1. Random remarks about random walks
D. Volchenkov
4 Sketches for discussion @
MPI for Mathematics in the Sciences Leipzig,

2. 1. Identifying important scales by localization of random walks

3. Disorder of steps vs. disorder of paths
Disorder of (infinite)
paths:

4. Disorder of steps vs. disorder of paths
Scale of RW,
as # transitions increases
Disorder of steps:
As centrality falls next to boundaries & obstacles
Disorder of (infinite)
paths:

5. Disorder of steps vs. disorder of paths

6. Random walks of different scales
Maximal complexity RW
Maximal entropy RW
Maximal paths disorder
Maximal steps disorder
Minimal steps disorder
Minimal paths disorder

7. Random walks of different scales
Maximal complexity RW
Maximal entropy RW
Maximal paths disorder
Maximal steps disorder
Minimal steps disorder
Minimal paths disorder

8. Random walks of different scales
Maximal complexity RW
Maximal entropy RW
Maximal paths disorder
Maximal steps disorder
Minimal steps disorder
Minimal paths disorder

9. Random walks of different scales
Maximal complexity RW
Maximal entropy RW
Maximal paths disorder
Maximal steps disorder
Minimal steps disorder
Minimal paths disorder

10. Random walks of different scales
Maximal complexity RW
Maximal entropy RW
Maximal paths disorder
Maximal steps disorder
Minimal steps disorder
Minimal paths disorder

11. Random walks of different scales
Maximal complexity RW
Maximal entropy RW
Maximal paths disorder
Maximal steps disorder
Minimal steps disorder
Minimal paths disorder

12. Random walks of different scales
Maximal complexity RW
Maximal entropy RW
Maximal paths disorder
Maximal steps disorder
Minimal steps disorder
Minimal paths disorder

13. Random walks of different scales
Maximal complexity RW
Maximal entropy RW
Maximal paths disorder
Maximal steps disorder
Minimal steps disorder
Minimal paths disorder

14. Random walks of different scales
Maximal complexity RW
Maximal entropy RW
Maximal paths disorder
Maximal steps disorder
Minimal steps disorder
Minimal paths disorder

15. Random walks of different scales
Maximal complexity RW
Maximal entropy RW
Maximal paths disorder
Maximal steps disorder
Minimal steps disorder
Minimal paths disorder

16. Structure reinforces order
A.)
B.)
C.)
Maximal complexity RWs: Localization within small scale structures
Disorder of steps:
The complexity-entropy diagram shows how information is stored, organized, and transformed across different scales of the structure.

17. Structure reinforces order
A.)
B.)
C.)
Maximal complexity RWs: Localization within small scale structures
Disorder of steps:
The complexity-entropy diagram shows how information is stored, organized, and transformed across different scales of the structure.

18. 2. Behaviour of anticipatory systems and random walks

19. acetylcholine (triggered by known unreliability of predictive cues)
Expected uncertainty
acetylcholine (triggered by known unreliability of predictive cues)
Unexpected uncertainty
norepinephrine (triggered by strongly unexpected observations)

20. acetylcholine (triggered by known unreliability of predictive cues)
Lévy flights:
Extremely long recurrence times;
Short first passage times;
Expected uncertainty
acetylcholine (triggered by known unreliability of predictive cues)
Unexpected uncertainty
norepinephrine (triggered by strongly unexpected observations)
Brownian walks:
Short recurrence times;
Long first passage times;

21. acetylcholine (triggered by known unreliability of predictive cues)
ballistic relocations
Lévy flights:
Extremely long recurrence times;
Short first passage times;
Expected uncertainty
acetylcholine (triggered by known unreliability of predictive cues)
Unexpected uncertainty
norepinephrine (triggered by strongly unexpected observations)
Brownian walks:
Short recurrence times;
Long first passage times;
a meticulous search

22. Visual intelligibility of environments
ballistic relocations
Lévy flights:
Extremely long recurrence times;
Short first passage times;
Visual intelligibility
of environments
Expected uncertainty
acetylcholine (triggered by known unreliability of predictive cues)
Unexpected uncertainty
norepinephrine (triggered by strongly unexpected observations)
Confidence in the eventual success
Brownian walks:
Short recurrence times;
Long first passage times;
a meticulous search

23. Structural intelligibility
Recurrence time
First-passage time
Tonality: the hierarchy of harmonic intervals
The basic pitches for the E minor scale are "E", "F#", "G", "A", "B".
The recurrence time vs. the first passage time over 804 compositions of 29 Western composers.
Tonality of music

24. Structural intelligibility
Recurrence time
First-passage time
Tonality: the hierarchy of harmonic intervals
The basic pitches for the E minor scale are "E", "F#", "G", "A", "B".
The recurrence time vs. the first passage time over 804 compositions of 29 Western composers.
Tonality of music

25. 3. The Schläfli formula for deforming polyhedra and network stability

26. Path integral distance for graphs & databases
Path integral is an analytic continuation of RW summation.
Path integral: a single classical trajectory is replaced with a sum over an infinity of possible trajectories to compute a propagator;
Propagator is the Green’s function of the diffusion operator (the Schrödinger equation is a diffusion equation with an imaginary diffusion constant);
Removal of ambiguities
The Laplace operator diverges, the Green function is not unique:
The Drazin generalized inverse (the group inverse w.r.t. matrix multiplication) preserves symmetries of the Laplace operator:
From path integral to the Riemannian geometry
Given two distributions x,y, their scalar product:
The (squared) norm of a distribution:
The Euclidean distance between two distributions :
Feynman path integral: Removal of point-loops ambiguities trough finite part renormalization
Transition to self-avoiding random walks (“no loops”).

27. Path integral distance for graphs & databases
Path integral is an analytic continuation of RW summation.
Path integral: a single classical trajectory is replaced with a sum over an infinity of possible trajectories to compute a propagator;
Propagator is the Green’s function of the diffusion operator (the Schrödinger equation is a diffusion equation with an imaginary diffusion constant);
Removal of ambiguities
The Laplace operator diverges, the Green function is not unique:
The Drazin generalized inverse (the group inverse w.r.t. matrix multiplication) preserves symmetries of the Laplace operator:
From path integral to the Riemannian geometry
Given two distributions x,y, their scalar product:
The (squared) norm of a distribution:
The Euclidean distance between two distributions :
Feynman path integral: Removal of point-loops ambiguities trough finite part renormalization
Transition to self-avoiding random walks (“no loops”).

28. Path integral distance for graphs & databases
Path integral is an analytic continuation of RW summation.
Path integral: a single classical trajectory is replaced with a sum over an infinity of possible trajectories to compute a propagator;
Propagator is the Green’s function of the diffusion operator (the Schrödinger equation is a diffusion equation with an imaginary diffusion constant);
Removal of ambiguities
The Laplace operator diverges, the Green function is not unique:
The Drazin generalized inverse (the group inverse w.r.t. matrix multiplication) preserves symmetries of the Laplace operator:
From path integral to the Riemannian geometry
Given two distributions x,y, their scalar product:
The (squared) norm of a distribution:
The Euclidean distance between two distributions :
Feynman path integral: Removal of point-loops ambiguities trough finite part renormalization
Transition to self-avoiding random walks (“no loops”).

29. A finite connected undirected weighted connected graph is a N- simplex, in which:
Squared norms of vectors are first-passage times;
Squared distances between vectors are commute times;

30. Fi, the face of the co-dimension 1 opposing i
Fij, the face of the co-dimension 2 opposing (i,j)
Fj, the face of the co-dimension 1 opposing j

31. The angle Gram matrix of the simplex

32. The angle Gram matrix of the simplex
Hyperbolic simplex
Spherical simplex
Euclidean simplex

33. The angle Gram matrix of the simplex
Hyperbolic simplex
Spherical simplex
Euclidean simplex

34. The angle Gram matrix of the simplex
The Schläfli formula
for polyhedra:
(continuity of volume)

35. The angle Gram matrix of the simplex
The Schläfli formula
for polyhedra:
(continuity of volume)
The integral of motion:

36. Another way of formalizing that
Another way of writing
the Schläfli formula:
“Stress tensor”
The eigenvalues are the “principal stresses”;
The eigenvectors are the principal directions;
The principal invariants are the deviatoric stress invariants;

37. 4. First attaining times manifolds.
The Morse theory

38. 3. First attaining times manifold
The first-passage time can be calculated as the mean of all first hitting times
with respect to the stationary distribution of random walks
For any given starting distribution that differs from the stationary one, we can
calculate the analogous quantity,
We call it the first attaining time to the node j by the random walks starting at the distribution ϕ1.

39. First attaining times manifold
ek are the direction cosines
A manifold locally homeomorphic to Euclidean space

40. First attaining times manifold. The Morse eory
Each node j is a critical point of the manifold of first attaining times, and the first passage times fj are the correspondent critical values.

41. First attaining times manifold
Following the ideas of the Morse theory, we can perform the standard classification of the critical points, introducing the index g j of the critical point j as the number of negative eigenvalues of at j.
The index of a critical point is the dimension of the largest subspace of the tangent space to the manifold at j on which the Hessian is negative definite).

42. First attaining times manifold. The Morse theory
The Euler characteristic c is an intrinsic property of a manifold that describes its topological space’s shape regardless of the way it is bent.
It is known that the Euler characteristic can be calculated as the alternating sum of Cg , the numbers of critical points of index c of the Hessian function,

43. First attaining times manifold. The Morse theory
Amsterdam (57 canals)
Venice (96 canals)
The negative Euler characteristics could either come from a pattern of symmetry in the hyperbolic surfaces, or from a manifold homeomorphic multiple tori.
The large positive value of the Euler characteristic can arise due to the well-known product property of Euler characteristics
for any product space M ×N, or, more generally, from a fibration, when one topological space (called a fiber) is being ”parameterized” by another topological space (called a base).