Random Matrices

While looking for lexicon keywords on arxiv saw  this unrelated paper: random matrix approach to language acquisition.

Which is weird because random matrices show up in prime theory and modern Hamiltonians 

 http://arxiv.org/abs/0909.3912

Strange connection.

 Polya and Hilbert suggested that the Riemann Hypothesis could be solved by finding a linear self-adjoint operator whose eigenvalues are given by the Riemann zeros. This idea received a great boost when it was discovered that the Riemann zeros have the same statistical properties as eigenvalues of large random matrices from the Gaussian Unitary Ensemble (21, 22). This ensemble happens also to be responsible for the statistics of chaotic quantum systems without time-reversal symmetry, which led to speculations that the Polya–Hilbert operator might be the Hamiltonian of a quantum system with a chaotic classical limit. Riemann's explicit formula, which connects the zeros with prime numbers, could then be viewed as a special case of Gutzwiller's trace formula, where the prime numbers are interpreted as logarithms of classical actions of the (unknown) classical dynamics (23). Although the connection to random matrix theory is still rather mysterious in the case of the Riemann zeta function, there have been recent exciting developments in the case of zeta functions of curves over finite fields, where the relation with the spectral measures of the classical groups is now well established (5, 12).

 http://www.pnas.org/content/98/19/10531.full

Just skip to the conclusion on the arxiv paper also it has a good summation.

We are dealing with language and it is not appropriate to consider it as an isolated

system. Rather we hope to capture aspects of the complex linguistic phenomenon by

resorting to a highly interdisciplinary method. In our paper we suggested that models

and techniques developed within physics might be useful in deciphering the language

riddle. The rationale behind the indicated course is that since language is strongly tied

to cognition, we expect the linguistic structures to reflect structures and patterns we

encounter in nature and analyzed by physics. This profound interrelationship nature

- human language is a permanent and continuous one and lies at the very foundation

of the “intelligibility” of the universe. As a first step we considered the most simple

A Random Matrix Approach to Language Acquisition 11

language, a protolanguage, which is essentially a mapping between sounds and objects.

This mapping is represented by a matrix and the language interaction is simulated by

random matrix mechanics. The suggested interaction Hamiltonian between the matrices

is (see eq. 2)

H = (1/2)Tr(PQ′ + P′Q) (23)

Our simple model bears great resemblance to a well known and extensively studied

problem in physics, magnet-magnet interaction. A magnet may have one direction in

space, chosen among a given set of possible directions. When many magnets are brought

together, it is expected that the interaction among the magnets to lead the magnets to

acquire a common direction in space, rather than each magnet having its own direction.

This common field is described as mean field and an individual field (a magnet, or

a particle) interacts with this average mean field. A particular matrix version of the

mean field technique may be found in ref.[43] , and our model Hamiltonian is very

similar to theirs. In a similar vein, a protolanguage appears as a specific choice among a

huge number (N!) of possibilities. Social interaction among the different partners, each

using its own protolanguage, will lead eventually to the adoption of a unique collective

“mean protolanguage”, L(P,Q) in our case. It is with this “mean protolanguage”

that an individual will interact, the interaction being described by eq. 23. Random

matrices have been widely used in Nuclear and Particle Physics and in general in systems

involving large numbers of degrees of freedom [44, 45]. Matrix models are directly linked

to string theory [46], the theory unifying all interactions in nature [47]. Also it has been

shown recently that relational logic and category theory are expressed by matrix models

[48]. Thus, our proposal opens the possibility for a fruitful interaction between linguistics

and advanced sectors of theoretical and mathematical physics

Random matrices show up in the latest QFT Lagrangian and in the Riemann Zeta prime number function thing and in game theory (which strangely uses the same Lagrangian formalism as modern QFT with different functional approaches.)

aside from just the obvious reason of trying to understand them for the purpose of understanding Lagrangians, isn't it just so weird that there is a single connection between matrix math, the random numbers, the primes, GAME theory! and QFT?!

I was reading some book on poker psychology yesterday and I kept thinking how bizarre it was that poker is a game fundamentally described by the math of game theory. And if that math is following a Lagrangian model, then insights into how to solve poker, from a psychological perspective, are approaching the same problem in a different isomorphism --

How can a psychology problem and a Lagrangian formalism share common traits?