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This paper investigates the spectral form factor (SFF) of the quadratic $R$-para-particle Sachdev-Ye-Kitaev ($R$-PSYK$_2$) model with various random matrix ensemble couplings. We generalize previous work on Gaussian Unitary Ensemble (GUE)…
Various results ensure the existence of large complete bipartite graphs in properly colored graphs when some condition related to a topological lower bound on the chromatic number is satisfied. We generalize three theorems of this kind,…
A vertex coloring of a given simple graph $G=(V,E)$ with $k$ colors ($k$-coloring) is a map from its vertex set to the set of integers $\{1,2,3,\dots, k\}$. A coloring is called perfect if the multiset of colors appearing on the neighbours…
In this article we describe a canonical way to expand a certain kind of $(\mathbb Z_2)^{n+1}$-colored regular graphs into closed $n$-manifolds by adding cells determined by the edge-colorings inductively. We show that every closed…
We discover new linear relations between the chromatic symmetric functions of certain sequences of graphs and apply these relations to find new families of e-positive unit interval graphs. Motivated by the results of Gebhard and Sagan, we…
Nonperturbative formulation of the Sachdev-Ye-Kitaev (SYK) model is discussed. The partition function of the model can be represented as a functional integral over the Grassmann variables in Euclidean time which is well defined but it…
The convolution of indicators of two conjugacy classes on the symmetric group S_q is usually a complicated linear combination of indicators of many conjugacy classes. Similarly, a product of the moments of the Jucys--Murphy element involves…
We review the fast developing subject of tensor models for the NAdS$_2$/NCFT$_1$ holographic correspondence. We include a brief review of the Sachdev-Ye-Kitaev (SYK) model and then focus on the associated quantum mechanical tensor models…
We obtain bivariate asymptotics for the number of (unicellular) combinatorial maps (a model of discrete surfaces) as both the size and the genus grow. This work is related to two research topics that have been very active recently:…
The Four color problem is closely related to other branches of mathematics and practical applications. More than 20 of its reformulations are known, which connect this problem with problems of algebra, statistical mechanics and planning.…
A central question on Kitaev materials is the effects of additional couplings on the Kitaev model which is proposed to be a candidate for realizing topological quantum computations. However, two spatial dimension typically suffers the…
We study a combinatorial model of the quantum scalar field with polynomial potential on a graph. In the first quantization formalism, the value of a Feynman graph is given by a sum over maps from the Feynman graph to the spacetime graph…
We discuss a supersymmetric generalization of the Sachdev-Ye-Kitaev model. These are quantum mechanical models involving $N$ Majorana fermions. The supercharge is given by a polynomial expression in terms of the Majorana fermions with…
In this article we associate a combinatorial differential graded algebra to a cubic planar graph G. This algebra is defined combinatorially by counting binary sequences, which we introduce, and several explicit computations are provided. In…
A colored Gaussian graphical model is a linear concentration model in which equalities among the concentrations are specified by a coloring of an underlying graph. The model is called RCOP if this coloring is given by the edge and vertex…
We present a detailed quantitative analysis of spectral correlations in the Sachdev-Ye-Kitaev (SYK) model. We find that the deviations from universal Random Matrix Theory (RMT) behavior are due to a small number of long-wavelength…
Let $2\le k\in\mathbb{Z}$. A total coloring of a simple connected regular graph via color set $ \{0,1,\ldots, k\}$ is said to be {\it efficient} if each color yields an efficient dominating set, where the efficient domination condition…
We consider an algebraic formulation of Quantum Theory and develop a combinatorial model of the Heisenberg-Weyl algebra structure. It is shown that by lifting this structure to the richer algebra of graph operator calculus, we gain a simple…
The Sachdev-Ye-Kitaev (SYK) model, a theory of N Majorana fermions with q-body interactions, becomes in the large q limit a conformally-broken Liouville field theory. Taking this limit preserves many interesting properties of the model, yet…
We examine $t$-colourings of oriented graphs in which, for a fixed integer $k \geq 1$, vertices joined by a directed path of length at most $k$ must be assigned different colours. A homomorphism model that extends the ideas of Sherk for the…