On statistical models on super trees

Abstract

We consider a particular example of interplay between statistical models related to CFT on one hand, and to the spectral properties of ODE, known as ODE/IS correspondence, on the other hand. We focus at the representation of wave functions of Schrödinger operators in terms of spectral properties of associated transfer matrices on "super trees" (the trees whose vertex degree changes with the distance from the root point). Such trees with varying branchings encode the structure of the Fock space of the model. We discuss basic spectral properties of "averaged random matrix ensembles" in terms of Hermite polynomials for the transfer matrix of super trees. At small "branching velocities" we have related the problem of paths counting on super trees to the statistics of area-weighted one-dimensional Dyck paths. We also discuss the connection of the spectral statistics of random walks on super trees with the Kardar-Parisi-Zhang scaling.

Lifshitz tails at spectral edge and holography with a finite cutoff

“…The question to be answered behind this formal relation is as follows: could we see a Griffiths-like behavior (4.2) of the partition function in the regime which we have described above? We provide some arguments, using recent results derived in [48][49][50] concerning the KPZ scaling of fluctuations in the restricted random walks on the plane. To justify the Griffiths-like behavior we first regard the model A of restricted random walk of fixed length, L, where the relation between the length, L, and the cutoff scale, R, is imposed by hands.…”

Equilibrium mean-field-like statistical models with KPZ scaling

“…The symmetric matrix M (with x ij = x ji ) allows a straightforward interpretation as the transfer matrix of a path counting problem on a random symmetric tree [12]. To make the statement more transparent, write M together with the "shifted" matrix M :…”

“…The partition function, Z N (k), of N -step trajectories starting at the root point of a finite ascending tree T + satisfies the recursion (12). To rewrite (12) in a matrix form, make a shift k → k + 1 and construct the K-dimensional vector Z N = (Z N (1), Z N (2), ...Z N (K)) . Then (13) sets the evolution of Z N in N :…”

“…Here we consider the limit of small branching velocity |a| 1. In [12] we have argued that the random walk on such trees is closely related to the area-weighted Dyck paths, where the effective magnetic field on the lattice is related to the branching velocity. Consider a N × N square lattice and enumerate all N -step trajectories (Dyck paths) starting at (0, 0), ending at (N, N ) and staying above the diagonal of the square (the path can touch the diagonal, but cannot cross it).…”

“…The determinant representation of the Hermite polynomials is closely related to the characteristic polynomial for the transfer matrix on a supertree. When the "vertex degree velocity" (the branching increment between neighboring tree levels) is small, we can identify the corresponding model with the (1+1)D lattice random walk in the transverse constant magnetic field [12]. The same problem can be formulated in the symmetric Riemann space with the non-constant radially-dependent curvature, which is the generalization of the space of constant negative curvature (the hyperbolic space).…”