Three deformations of generalized Leibniz triangles

Gabriele Sicuroin collaboration with P. Tempesta (UCM), A. Rodríguez (UPM) and C. Tsallis (CBPF)

Centro Brasileiro de Pesquisas FísicasRio de Janeiro, Brazil

19 October 2015

An elementary model

Consider a set of N identical binary random variables {xi}i=1,...,N, xi ∈ {0, 1}

N = 30

Let us call rN,n the probability of having∑

i xi = n in a certain configuration. Due toindistinguishability, the probability of having

∑i xi = n is

pN,n =

(Nn

)rN,n.

Pascal triangleOr the Meru–prastaara, the Staircase of Mount Meru

It is well known that the binomial coefficients can be arranged in a triangle:

1

1 1

1(2

1

)1

1(3

1

) (32

)1

1(4

1

) (42

) (43

)1

1(5

1

) (52

) (53

) (54

)1

1(6

1

) (62

) (63

) (64

) (65

)1

Pingala, II century bC, India

Varahamirhira, 505 aD, India

Mahavira, 850 aD, India

Halayudha, 975 aD, India

Bhat.t.otpala, 1086 aD, India

Al-Karaji, XI century aD, Iran

Omar Khayyám, XI–XII century aD, Iran

Jia Xian, XI century aD, China

Yang Hui, XIII century aD, China

Levi ben Gershon, XIV century aD, France

Peter Apian, 1527 aD, Germany

Michael Stifel, 1544 aD, Germany

Niccolò Tartaglia, 1556 aD, Italy

Gerolamo Cardano, 1570 aD, Italy

Blaise Pascal, 1665 aD, France

Independent variables

For independent variables, rN,n = ρn(1− ρ)N−n for ρ ∈ (0, 1) and therefore pN,n is thebinomial distribution.

1

r1,0 r1,1

r2,0 r2,1 r2,2

r3,0 r3,1 r3,2 r3,3

r4,0 r4,1 r4,2 r4,3 r4,4

r5,0 r5,1 r5,2 r5,3 r5,4 r5,5

=

1

ρ1(1−ρ)0 ρ0(1−ρ)1

ρ2(1−ρ)0 ρ1(1−ρ)1 ρ0(1−ρ)2

ρ3(1−ρ)0 ρ2(1−ρ)1 ρ1(1−ρ)2 ρ0(1−ρ)3

ρ4(1−ρ)0 ρ3(1−ρ)1 ρ2(1−ρ)2 ρ1(1−ρ)3 ρ0(1−ρ)4

ρ5(1−ρ)0 ρ4(1−ρ)1 ρ3(1−ρ)2 ρ2(1−ρ)3 ρ1(1−ρ)4 ρ0(1−ρ)5

pN,n =N�1∼ 1√

2πNρ(1− ρ)e−

N2ρ(1−ρ) ( n

N−ρ)2

.

Introducing correlations: Leibniz triangle

Leibniz considered the set of triangles satisfying the following scale–invarianceproperty:

rN,n+1 + rN,n = rN−1,n

Moreover, this constraint uniquely determines the entries of the triangle if thevalues rN,0 are given.

1

r1,0 r1,1

r2,0 r2,1 r2,2

r3,0 r3,1 r3,2 r3,3

r4,0 r4,1 r4,2 r4,3 r4,4

r5,0 r5,1 r5,2 r5,3 r5,4 r5,5

Leibniz choser(1)

N,0 =1

N + 1=⇒ r(1)

N,n =1

N + 11(Nn

) .Obviously, the limiting distribution is the uniform distribution.

A model for q–GaussiansRodríguez, Schwämmle, Tsallis – J. Stat. Mech., 2008(09), P09006 (2008).

1 N = 0r(1)N,n

r(1)1,0 r(1)

1,1 N = 1

r(1)2,0 r(1)

2,1 r(1)2,2 N = 2

r(1)3,0 r(1)

3,1 r(1)3,2 r(1)

3,3 N = 3

r(1)4,0 r(1)

4,1 r(1)4,2 r(1)

4,3 r(1)4,4 N = 4

r(1)5,0 r(1)

5,1 r(1)5,2 r(1)

5,3 r(1)5,4 r(1)

5,5 N = 5

r(1)6,0 r(1)

6,1 r(1)6,2 r(1)

6,3 r(1)6,4 r(1)

6,5 r(1)6,6 N = 6

r(1)7,0 r(1)

7,1 r(1)7,2 r(1)

7,3 r(1)7,4 r(1)

7,5 r(1)7,6 r(1)

7,7 N = 7

r(1)8,0 r(1)

8,1 r(1)8,2 r(1)

8,3 r(1)8,4 r(1)

8,5 r(1)8,6 r(1)

8,7 r(1)8,8 N = 8

r(1)9,0 r(1)

9,1 r(1)9,2 r(1)

9,3 r(1)9,4 r(1)

9,5 r(1)9,6 r(1)

9,7 r(1)9,8 r(1)

9,9 N = 9

A model for q–GaussiansRodríguez, Schwämmle, Tsallis – J. Stat. Mech., 2008(09), P09006 (2008).

1

r(1)1,0 r(1)

1,1

r(1)2,0 r(1)

2,1 r(1)2,2

r(1)3,0 r(1)

3,1 r(1)3,2 r(1)

3,3

r(1)4,0 r(1)

4,1 r(1)4,2 r(1)

4,3 r(1)4,4

r(1)5,0 r(1)

5,1 r(1)5,2 r(1)

5,3 r(1)5,4 r(1)

5,5

r(1)6,0 r(1)

6,1 r(1)6,2 r(1)

6,3 r(1)6,4 r(1)

6,5 r(1)6,6

r(1)7,0 r(1)

7,1 r(1)7,2 r(1)

7,3 r(1)7,4 r(1)

7,5 r(1)7,6 r(1)

7,7

r(1)8,0 r(1)

8,1 r(1)8,2 r(1)

8,3 r(1)8,4 r(1)

8,5 r(1)8,6 r(1)

8,7 r(1)8,8

r(1)9,0 r(1)

9,1 r(1)9,2 r(1)

9,3 r(1)9,4 r(1)

9,5 r(1)9,6 r(1)

9,7 r(1)9,8 r(1)

9,9

r(3)N,n =

r(1)N+4,n+2

r(1)4,2

and in general

r(ν)N,n =

r(1)N+2(ν−1),n+ν−1

r(1)2(ν−1),ν−1

scale–invariant

p(ν)N,n =

(Nn

)r(ν)

N,n

q–Gaussian Pq1(ν)(x)

q1(ν) = 1 − 1ν − 1

x = 2√1−q1(ν)

( nN −

12

)

The q–Gaussian distribution

The q–Gaussian distribution is defined as follows:

Pq(x) :=

3−q2

√1−qπ

Γ(

3−q2−2q

)Γ(

11−q

) [1− (1− q)x2] 11−q+ for q < 1,

e−x2√π

for q = 1,√q−1π

Γ(

1q−1

)Γ(

3−q2q−2

) [1− (1− q)x2] 11−q for 1 < q < 3.

In the previous definition

[x]+ :=

{x x > 0,0 x ≤ 0.

−6 −4 −2 0 2 4 6

0

0.2

0.4

0.6

x

P q

q = 12

q = 1q = 2

The α–numbers and the β–numbersSicuro, Tempesta, Rodríguez, Tsallis — Annals of Physics, in press (ArXiv 1506.02136)

Let us now deform the generalized triangles above introducing two deformation ofthe natural numbers.

The α–numbersGiven n ∈ N ∪ 0, and α > 0, α 6= 1,

{n}α := (n + 1)(

1− 1− α1− αn+1

).

We define the α–binomial as{Nn

}α

:=

∏Nk=1{k}α∏N−n

k=1 {k}α∏n

k=1{k}α.

limα→1{n}α = n.

The β–numbersGiven n ∈ N ∪ 0, and β > 0, β 6= 1,

[n]β := n(

1− 1− β1− βn

)+ 1.

We define the β–binomial as[Nn

]β

:=

∏Nk=1[k]β∏N−n

k=1 [k]β∏n

k=1[k]β.

limβ→1

[n]β = n.

limn→∞

[n]α{n}α

= 1.

The α–triangles and the β–trianglesSicuro, Tempesta, Rodríguez, Tsallis — Annals of Physics, in press (ArXiv 1506.02136)

The Leibniz α–triangles

r(1)N,n,α :=

1{N + 1}α { N

n }αFor ν ∈ N:

r(ν)N,n,α :=

r(1)N+2(ν−1),n+ν−1,α∑N

k=0

(Nk

)r(1)

N+2(ν−1),k+ν−1,α︸ ︷︷ ︸p(ν)N,k,α

The Leibniz β–triangles

r(1)N,n,β :=

1[N + 1]β [ N

n ]β

For ν ∈ N:

r(ν)N,n,β :=

r(1)N+2(ν−1),n+ν−1,β∑N

k=0

(Nk

)r(1)

N+2(ν−1),k+ν−1,β︸ ︷︷ ︸p(ν)N,k,β

The asymptotic scale invarianceSicuro, Tempesta, Rodríguez, Tsallis — Annals of Physics, in press (ArXiv 1506.02136)

Both the deformed triangles satisfy the scale invariance condition asymptotically!

limN→∞

nN≡η

r(ν)N−1,n,•

r(ν)N,n+1,• + r(ν)

N,n,•︸ ︷︷ ︸t(ν)N,η,•

= 1

0 0.2 0.4 0.6 0.8 11

1.01

η

t(2)

50,η,α

=1 2

0 0.2 0.4 0.6 0.8 11

1.01

η

t(2)

50,η,β

=1 2

The main theoremsSicuro, Tempesta, Rodríguez, Tsallis — Annals of Physics, in press (ArXiv 1506.02136)

Robustness of α–trianglesWe have that

N2√ν − δα,1

p(ν)N,n,α

N�1∼ Pqα(ν)(x), x := 2√ν − δα,1

( nN− 1

2

),

where

qα(ν) :=

{1− 1

νfor α 6= 1,

1− 1ν−1 for α = 1.

0 0.5 1 1.5 20

0.5

1

α

q α(2

)

0 0.2 0.4 0.6 0.8 10

1

2×10−4

nN

p(2)

N,n,α

α = 13

α = 23

α = 1

α = 2

α = 3

The main theoremsSicuro, Tempesta, Rodríguez, Tsallis — Annals of Physics, in press (ArXiv 1506.02136)

Robustness of β–trianglesWe have that

N2√ν − χ(β)

p(ν)N,n,β

N�1∼ Pqβ(ν)(x), x :=2

√ν−δβ,1 +

1−min{β, 1}min{β, 1}

( nN− 1

2

),

where

qβ(ν) =

1− 1

νfor β > 1,

1− 1ν−1 for β = 1,

1− ββν+1−β for 0 < β < 1.

0 0.5 1 1.5 20

0.5

1

β

q β(2

)

0 0.2 0.4 0.6 0.8 10

1

2

×10−4

nN

p(2)

N,n,β

β = 13

β = 23

β = 1

β = 2

β = 3

A recurrent algebraSicuro, Tempesta, Rodríguez, Tsallis — Annals of Physics, in press (ArXiv 1506.02136)

α–triangles

11− qα(ν)

=1

1− q1(ν)+ 1, α 6= 1

β–triangles

min{β, 1}1− qβ(ν)

=min{β, 1}1− q1(ν)

+ 1, β 6= 1

β–triangles

∆(ν)N (β) :=

√N∑N

k=0

∣∣∣p(ν)N,k,β − p(ν)

N,k

∣∣∣2

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

Nonuniform convergence

β

∆(2

)N

(β)

N = 50N = 400N →∞

A non asymptotically scale–invariant deformationSicuro, Tempesta, Rodríguez, Tsallis — Annals of Physics, in press (ArXiv 1506.02136)

Inspired by the Q–calculus, we considered an alternative deformation of the Leibniztriangle based on the so called Q–numbers:

JnKQ :=1− Qn

1− Q, Q ∈ (0,∞) \ {1}, lim

Q→1JnKQ = n.

We can introduce also the Gauss binomial coefficients

sNn

{

Q

:=

∏Nk=1JkKQ∏N−n

k=1 JkKQ∏n

k=1JkKQ

Q→1−−−→

(Nn

).

Q–triangles

r(1)N,n,Q :=

1JN + 1KQ

1J N

n KQ=⇒ r(ν)

N,n,Q :=r(1)

N+2(ν−1),n+ν−1,Q∑Nk=0

(Nk

)r(1)

N+2(ν−1),k+ν−1,Q︸ ︷︷ ︸p(ν)N,k,Q

A non asymptotically scale–invariant deformationSicuro, Tempesta, Rodríguez, Tsallis — Annals of Physics, in press (ArXiv 1506.02136)

The Q–triangles are not asymptotically scale invariant! Indeed

limN→∞

nN≡η fixed

r(ν)N−1,n,Q

r(ν)N,n,Q + r(ν)

N,n+1,Q

=

{12 for Q < 1,0 for Q > 1.

Robustness of Q–trianglesWe have that

p(ν)N,n,Q

N�1∼

{√2πN e−2N(η− 1

2 )2if 0 < Q < 1,

δn,0+δn,N2 if Q > 1.

−5 0 510−24

10−11

102

1√N

(n− N

2

)

√N

p(2)

N,n,Q

Q = 13

Q = 23

0 0.5 10

0.5

nN

p(2)

N,n,

3 2

N = 5N = 10N = 30

Final remarksSicuro, Tempesta, Rodríguez, Tsallis — Annals of Physics, in press (ArXiv 1506.02136)

We deformed the generalized Leibniz triangle to evaluate the robustness of thelimiting distributions in three different ways. We obtained the following results

The asymptotically scale invariant deformed triangles have still q–Gaussians aslimiting distribution, but the limiting value of q depends in general on theform of the perturbation, even if two asymptotically equivalent perturbations areconsidered.

Switching from exact scale invariance to asymptotically scale invariance canmake a discontinuity appear in the limiting value of q as function of theperturbation parameter.

The not-asymptotically scale invariant deformed triangle can have a limitingdistribution outside the family of q–Gaussians.

These results suggest that the (asymptotic) scale–invariance property plays a centralrole in the robustness of the set of q–Gaussian distributions as limitingdistributions. More specifically, the set of q–Gaussians appears to be robust underasymptotically scale–invariant deformations.

Thank youfor your attention!