(−1)e(G)

CRM, Barcelona Olivier Bernardi – p.26/37

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Rearranging the sum

  • G contained in S

(−1)e(G) 2 3 −1 1

CRM, Barcelona Olivier Bernardi – p.27/37

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A killing involution

We define an involution Φ on the set of connected graphs contained in S :

  • Order the edges of Kn lexicographicaly.
  • Define E∗(G) = {e = (i, j) / i and j are connected by G>e},

Φ(G) = G

if E∗(G) = ∅

G ⊕ min(E∗(G))

  • therwise.

CRM, Barcelona

▽Olivier Bernardi – p.28/37

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SLIDE 59

A killing involution

We define an involution Φ on the set of connected graphs contained in S :

  • Order the edges of Kn lexicographicaly.
  • Define E∗(G) = {e = (i, j) / i and j are connected by G>e},

Φ(G) = G

if E∗(G) = ∅

G ⊕ min(E∗(G))

  • therwise.

Proposition [B.] : The only remaining graphs are the increasing spanning trees. Corrolary :

  • G contained in S

(−1)e(G) = (−1)n−1#{increasing tree on S}.

CRM, Barcelona Olivier Bernardi – p.28/37

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A killing involution

CRM, Barcelona

▽Olivier Bernardi – p.29/37

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A killing involution

CRM, Barcelona Olivier Bernardi – p.29/37

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SLIDE 62

Bijection with Cayley trees

Theorem [B.] :

  • S scheme

{increasing tree} are in bijection with rooted Cayley trees.

CRM, Barcelona

▽Olivier Bernardi – p.30/37

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SLIDE 63

Bijection with Cayley trees

Theorem [B.] :

  • S scheme

{increasing tree} are in bijection with rooted Cayley trees.

CRM, Barcelona

▽Olivier Bernardi – p.30/37

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SLIDE 64

Bijection with Cayley trees

Theorem [B.] :

  • S scheme

{increasing tree} are in bijection with rooted Cayley trees.

1 2 3 4 6 7 8 9 5

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▽Olivier Bernardi – p.30/37

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Bijection with Cayley trees

Theorem [B.] :

  • S scheme

{increasing tree} are in bijection with rooted Cayley trees.

1 2 3 4 6 7 8 9 5

CRM, Barcelona Olivier Bernardi – p.30/37

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Bijection with Cayley trees

Corollary [B.] :

  • G⊆Kn

connected

W(G) =

  • S scheme
  • G contained in S

(−1)e(G) = (−1)n−1

  • S scheme

#{increasing tree on S} = (−1)n−1 nn−1.

CRM, Barcelona Olivier Bernardi – p.31/37

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Bijection with Cayley trees

3 1 2 2 3 1 1 2 3 2 3 1 2 3 3 1 2 1 3 2 1 2 3 2 1 3 1

CRM, Barcelona Olivier Bernardi – p.32/37

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Concluding remarks

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Mayer’s transformation

Producing graph weights Mayer

Z(Ω, T, z)

  • G

W(G)z|G| |G|!

CRM, Barcelona

▽Olivier Bernardi – p.34/37

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SLIDE 70

Mayer’s transformation

Producing graph weights Producing nasty identities Mayer

Z(Ω, T, z) log A B log =

  • G

W(G)z|G| |G|!

CRM, Barcelona Olivier Bernardi – p.34/37

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SLIDE 71

Discrete hard-core gas (colorings)

Mayer

Z(Ω, T, z) log log =

  • G

W(G)z|G| |G|! (−1)n−1(n − 1)!

  • G⊆Kn

connected

(−1)e(G)

CRM, Barcelona

▽Olivier Bernardi – p.35/37

slide-72
SLIDE 72

Discrete hard-core gas (colorings)

Mayer

Z(Ω, T, z) log log =

  • G

W(G)z|G| |G|! (−1)n−1(n − 1)!

  • G⊆Kn

connected

(−1)e(G)

Potts model ⇐

⇒

Subgraph expansion Spanning tree expansion ⇐

⇒

Subgraph expansion

CRM, Barcelona Olivier Bernardi – p.35/37

slide-73
SLIDE 73

Continuous hard-core gas

Mayer

log log =

  • G

W(G)z|G| |G|! Z(T, z)

  • G⊆Kn W∞(G)

(−1)n−1nn−1

CRM, Barcelona Olivier Bernardi – p.36/37

slide-74
SLIDE 74

Thanks.

CRM, Barcelona Olivier Bernardi – p.37/37