Course  X.V.  IMSc 2017


Ch7   Heaps  in  statistical  mechanics

    For each item, the first number refers to the page of the slides, clicking on «video» takes you directly to the corresponding slide in the video. (to be completed)


                     2  March  2017

                                        slides first part          (pdf   23 Mo )       video

                                        slides second part     (pdf  18 Mo)

first part of slides

a few words about statistical mechanics  3

example 1:  the Ising model   6

example 2: gas model  13

statistical mechanics   22

example 3:  percolation  26

        polyomino  29

        animal  31

the directed animal problem  34

        directed animal: definition  36

        critical exponents for the length and the width  40

        directed animals on a cylinder: the formula of Derrida, Nadal, Vannimenus  44

directed animals and heaps of dimers  47

        bijection directed animal (square lattice) and strict pyramids of dimers  49

        algebraic equations for directed animals  50, 51

Motzkin paths  52

        algebraic equations for prefix of Motzkin paths  55, 56

        random directed animals  61, 62

complements: compact source size directed animals  63

        algebraic system of equations for compact source size directed animals  69

        the formula  3^n  70

end of the first part of slides of Ch7a  73

second part of slides

directed animals on a triangular lattice  2

        bijection  directed animal (triangular lattice) and pyramids of dimers  4

directed animals on a bounded strip  6

        generating function for directed animal on a bounded circular strip

                and proof a Derrida, Nadal, Vannimenus formula   10

combinatorial understanding of the thermodynamic limit with 1D gas model  11

        density of the gas: definition  17

        combinatorial interpretation of the density  18

        research problem: combinatorial interpretation of the partition function  Z(t)   21

the hard hexagons gas model  23

        interpretation of the density of the gas with pyramids of hexagons  29

combinatorial understanding of the thermodynamic limit  34

proof of the interpretation with pyramids of hexagons  37

        a proposition related to the limit of the domain D   40

Baxter’s solution of the hard hexagons model  41

research problem about the hard hexagons partition function Z(t)  52

end of the second part of slides of Ch7a  57


                     13  March  2017

                                         slides first part          (pdf  22 Mo)      video    

                                         slides second part     (pdf    21 Mo)

first part of slides

from the previous lecture  3

algebricity of the density for hard hexagons  15

        research problem (5+++)    16, 17

Lorentzian triangulations in 2D quantum gravity  20

a brief introduction to quantum gravity  21

        classical ...  22

        general relativity  24

        the quantum world  27

        string theory  32

        Alain Connes non-commutative geometry  34

        loop quantum gravity  35

        causal sets  38

        causal dynamical triangulations  39

end of the first part of slides of Ch7b  47

second part of slides

2D Lorentzian triangulations  3

        path integral amplitude for the propagation of the geometry   11

                (the four parameters for the enumeration of 2D Lorentzian triangulations)

        Lorentzian triangulations on a cylinder  12

        border conditions for Lorentzian triangulations  14, 15

        bijection heaps of dimers Lorentzian triangulations  16

the four parameters generating function for Lorentzian triangulations with border conditions  26

        bijection double semi-pyramids Lorentzian triangulations with left-right border conditions  36

        exercise: bijection double semi-pyramids --  (general) heaps of dimers  37

the curvature parameter of the 2D space-time  41

        interpretation of the up and down curvature on the heaps of dimers  46

        an example with the stairs decomposition  47-53

        characterization of heaps of dimers with zero up-curvature and zero total curvature  56-60

the nordic decomposition of a heap of dimers  61

        connected heap of dimers  64

        multi-directed animal  (Bousquet-Mélou, Rechnitzer)  65-67

        Bousquet-Mélou--Rechnitzer formula for connected heaps of dimers  70

        bijective proof of this formula with the nordic decomposition of a connected heap  72-75

end of the bijective proof: Fibonacci polynomials and Catalan generating function  54

                (solution of exercise  Ch2b, p103)

        application of the nordic decomposition for partially directed animals  (Bacher)  91

extensions: Lorentzian quantum gravity in (1+1)+1 dimension  92

end of the second part of slides of Ch7b  99

Ch7c    q-Bessel functions in physics

                     16  March  2017

                                       slides first part           (pdf  23  Mo)          video

                                       slides second part      (pdf  15 Mo)                                          

Epilogue                      slides                            (pdf  8 Mo)             


first part of slides

Bessel functions and q-Bessel functions  2

from the previous lecture  6

parallelogram polyominoes (staircase polygons) and q-Bessel functions  12

        the 3 parameters generating function  16

bijection  parallelogram polyominoes  -- semi-pyramids of segments  17

proof of the 3 parameters generating function for parallelogram polyominoes  31

from integers partitions to q-Bessel functions   35

        q-Bessel functions as trivial heaps of segments  40

random parallelogram polyominoes  41

the Catalan garden  44

A festival of bijections  47

other description of the bijection  (parallelogram polyominoes  -- semi-pyramids of segments)

    with the stairs decomposition of a heap of dimers  49

        bijection  parallelogram polyominoes -- Dyck paths

        bijection Dyck paths -- semi-pyramids of dimers  56

                video with violin  57

        bijection semi-pyramids of dimers -- semi-pyramids of segments  65

other description of the bijection  (parallelogram polyominoes  -- semi-pyramids of segments)

    with Lukasiewicz paths  69

        bijection Dyck paths -- (reverse) Lukasiewicz paths   78

        bijection  (reverse) Lukasiewicz paths -- semi-pyramids of segments  83 

other description of the bijection  (parallelogram polyominoes  -- semi-pyramids of segments)

    with the bijection Psi  (paths -- heaps of oriented loops + trail)  87

            bijection Dyck paths -- heaps of  oriented loops  89

end of the first part of slides of Ch7c  106

second part of slides

Complements: q-Bessel functions and SOS (Solid-on-Solid) model  3

        definition of the SOS path 5

        weight of the SOS path  6, 7

        the 3 parameters generating function for SOS paths  8

        from SOS paths to heaps of segments  11-14

        an involution for  the term  x(1-y^2)

        partially directed paths with interactions  16

particular case: weighted heaps of dimers and Ramanujan continued fraction  17

        area: q-Catalan  19

Rogers-Ramanujan identities  23

        D-partitions  26

        from partitions to D-partitions  27-30

        generating function for weighted semi-pyramids of dimers  33

        interpretations of the Rogers-Ramanujan identities with intergers partitions   35, 36

Ramanujan continued fraction  37

        interpretation of Ramanujan continued fraction as weighted semi-pyramids of dimers  38

        decomposition of a semi-pyramids as sequence of primitive semi-pyramids  40-53

        end of the proof  56

        Ramanujan continued fraction as the ratio N/D  58

        back to the system of q-equations for the partition function Z(t) of the hard gas model  59

Andrews ‘s interpretation of the reciprocal of Ramanujan identities  60

other future chapters  66

end of the second part of slides of Ch7c  70

Epilogue: Kepler towers 

Kepler towers  2

        definition: system of Kepler towers  4

        proposition: enumeration of systems of Kepler towers with Catalan numbers  11

        Kepler disks  13

        why Kepler towers ?  15

        Kepler mysterium cosmographicum  19

        Kepler towers and Strahler number of a binary tree  23

        logarithmic height of a Dyck path  26

        Programs to read by D. Knuth  29

Many Thanks 30

the end of  the Epilogue and the end of the course !  31


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