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Walks

Universality

Universality is unconditionality is how God thinks. I am interested not in how the various mathematicians think but how God thinks.

Universality. The tendency of the eigenvalues of random matrices to space themselves out uniformly. Similarly, in nature, physical phenomena space themselves out in different orders of magnitude. Similarly, we have orders of scale in Alexander's theory.

• Natalie Wolchover. In Mysterious Pattern, Math and Nature Converge.
• 维基百科: Tracy-Widom distribution There are Tracy-Widom distributions for orthogonal {$F_1$}, unitary {$F_2$} and symplectic {$F_4$} random matrices.
• Painlevé transcendents
• At the far ends of a new universal law.
• Quanta Magazine articles on universality
• V: What is Universality?
• Does the number {$\sqrt{2n}$} relate to the root systems with {$2n$} simple roots?

Shift from objects (n) to relations (n2) as in the Tracy-Widom distribution? Can the Yoneda lemma help model such a phase transition?

• There may be a shift from relations (n2) to objects (n) when n grows large so as to reduce the overhead. This would break a system down into subsystems, which can operate independently, line by line. So there should be a pressure for a system to break down into subsystems.
• The Yoneda Lemma establishes the underlying isomorphism. But how can such naturally isomorphic structures be meaningfully different? It is because the interpreting perspective is different. As with entropy, the perspective matters, the coordinate system we choose. This perspective is inherent in whether we have a system or a subsystem.
• Think also of the orders of magnitude in the universe. And orders of scale in Alexander's principles of life.
• Switching over from a wave (or boson) point of view to a particle (or fermion) point of view. Note that multiple morphisms can have the same location, but multiple objects cannot.
• Consider how the identity morphism becomes distinguished. Note that the Yoneda lemma can be explaining how an identity morphism gets inferred. Because perhaps there aren't identity morphisms in the beginning.
• Consider how a system of morphisms unfolds from a single morphism, consider how a new morphism arises and consider the conditions that it has to satisfy to be consistent with the system, especially with composition.
• Consider how symmetric functions of eigenvalues are relevant here.

Quanta magazine. The Universal Pattern Popping Up in Math, Physics and Biology.

• The peak of the Tracy-Widon distribution is at {$\sqrt{2N}$}. How might that be related to the eigenvalues of raising {$\sqrt{N+1}$} and lowering {$\sqrt{N}$} operators?
• How does the distribution of the largest eigenvalue of a random matrix, and whether it is positive or negative, relate to the idea that "God doesn't have to be good?"

Morally, the phase transition can model good and bad behavior. Bad behavior - such as slowing down your bus so that you are followed by another bus and take all of its customers - is behavior that complicates the phase transition.

Also, the overhead of words: if you think in words (based on the world) rather than concepts (rooted in the features of your mind) then you will be taxed for that and at a certain point it won't be sustainable. In general, this is modeling nonsustainability.

Random matrices are related to random walks and other symmetric functions of the eigenvalues of matrices.

If we think of a hook as a walk, within a rim hook or a special rim hook, then we can think of these tableaux as ways of assembling walks, and thus of assembling systems from relationships.

Dyson's Threefold Way:

• F.J. Dyson, “The threefold way: algebraic structure of symmetry groups and ensembles in quantum mechanics”, J. Math. Phys.3(1962) 1199-1215
• "the most general matrix ensemble, defined with a symmetry group which may be completely arbitrary, reduces to a direct product of independent irreducible ensembles each of whichbelongs to one of the three known types": Complex Hermitian, real symmetric, or quaternion self-dual.

Bott periodicity and symmetry classes are related in the periodic table of topological insulators.

Martin R. Zirnbauer. Symmetry Classes.

Rubio. Random Matrix Symmetries

Energy levels are eigenvalues. Thus we consider the distribution of energy levels and their separation.

On the analytic wing of the house of knowledge, we can think of these as the energy levels, the eigenvalues, that must be kept separate, kept distinguished, like fermions. And in the case of a long tail we have a low energy extremes.

V: Course on topology in condensed matter

Zero energy excitations - "do nothing" - whether or not they exist - basis for topology in condensed matter. Topology and symmetry intro (by Anton Akhmerov) Without zero energy excitations, one can't transform certain systems into other systems. Thus we can group these systems into classes.

Quantum dots - zero dimensional systems.

维基百科: Chern class

维基百科: Majorana fermion - Majorana modes - Kitaev chains - related to Dynkin diagrams - and propagation of a signal in the Cartan matrix? Bulk-edge correspondence outlook (by Jay Sau) Video domain walls between Strong coupling limit - Kitaev chain, Dirac limit - produces delocalized Majorana modes in weakly gapped system. Different sides of the same coin. Bulk topological invariant connects topology in condensed matter with mathematics. Block Hamiltonian associated with crystal momentum K constrained on a circle. Difference between cylinder and a Mobius strip. K-theory worked this out.

Rimhook - is the growth of a tableaux by a continuous set of sells: a rimhook, as opposed to a row or a special rim hook, etc.

V: P. Vivo. Random Matrices: Theory and Practice.

维基百科: Random matrix

维基百科: Painlevé transcendents

Calculate and interpret symmetric functions of the eigenvalues of random matrices, starting with the determinant and the trace.

• Terence Tao. Singularity and determinant of random matrices
• Math StackExchange: Expected determinant of a random nxn matrix
• Nguyen, Vu. Random matrices: Law of the determinants.
• Math StackExchange: Expected determinant of random symmetric matrix with different Gaussian distributions of the diagonal and non-diagonal elements
• Asaph Keikara Muhumuza. Extreme points of the Vandermonde determinant in numerical approximation, random matrix theory and financial mathematics.
• Random walks on trees may express how to love and foster consciousness. The tree is the underlying unfolding of everything. The walk is what develops from nothing.

Walks on trees

See: Binomial theorem, Coincidences, Divisions, Entropy, Lie theory, Math connections, P vs NP

• Look at tree shaped categories.
• Penrose, page 675. 26.8 Constructing Feynman graphs; the S-matrix. Tree graphs describe the classical theory.

Leonard Susskind. Three Lectures on Complexity and Black Holes, Video: Complexity and gravity. Complexity theory is the geometry of {$SU(2)^k$}. 49:00 mentions the subject of random walks on trees.

Tracy-Widon distribution - check the article

• loosely coupled - particles? = inside? bound?
• tightly coupled - plasma = outside? space?

Perron-Frobenius theorem asserts that a real square matrix with positive entries has a unique largest real eigenvalue and that the corresponding eigenvector can be chosen to have strictly positive components, and also asserts a similar statement for certain classes of nonnegative matrices.

• Self-avoiding walk. One of the phenomena associated with self-avoiding walks and statistical physics models in general is the notion of universality, that is, independence of macroscopic observables from microscopic details, such as the choice of the lattice. One important quantity that appears in conjectures for universal laws is the connective constant, defined as follows. Let cn denote the number of n-step self-avoiding walks. Since every (n + m)-step self avoiding walk can be decomposed into an n-step self-avoiding walk and an m-step self-avoiding walk, it follows that cn+m ≤ cncm. Therefore, the sequence {log cn} is subadditive and we can apply Fekete's lemma to show that the following limit exists:
• μ = lim n → ∞ c n 1 n . {\displaystyle \mu =\lim _{n\to \infty }c_{n}^{\frac {1}{n}}.} \mu =\lim _{n\to \infty }c_{n}^{\frac {1}{n}}.
• μ is called the connective constant, since cn depends on the particular lattice chosen for the walk so does μ.

Wigner's classification

• Wigner found that massless particles are fundamentally different from massive particles.

Random matrix theory

• https://www.researchgate.net/publication/2091436_Symmetry_Classes_in_Random_Matrix_Theory
• Denis Bernard, Andre LeClair. A Classification of Non-Hermitian Random Matrices.

Perron-Frobenius theorem about the largest eigenvalue

• The fact that there is a real eigenvalue as large or larger than any of the complex eigenvalues reflects the chance that the matrix elements could all have zero imaginary component, and all positive components, thus be positive (or nonnegative) real numbers.

笔记

Walks

• Independent entries vs. Rotational invariance yield {$P[X]\propto e^{-\frac{1}{2}\textrm{Tr}X^2}$}.
• Relate walks on trees to covering groups. What do conjugates (paths) mean? What is the homotopy group?

Emergence of Random Structure Random structure (such as divisions of everything?) arises at certain thresholds.

Walks on binary trees: The Bruhat-Tits tree for the 2-adic Lie group {$SL(2,Q_2)$}. See Building.