Combinatorics of exclusion processes with open boundaries Sylvie - - PowerPoint PPT Presentation

Combinatorics of exclusion processes with open boundaries

Sylvie Corteel (CNRS Paris 7)

GGI, Florence, May 19th 2015

Koornwinder moments and the two species ASEP

Lauren Williams (Berkeley)

Sylvie Corteel (CNRS Paris 7)

. . .

Triangular staircase tableaux

Sylvie Corteel, Olya Mandelshtam (Berkeley) and Lauren Williams (Berkeley)

Binary trees, Paths and tableaux

Binary trees, Paths and tableaux

Binary trees, Paths and tableaux

Binary trees, Paths and tableaux

Binary trees, Paths and tableaux

Binary trees, Paths and tableaux

Binary trees, Paths and tableaux

⌧ 2 {, •}N B(⌧) number of trees of canopy ⌧ M(⌧) number of paths of shape ⌧ C(⌧) number of tableaux of shape ⌧

Binary trees, Paths and tableaux

⌧ 2 {, •}N B(⌧) number of trees of canopy ⌧ M(⌧) number of paths of shape ⌧ C(⌧) number of tableaux of shape ⌧

B(τ) = M(τ) = C(τ)

P

⌧ C(⌧) = Cn+1 Catalan numbers

Binary trees, Paths and tableaux

⌧ 2 {, •}N B(⌧) number of trees of canopy ⌧ M(⌧) number of paths of shape ⌧ C(⌧) number of tableaux of shape ⌧

B(τ) = M(τ) = C(τ)

P

⌧ C(⌧) = Cn+1 Catalan numbers

Binary trees, Paths and tableaux

⌧ 2 {, •}N B(⌧) number of trees of canopy ⌧ M(⌧) number of paths of shape ⌧ C(⌧) number of tableaux of shape ⌧

B(τ) = M(τ) = C(τ)

P

⌧ C(⌧) = Cn+1 Catalan numbers

Binary trees, Paths and tableaux

⌧ 2 {, •}N B(⌧) number of trees of canopy ⌧ M(⌧) number of paths of shape ⌧ C(⌧) number of tableaux of shape ⌧

B(τ) = M(τ) = C(τ)

P

⌧ C(⌧) = Cn+1 Catalan numbers

Binary trees, Paths and tableaux

⌧ 2 {, •}N B(⌧) number of trees of canopy ⌧ M(⌧) number of paths of shape ⌧ C(⌧) number of tableaux of shape ⌧

B(τ) = M(τ) = C(τ)

B(⌧)/Cn+1 is the probability to be in state ⌧ of the TASEP with open boundaries and N sites P

⌧ C(⌧) = Cn+1 Catalan numbers

Binary trees, Paths and tableaux

⌧ 2 {, •}N B(⌧) number of trees of canopy ⌧ M(⌧) number of paths of shape ⌧ C(⌧) number of tableaux of shape ⌧

B(τ) = M(τ) = C(τ)

B(⌧)/Cn+1 is the probability to be in state ⌧ of the TASEP with open boundaries and N sites P

⌧ C(⌧) = Cn+1 Catalan numbers

Matrix Ansatz [Derrida et al 93]

• hW|E = hW|
• D|V i = |V i
• DE = D + E

Matrices D and E, and vectors hW| and |V i ZN = hW|(D + E)N|V i.

P(τ) = hW| QN

i=1[⌧iD+(1⌧i)E]|V i

ZN

.

Steady state ⌧ 2 {, •} = {0, 1}N

Matrix Ansatz [Derrida et al 93]

• hW|E = hW|
• D|V i = |V i
• DE = D + E

Matrices D and E, and vectors hW| and |V i ZN = hW|(D + E)N|V i.

P(τ) = hW| QN

i=1[⌧iD+(1⌧i)E]|V i

ZN

.

Steady state ⌧ 2 {, •} = {0, 1}N Solution: hW| = (1, 0, . . .), |V i = (1, 0, . . .)T D = B B B B B @ 1 1 . . . 1 1 . . . 1 1 . . . 1 . . . . . . . . . 1 C C C C C A E = B B B B B @ 1 . . . 1 1 . . . 1 1 . . . 1 1 . . . . . . . . . 1 C C C C C A Motzkin paths [Zeilberger, Duchi and Schaeffer, Brak and Essam]

Matrix Ansatz [Derrida et al 93]

• hW|E = hW|
• D|V i = |V i
• DE = D + E

Matrices D and E, and vectors hW| and |V i ZN = hW|(D + E)N|V i.

P(τ) = hW| QN

i=1[⌧iD+(1⌧i)E]|V i

ZN

.

Steady state ⌧ 2 {, •} = {0, 1}N Solution: hW| = (1, 0, . . .), |V i = (1, 1, . . .)T D = B B B B B @ 1 . . . 1 . . . 1 . . . . . . . . . . . . 1 C C C C C A E = B B B B B @ 1 . . . 1 1 . . . 1 1 1 . . . 1 1 1 1 . . . . . . . . . 1 C C C C C A Lukasiewicz paths, Catalan tableaux

Asymmetric exclusion process with 5 parameters

Asymmetric exclusion process with 5 parameters α α β β γ γ δ δ

q

Asymmetric exclusion process with 5 parameters α α β β γ γ δ δ

q

• hW|(↵E D) = hW|
• (D E)|V i = |V i
• DE = qED + D + E

Matrix Ansatz

Asymmetric exclusion process with 5 parameters α α β β γ γ δ δ

q

• hW|(↵E D) = hW|
• (D E)|V i = |V i
• DE = qED + D + E

Matrix Ansatz = = 0

• Paths ) moments of AlSalam-Chihara Polynomials
• Trees ) tree like tableaux
• Tableaux ) Permutation tableaux, Alternative tableaux

Asymmetric exclusion process with 5 parameters α α β β γ γ δ δ

q

• hW|(↵E D) = hW|
• (D E)|V i = |V i
• DE = qED + D + E

Matrix Ansatz = = 0

• Paths ) moments of AlSalam-Chihara Polynomials
• Trees ) tree like tableaux
• Tableaux ) Permutation tableaux, Alternative tableaux

[Aval, Boussicault, C. Josuat-Verg` es, Nadeau, Viennot, Williams. . . ]

Asymmetric exclusion process with 5 parameters α α β β γ γ δ δ

q General model

• Moments of Askey Wilson polynomials [Uchiyama, Sasamoto, Wadati 04]
• Staircase tableaux [C., Williams 10]

Askey Wilson polynomials symmetric in a, b, c, d

Pn+1(x) = (x bn)Pn(x) nPn1(x) bn = 1/2(a + 1/a An Cn) n = An1Cn/4 An = (1abqn)(1acqn)(1adqn)(1abcdqn−1)

a(1abcdq2n)(1abcdq2n−1)

Cn = (1abqn−1)(1bcqn−1)(1bdqn−1)(1qn)

a(1abcdq2n−2)(1abcdq2n−1)

Askey Wilson polynomials symmetric in a, b, c, d

• rthogonal

H

C dz 4⇡izw

⇣

z+z−1 2

⌘ Pm ⇣

z+z−1 2

⌘ Pn ⇣

z+z−1 2

⌘ = hnδmn, w(x) =

(z2,z−2;q)∞ (az,a/z,bz,b/z,cz,c/z,dz,d/z;q)∞, x = (z + z1)/2

hn = (1qn−1abcd)(q,ab,ac,ad,bc,bd,cd;q)n

(1q2n−1abcd)(abcd;q)n

Pn+1(x) = (x bn)Pn(x) nPn1(x) bn = 1/2(a + 1/a An Cn) n = An1Cn/4 An = (1abqn)(1acqn)(1adqn)(1abcdqn−1)

a(1abcdq2n)(1abcdq2n−1)

Cn = (1abqn−1)(1bcqn−1)(1bdqn−1)(1qn)

a(1abcdq2n−2)(1abcdq2n−1)

Askey Wilson polynomials symmetric in a, b, c, d

• rthogonal

H

C dz 4⇡izw

⇣

z+z−1 2

⌘ Pm ⇣

z+z−1 2

⌘ Pn ⇣

z+z−1 2

⌘ = hnδmn, w(x) =

(z2,z−2;q)∞ (az,a/z,bz,b/z,cz,c/z,dz,d/z;q)∞, x = (z + z1)/2

hn = (1qn−1abcd)(q,ab,ac,ad,bc,bd,cd;q)n

(1q2n−1abcd)(abcd;q)n

Moments µAW

N

= H

C dz 4⇡izw

⇣

z+z−1 2

⌘ ⇣

z+z−1 2

⌘N

Pn+1(x) = (x bn)Pn(x) nPn1(x) bn = 1/2(a + 1/a An Cn) n = An1Cn/4 An = (1abqn)(1acqn)(1adqn)(1abcdqn−1)

a(1abcdq2n)(1abcdq2n−1)

Cn = (1abqn−1)(1bcqn−1)(1bdqn−1)(1qn)

a(1abcdq2n−2)(1abcdq2n−1)

Combinatorics of moments

[Flajolet, Viennot 80s] Pn+1(x) = (x bn)Pn(x) nPn1(x) (0, 0) (N, 0) b1 b2 1 1 2 2

Combinatorics of moments

[Flajolet, Viennot 80s] Pn+1(x) = (x bn)Pn(x) nPn1(x) (0, 0) (N, 0) b1 b2 1 1 2 2 µN = P

P W(p)

W(P) = b1b212

2

Combinatorics of moments

[Flajolet, Viennot 80s] Pn+1(x) = (x bn)Pn(x) nPn1(x) (0, 0) (N, 0) b1 b2 1 1 2 2 µN = P

P W(p)

W(P) = b1b212

2

(0, 0) (N, r) b1 b1 b0 2 1

µN,r = H

C dz 4⇡izw

⇣

z+z−1 2

⌘ Pr ⇣

z+z−1 2

⌘ ⇣

z+z−1 2

⌘N

Solution of the 5 parameter model [USW 04]

d[

n = qnbd (1qnac)(1qnbd)n

e[

n = 1 (1qnac)(1qnbd)n

d]

n = 1 e] n = qnac

d = B B B B @ d\ d] · · · d[ d\

1

d]

1

d[

1

d\

2

... . . . ... ... 1 C C C C A hW| = (1, 0, . . .), |V i = (1, 0, . . .)T

Solution of the 5 parameter model [USW 04]

d[

n = qnbd (1qnac)(1qnbd)n

e[

n = 1 (1qnac)(1qnbd)n

d]

n = 1 e] n = qnac

d = B B B B @ d\ d] · · · d[ d\

1

d]

1

d[

1

d\

2

... . . . ... ... 1 C C C C A hW| = (1, 0, . . .), |V i = (1, 0, . . .)T

µAW

N

= hW|(d + e)N|V i d\

n + e\ n = bn

Solution of the 5 parameter model [USW 04]

a =

1q↵++p (1q↵+)2+4↵ 2↵

b =

1q++p (1q+)2+4 2

d[

n = qnbd (1qnac)(1qnbd)n

e[

n = 1 (1qnac)(1qnbd)n

d]

n = 1 e] n = qnac

d = B B B B @ d\ d] · · · d[ d\

1

d]

1

d[

1

d\

2

... . . . ... ... 1 C C C C A hW| = (1, 0, . . .), |V i = (1, 0, . . .)T

µAW

N

= hW|(d + e)N|V i ZN = hW|(D + E)N|V i D = 1+d

1q, E = 1+e 1q

d\

n + e\ n = bn

Koorwinder polynomials

Multivariate version of the AW polynomials

P(z; a, b, c, d|q, q) = const ·

det(pm−j+λj (zi;a,b,c,d|q))m

i,j=1

det(pm−j(zi;a,b,c,d|q))m

i,j=1

at q = t P(z1, . . . , zm; a, b, c, d|q, t) Density Q

1i<jm(1 zizj)(1 zi/zj)(1 zj/zi)(1 1/zizj) Q 1im w

⇣

zi+z−1

i

2

⌘

Koorwinder polynomials

Multivariate version of the AW polynomials

P(z; a, b, c, d|q, q) = const ·

det(pm−j+λj (zi;a,b,c,d|q))m

i,j=1

det(pm−j(zi;a,b,c,d|q))m

i,j=1

at q = t P(z1, . . . , zm; a, b, c, d|q, t) Density Q

1i<jm(1 zizj)(1 zi/zj)(1 zj/zi)(1 1/zizj) Q 1im w

⇣

zi+z−1

i

2

⌘ AW-density Possible definition of moments M = Ik(s(x1, . . . , xm); a, b, c, d; q, q). AW-polynomials

Koorwinder polynomials

Multivariate version of the AW polynomials

P(z; a, b, c, d|q, q) = const ·

det(pm−j+λj (zi;a,b,c,d|q))m

i,j=1

det(pm−j(zi;a,b,c,d|q))m

i,j=1

at q = t P(z1, . . . , zm; a, b, c, d|q, t) Density Q

1i<jm(1 zizj)(1 zi/zj)(1 zj/zi)(1 1/zizj) Q 1im w

⇣

zi+z−1

i

2

⌘ AW-density Possible definition of moments M = Ik(s(x1, . . . , xm); a, b, c, d; q, q). Schur functions Integrate with respect to the Koorwinder density AW-polynomials

Koorwinder polynomials

Multivariate version of the AW polynomials

P(z; a, b, c, d|q, q) = const ·

det(pm−j+λj (zi;a,b,c,d|q))m

i,j=1

det(pm−j(zi;a,b,c,d|q))m

i,j=1

at q = t P(z1, . . . , zm; a, b, c, d|q, t) Density Q

1i<jm(1 zizj)(1 zi/zj)(1 zj/zi)(1 1/zizj) Q 1im w

⇣

zi+z−1

i

2

⌘ AW-density Possible definition of moments M = Ik(s(x1, . . . , xm); a, b, c, d; q, q). Rains

Mλ =

det(µi+mi+mj)m

i,j=1

det(µ2mij)m

i,j=1 Lemma AW-polynomials

Koorwinder moments

Mλ =

det(µi+mi+mj)m

i,j=1

det(µ2mij)m

i,j=1 Path interpretation det(µ2mij)m

i,j=1

Lindstr¨

• m, Gessel, Viennot

Qm

i=1 mi i

Koorwinder moments

Mλ =

det(µi+mi+mj)m

i,j=1

det(µ2mij)m

i,j=1 Path interpretation det(µ2mij)m

i,j=1

Lindstr¨

• m, Gessel, Viennot

Qm

i=1 mi i

det(µi+mi+mj) j + 2(j 1)

Koorwinder moments

Mλ =

det(µi+mi+mj)m

i,j=1

det(µ2mij)m

i,j=1 Path interpretation det(µ2mij)m

i,j=1

Lindstr¨

• m, Gessel, Viennot

Qm

i=1 mi i

det(µi+mi+mj) Frozen j + 2(j 1)

Koorwinder moments

Mλ =

det(µi+mi+mj)m

i,j=1

det(µ2mij)m

i,j=1 Path interpretation det(µ2mij)m

i,j=1

Lindstr¨

• m, Gessel, Viennot

Qm

i=1 mi i

det(µi+mi+mj) Frozen j + 2(j 1) M = det(µi+ni+mj,j)

More Koornwinder moments

Kλ =

det(Zi+mi+mj)m

i,j=1

det(Z2mij)m

i,j=1

Conjecture [C., Rains, Williams 14] The Koornwinder moment K is a polynomial in α, β, γ, δ, q with positive coefficients (up to a normalizing factor).

1 . . . m 0

K = det(K(i+ji,0,0,...,0))n

i,j=1

More Koornwinder moments

Kλ =

det(Zi+mi+mj)m

i,j=1

det(Z2mij)m

i,j=1

Conjecture [C., Rains, Williams 14] The Koornwinder moment K is a polynomial in α, β, γ, δ, q with positive coefficients (up to a normalizing factor).

True for = (N r, 0, . . . , 0 | {z }

r

) 1 . . . m 0

Theorem [C., Williams 15; Cantini 15] K(Nr.0,...,0) Partition function of the two species ASEP K = det(K(i+ji,0,0,...,0))n

i,j=1

Two species ASEP

N sites r particles equal to ⇥ 1 1 q ↵ q

Two species ASEP

N sites r particles equal to ⇥ 1 q 1 q 1 q 1 1 q ↵ q

Two species ASEP

N sites r particles equal to ⇥ 1 1 q ↵ q

• Matrix Ansatz [Uchiyama 08]
• hW|(↵E D) = hW|
• (D E)|V i = |V i
• DE qED = D + E
• DA = qAD + A
• AE = qEA + A.

ZN,r = [yr]hW|(D+E+yA)N|V i

hW|Ar|V i

.

Partition function

Two species ASEP

ZN,r = [yr]hW|(D+E+yA)N|V i

hW|Ar|V i

.

K(Nr.0,...,0) = hW|(D + E)N|V ri (0, 0) (N, r) b1 b1 b0 2 1

K(Nr,0...,0) = µN,r = H

C dz 4⇡izw

⇣

z+z−1 2

⌘ Pr ⇣

z+z−1 2

⌘ ⇣

z+z−1 2

⌘N

|V ri = (0, . . . , 0, 1, 0, . . .)T

• Theorem. ZN,r = ↵r(1q)r

↵+qi

⇥ K(nr,0,...,0)

Sketch of proof

• Lemma. The theorem is true if hW|DN|V ri↵r(1 q)r = [yr] hW |(D+yA)N|V i

hW |Ar|V i

.

Sketch of proof

• Lemma. The theorem is true if hW|DN|V ri↵r(1 q)r = [yr] hW |(D+yA)N|V i

hW |Ar|V i

.

• Proof. Matrix Ansatz

Sketch of proof

• Lemma. The theorem is true if hW|DN|V ri↵r(1 q)r = [yr] hW |(D+yA)N|V i

hW |Ar|V i

.

• Proof. Matrix Ansatz
• Lemma. The theorem is true if hW|dN|V ri =

 N r

• q

hW |ArdN−r|V i hW |Ar|V i

. D = (1 + d)/(1 q)

• Proof. Matrix Ansatz

Sketch of proof

• Lemma. The theorem is true if hW|DN|V ri↵r(1 q)r = [yr] hW |(D+yA)N|V i

hW |Ar|V i

.

• Proof. Matrix Ansatz
• Lemma. The theorem is true if hW|dN|V ri =

 N r

• q

hW |ArdN−r|V i hW |Ar|V i

. D = (1 + d)/(1 q)

• Proof. Matrix Ansatz

”Guess and check” Proposition

hW|ArdN−r|V i hW|Ar|V i

=

PN−r

i=0 (1)i

2 4 N r

i

3 5

q

q(i

2)(bdqr)iBN−r−i(b,d,q)Bi(a,c,1/q)

QN−r−1

i=0

(1abcdq2r+i)

Bm(b, d, q) = Pm

j=0

 m j

• q

bjdmj !

Enumeration formula

• Theorem. [Stanton 15]

ZN,r = PN

k=0

Pk

j=0 Fk,rqk q−j2a−2j (q,q1−2j/a2;q)j(q,a2q1+2j;q)k−j (1 + aqj + 1/(aqj))N/2N

Fk,r = (a)r k r

• q

(abqr,acqr,adqr,q)k−r (abcdq2r,q)k−r (q;q)r (abcd;q)2r (ab, ac, ad, bc, bd, cd; q)rq(

r 2)

a =

1q↵++p (1q↵+)2+4↵ 2↵

, b =

1q++p (1q+)2+4 2

Enumeration formula

• Theorem. [Stanton 15]

ZN,r = PN

k=0

Pk

j=0 Fk,rqk q−j2a−2j (q,q1−2j/a2;q)j(q,a2q1+2j;q)k−j (1 + aqj + 1/(aqj))N/2N

Fk,r = (a)r k r

• q

(abqr,acqr,adqr,q)k−r (abcdq2r,q)k−r (q;q)r (abcd;q)2r (ab, ac, ad, bc, bd, cd; q)rq(

r 2)

a =

1q↵++p (1q↵+)2+4↵ 2↵

, b =

1q++p (1q+)2+4 2

• Remark. ZN,r is a polynomial with positive coefficients in ↵, , , and q

with 4Nr(n r)! n

r

2 terms

Can we extract the combinatorics of the two species ASEP?

Triangular alternative tableaux

= = 0 [Viennot, Mandelshtam 2015] q = 0 [Mandelshtam 14]

Triangular alternative tableaux

= = 0 [Viennot, Mandelshtam 2015] q = 0 [Mandelshtam 14]

Triangular alternative tableaux

= = 0 [Viennot, Mandelshtam 2015] q = 0 [Mandelshtam 14]

Triangular alternative tableaux

= = 0 [Viennot, Mandelshtam 2015] q = 0 [Mandelshtam 14]

Triangular alternative tableaux

= = 0 [Viennot, Mandelshtam 2015] ↵ ↵

• q = 0 [Mandelshtam 14]

Triangular alternative tableaux

= = 0 [Viennot, Mandelshtam 2015] ↵ ↵

• q = 0 [Mandelshtam 14]

Triangular alternative tableaux

= = 0 [Viennot, Mandelshtam 2015] ↵ ↵

• q

q q q q q q qq q = 0 [Mandelshtam 14]

Triangular alternative tableaux

= = 0 [Viennot, Mandelshtam 2015] ↵ ↵

• q

q q q q q q qq Weight= ↵j` Q entries ` j q = 0 [Mandelshtam 14]

Triangular alternative tableaux

= = 0 [Viennot, Mandelshtam 2015] ↵ ↵

• q

q q q q q q qq Weight= ↵j` Q entries Fix a tiling t Z(⌧, t) = P

T weight(T)

` j q = 0 [Mandelshtam 14]

Triangular alternative tableaux

= = 0 [Viennot, Mandelshtam 2015] ↵ ↵

• q

q q q q q q qq Weight= ↵j` Q entries Fix a tiling t Z(⌧, t) = P

T weight(T)

P(⌧) = Z(⌧, t)/ P

⌧ Z(⌧, t)

` j q = 0 [Mandelshtam 14]

Triangular alternative tableaux

= = 0 [Viennot, Mandelshtam 2015] ↵ ↵

• q

q q q q q q qq Weight= ↵j` Q entries Fix a tiling t Z(⌧, t) = P

T weight(T)

P(⌧) = Z(⌧, t)/ P

⌧ Z(⌧, t)

` j For t and t0 tilings, Z(⌧, t) = Z(⌧, t0) q = 0 [Mandelshtam 14]

Triangular alternative tableaux

= = 0 [Viennot, Mandelshtam 2015] ↵ ↵

• q

q q q q q q qq Weight= ↵j` Q entries Fix a tiling t Z(⌧, t) = P

T weight(T)

P(⌧) = Z(⌧, t)/ P

⌧ Z(⌧, t)

` j q = 0 [Mandelshtam 14]

N

r

(N+1)!

(r+1)! tableaux

Triangular staircase tableaux [C., Mandelshtam, Williams 15]

Triangular staircase tableaux [C., Mandelshtam, Williams 15]

Triangular staircase tableaux [C., Mandelshtam, Williams 15]

x ↵

• r

x = uq, ↵u or q q2

• u2

Triangular staircase tableaux [C., Mandelshtam, Williams 15]

x ↵

• r

x = uq, ↵u or q q2

• u2

↵ or x x = uq, u or q u2 q2

Triangular staircase tableaux [C., Mandelshtam, Williams 15]

x ↵

• r

x = uq, ↵u or q q2

• ↵ or

x x = q, ↵, , or

• ↵ or

x x = 1, ↵, , or

• q
• u

u2 ↵ or x x = uq, u or q u2 q2 Staircase tableaux [C., Williams 09]

Triangular staircase tableaux [C., Mandelshtam, Williams 15]

x ↵

• r

x = uq, ↵u or q q2

• ↵ or

x x = q, ↵, , or

• ↵ or

x x = 1, ↵, , or

• q
• u

u2 ↵ or x x = uq, u or q u2 q2 Staircase tableaux [C., Williams 09]

• ↵

u2

• uq

q2 q u q2 uq u

• q

q u ↵u u2

Type

• ↵

u2

• uq

q2 q u q2 uq u

• q

q u ↵u u2

Zn(α, β, γ, δ, q, u) = P

⌧ Z⌧

Z⌧ = P

T type ⌧ W(T)

P(τ) = Z⌧/Zn Zn(α, β, γ, δ, 1, 1) = n

r

Qn1

i=r ((α + γ)(β + δ)i + α + β + γ + δ)

Bijective proof?

4nr(n r)! n

r

2 tableaux

More to do?

Links with Affine Hecke algebras? How to prove the general conjecture?

• Conj. K is a polynomial in ↵, , , , q with non-negative coefficients

More to do?

Thanks! Thanks! Thanks! Links with Affine Hecke algebras? How to prove the general conjecture?

• Conj. K is a polynomial in ↵, , , , q with non-negative coefficients