2. A scale invariant probabilistic model based on Leibniz-like pyramids Antonio Rodríguez 1,2 1 Dpto. Matemática Aplicada y Estadística. Universidad Politécnica de Madrid Universidad Politécnica de Madrid 2 Grupo Interdisciplinar de Sistemas Complejos

3. Outline One-dimensional model. One-dimensional model. Scale invariant triangles. Scale invariant triangles. q-entropy. q-entropy. Two-dimensional model. Two-dimensional model. Scale invariant tetrahedrons. Scale invariant tetrahedrons. Conditional and marginal distributions. Conditional and marginal distributions. Generalization to arbitrary dimension. Generalization to arbitrary dimension. Conclusions. Conclusions.

4. scale invariance extensivity q-gaussianity

5. A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008) joint probability distribution N-1 variables probability distribution variables marginal N-1 joint N scale invariance

6. Scale invariance A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008) joint probability distribution N-1 variables probability distribution variables marginal N-1 joint N

7. One-dimensional model. x1x1x1x1p 1-p 1 0 N=1 N distinguisable 1d-binary independent variables A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008) 1

8. One-dimensional model. x1x1x1x1 p2p2p2p2 p (1-p) (1-p) 2 1 0 1 0 x2x2x2x2 N=2 N distinguisable 1d-binary independent variables A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)

9. One-dimensional model. x1x1x1x1 p2p2p2p2 p (1-p) (1-p) 2 1 0 1 0 x2x2x2x2 p 1-p 1-p p 1-p N=2 N distinguisable 1d-binary independent variables A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)

10. p3p3p3p3 p 2 (1-p) p 2 (1-p) p 2 (1-p) p(1-p) 2 p(1-p) 2 N=3 One-dimensional model. p2p2p2p2 p(1-p) (1-p) 2 p 2 (1-p) p(1-p) 2 (1-p) 3 x 3 =1 x 3 =0 A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)

11. p3p3p3p3 p 2 (1-p) p 2 (1-p) p(1-p) 2 p(1-p) 2 p 2 (1-p) N=3 One-dimensional model. p2p2p2p2 p(1-p) p(1-p) (1-p) 2 p(1-p) (1-p) 3 p 2 (1-p) p(1-p) 2 1 p 1-p N=0 N=1 N=2 A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)

12. p3p3p3p3 p 2 (1-p) p 2 (1-p) p(1-p) 2 p(1-p) 2 N=3 One-dimensional model. p2 p2 p2 p2 p(1-p) p(1-p) (1-p) 2 (1-p) 2 (1-p) 3 (1-p) 3 1 p 1-p N=0 N=1 N=2 + + + Leibniz rule A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)

13. p3p3p3p3 p 2 (1-p) p 2 (1-p) p(1-p) 2 p(1-p) 2 N=3 One-dimensional model. p2 p2 p2 p2 p(1-p) p(1-p) (1-p) 2 (1-p) 2 (1-p) 3 (1-p) 3 1 p 1-p N=0 N=1 N=2 1 1 1 2 1 1 3 3 1 1 Pascal triangle CLT A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008) Binomial distribution Gaussian

14. p3p3p3p3 p 2 (1-p) p 2 (1-p) p(1-p) 2 p(1-p) 2 N=3 Scale invariant triangles p2 p2 p2 p2 p(1-p) p(1-p) (1-p) 2 (1-p) 2 (1-p) 3 (1-p) 3 1 p 1-p N=0 N=1 N=2 A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)

15. N=3 N=0 N=1 N=2 Scale invariant triangles Leibniz triangle A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)

16. N=3 N=0 N=1 N=2 Scale invariant triangles Leibniz triangle A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)

17. N=3 N=0 N=1 N=2 Scale invariant triangles Leibniz triangle A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)

18. N=3 N=0 N=1 N=2 Scale invariant triangles A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)

19. Scale invariant triangles A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)

20. Scale invariant triangles A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)

21. Scale invariant triangles A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)

22. Scale invariant triangles A. Rodríguez, C. Tsallis and V. Schammle, J. Stat. Mech: theor. exp. P09006 (2008)

23. Scale invariant triangles R. Hanel, S. Thurner and C. Tsallis. Eur. Phys. J. B 72, 263 (2009 )

24. Outline One-dimensional model. One-dimensional model. Scale invariant triangles. Scale invariant triangles. q-entropy. q-entropy. Two-dimensional model. Two-dimensional model. Scale invariant tetrahedrons. Scale invariant tetrahedrons. Conditional and marginal distributions. Conditional and marginal distributions. Generalization to arbitrary dimension. Generalization to arbitrary dimension. Conclusions Conclusions

25. scale invariance extensivity q-gaussianity ? for ?

26. q-entropy

27. scale invariance extensivity q-gaussianity ? for ?

28. scale invariance extensivity q-gaussianity ?

29. Outline One-dimensional model. One-dimensional model. Scale invariant triangles. Scale invariant triangles. q-entropy. q-entropy. Two-dimensional model. Two-dimensional model. Scale invariant tetrahedrons. Scale invariant tetrahedrons. Conditional and marginal distributions. Conditional and marginal distributions. Generalization to arbitrary dimension. Generalization to arbitrary dimension. Conclusions Conclusions

30. Two dimensional model N=1 N distinguisable independent variables 1 2d-ternary (x1, y1)(x1, y1)(x1, y1)(x1, y1)p q (1, 0) (0, 1) 1-p-q (0, 0) A. Rodríguez and C. Tsallis, J. Math. Phys 53, 023302 (2012)

31. Two dimensional model N=2 N distinguisable 2d-ternary independent variables (x1, y1)(x1, y1)(x1, y1)(x1, y1) p2p2p2p2 pq pq pq q2q2q2q2 (1, 0) (0, 1) (1, 0) (0, 1) (x2, y2)(x2, y2)(x2, y2)(x2, y2) (0, 0) p(1-p-q) q(1-p-q) (1-p-q) 2 p(1-p-q) q(1-p-q) p q 1-p-q p q 1-p-q 1-p-q A. Rodríguez and C. Tsallis, J. Math. Phys 53, 023302 (2012)

32. N=2 N=0 N=1 1-p-q p q p2p2p2p2 pq pq q2q2q2q2 p(1-p-q) (1-p-q) 2 N=3 1 q3q3q3q3 (1-p-q) 3 p3p3p3p3 p 2 q p 2 q pq 2 pq 2 p 2 (1-p-q) p(1-p-q) 2 q 2 (1-p-q) q (1-p-q) 2 pq (1-p-q) q(1-p-q)

33. N=2 p2p2p2p2 1 N=0 N=1 N=3 p 1-p-q q p(1-p-q) (1-p-q) 2 q(1-p-q) q2q2q2q2 pq pq q3q3q3q3 p 2 q p 2 q pq 2 pq 2 p 2 (1-p-q) q 2 (1-p-q) q (1-p-q) 2 pq (1-p-q) p3p3p3p3 p(1-p-q) 2 (1-p-q) 3 + + + + + + + + + + + + Generalized Leibniz rule

34. N=2 p2p2p2p2 1 N=0 N=1 q2q2q2q2 q q3q3q3q3 N=3 Pascal pyramid Trinomial distribution 1 1 1 CLT 2d-Gaussian 1-p-q 1 p(1-p-q) q(1-p-q) (1-p-q) 2 1 pq pq 2 p3p3p3p3 p 2 (1-p-q) p(1-p-q) 2 p 2 q p 2 q pq 2 pq 2 q (1-p-q) 2 q 2 (1-p-q) 3 6 3 3 pq (1-p-q) 1 3 2 p 1 2 1 3 3 1 (1-p-q) 3 1

35. N=2 p2p2p2p2 p 1 N=0 N=1 1-p-q p(1-p-q) (1-p-q) 2 q(1-p-q) q2q2q2q2 pq pq q q3q3q3q3 (1-p-q) 3 p3p3p3p3 p 2 q p 2 q pq 2 pq 2 p 2 (1-p-q) p(1-p-q) 2 q 2 (1-p-q) q (1-p-q) 2 pq (1-p-q) N=3

36. N=2 N=0 N=1 N=3 Leibniz-like pyramid

37. N=2 N=0 N=1 N=3 Leibniz-like pyramid

38. N=2 N=0 N=1 N=3 Leibniz pyramid

39. N=2 N=0 N=1 N=3

40. Scale invariant pyramids

41. 2D q-Gaussian ?

42. Outline One-dimensional model. One-dimensional model. Scale invariant triangles. Scale invariant triangles. q-entropy. q-entropy. Two-dimensional model. Two-dimensional model. Scale invariant tetrahedrons. Scale invariant tetrahedrons. Conditional and marginal distributions. Conditional and marginal distributions. Generalization to arbitrary dimension. Generalization to arbitrary dimension. Conclusions Conclusions

43. Conditional distributions Conditional distributions

45. Marginal distributions Marginal distributions N=3

46. Marginal distributions Marginal distributions The three directions yield identical nonsymmetric scale-invariant distributions. The three directions yield identical nonsymmetric scale-invariant distributions.

47. Marginal distributions Marginal distributions The three directions yield identical nonsymmetric scale-invariant distributions. The three directions yield identical nonsymmetric scale-invariant distributions.

48. Marginal distributions Marginal distributions

49. The direction yields a symmetric non scale-invariant distribution The direction yields a symmetric non scale-invariant distribution

50. Marginal distributions Marginal distributions The direction yields a symmetric The direction yields a symmetric non scale-invariant distribution non scale-invariant distribution

52. Joint distribution Joint distribution

53. scale invariance extensivity q-gaussianity ?

54. q-entropy

55. Outline One-dimensional model. One-dimensional model. Scale invariant triangles. Scale invariant triangles. q-entropy. q-entropy. Two-dimensional model. Two-dimensional model. Scale invariant tetrahedrons. Scale invariant tetrahedrons. Conditional and marginal distributions. Conditional and marginal distributions. Generalization to arbitrary dimension. Generalization to arbitrary dimension. Conclusions Conclusions

56. Scale invariant hyperpyramids N=1 N distinguisable independent variables 1 3d-cuaternary p1p1p1p1 (0, 1, 0) (0, 0, 1) (0, 0, 0) p2p2p2p2 p3p3p3p3 1-p 1 -p 2 -p 3 (x 1, y 1, y 1 ) (1, 0, 0) CLT 3D-Gaussian

57. Scale invariant hyperpyramids N distinguisable independent variables 3d-cuaternary

58. N=2 N=0 N=1 N=3 Leibniz-like hyperpyramid

59. (d + 1)-sided dice d-dimensional variable d-dimensional variable d-dimensional (d+1)-ary variable taking values d-dimensional (d+1)-ary variable taking values Leibniz hyperpyramid Leibniz hyperpyramid

60. Outline One-dimensional model. One-dimensional model. Scale invariant triangles. Scale invariant triangles. q-entropy. q-entropy. Two-dimensional model. Two-dimensional model. Scale invariant tetrahedrons. Scale invariant tetrahedrons. Conditional and marginal distributions. Conditional and marginal distributions. Generalization to arbitrary dimension. Generalization to arbitrary dimension. Conclusions Conclusions

61. Conclusions and future work We have generalized to an arbitrary dimension a one-dimensional discrete probabilistic model which, for one dimension, yields q-gaussians in the thermodynamic limit. We have generalized to an arbitrary dimension a one-dimensional discrete probabilistic model which, for one dimension, yields q-gaussians in the thermodynamic limit. Our two-dimensional model, though containing one-dimensional conditional distributions yielding q-gaussians doesn’t seem to yield bidimensional q-gaussians as limiting probability distributions for. Our two-dimensional model, though containing one-dimensional conditional distributions yielding q-gaussians doesn’t seem to yield bidimensional q-gaussians as limiting probability distributions for. The case of binary variables is special !!! The case of binary variables is special !!! The formulation of a probabilistic model yielding multidimensional q-gaussians in the thermmodynamic limit is still an open question. The formulation of a probabilistic model yielding multidimensional q-gaussians in the thermmodynamic limit is still an open question. The relationship between scale invariance, q-gaussianity and extensivity is still an open question. The relationship between scale invariance, q-gaussianity and extensivity is still an open question.

63. Thanks!