Related papers: Average Length of Cycles in Rectangular Lattice

We study a duality transformation from the gauge-invariant subspace of a $\mathbb{Z}_N$ lattice gauge theory on a two-leg ladder geometry to an $N$-clock model on a single chain. The main feature of this mapping is the emergence of a…

We compute the four-loop contributions to the $\beta$-function and the anomalous dimension of the field for the $O(N)$-invariant $N$-vector model. These results are used to compute the second analytic corrections to the correlation length…

We generalize the notion of length to an ordinal-valued invariant defined on the class of finitely generated modules over a Noetherian ring. A key property of this invariant is its semi-additivity on short exact sequences. We show how to…

We here elaborate on a quantitative argument to support the validity of the Collatz conjecture, also known as the (3x + 1) or Syracuse conjecture. The analysis is structured as follows. First, three distinct fixed points are found for the…

A flat of a matroid is cyclic if it is a union of circuits; such flats form a lattice under inclusion and, up to isomorphism, all lattices can be obtained this way. A lattice is a Tr-lattice if all matroids whose lattices of cyclic flats…

We determine the general structure of the partition function of the $q$-state Potts model in an external magnetic field, $Z(G,q,v,w)$ for arbitrary $q$, temperature variable $v$, and magnetic field variable $w$, on cyclic, M\"obius, and…

Azam and Richmond arXiv:2107.09149 obtained a recursion for the generating function of \(P_\lambda(y)\), itself a generating function enumerating by length partitions in the lower ideal \([0,\lambda]\) in the Young lattice. We show that…

We compute several matrix elements of dimension-six four-fermion operators and extract their B-parameters. The calculations have been performed with the tree-level Clover action at $\beta = 6.0$. The renormalization constants and mixing…

In their study of cyclic pattern containment, Domagalski et al. conjecture differential equations for the generating functions of circular permutations avoiding consecutive patterns of length 3. In this note, we prove and significantly…

An m-ballot path of size n is a path on the square grid consisting of north and east steps, starting at (0,0), ending at (mn,n), and never going below the line {x=my}. The set of these paths can be equipped with a lattice structure, called…

We study the spectral properties of the transfer matrix for a gonihedric random surface model on a three-dimensional lattice. The transfer matrix is indexed by generalized loops in a natural fashion and is invariant under a group of motions…

In the Stanley lattice defined on Dyck paths of size $n$, cover relations are obtained by replacing a valley $DU$ by a peak $UD$. We investigate a greedy version of this lattice, first introduced by Chenevi\`ere, where cover relations…

The Collatz function is defined as C(n) = n / 2 if n is even and C(n) = 3n + 1 if n is odd. The Collatz conjecture states that every sequence generated by the Collatz function ends with the cycle (4, 2, 1) after a finite number of…

We study the winding angles of random and self-avoiding walks on square and cubic lattices with number of steps $N$ ranging up to $10^7$. We show that the mean square winding angle $\langle\theta^2\rangle$ of random walks converges to the…

Recent work by Arpin, Chen, Lauter, Scheidler, Stange, and Tran counted the number of cycles of length $r$ in supersingular $\ell$-isogeny graphs. In this paper, we extend this work to count the number of cycles that occur along the spine.…

In this paper, we give some non-trivial geometric cycles of the space of long embeddings R^j --> R^n (n-j >= 2) modulo immersions. We construct a class of cycles from specific chord diagrams associated with the 2-loop or 3-loop hairy…

We study the effects of scattering lengths on L\'evy walks in quenched one-dimensional random and fractal quasi-lattices, with scatterers spaced according to a long-tailed distribution. By analyzing the scaling properties of the random-walk…

The recent realization of a "Levy glass" (a three-dimensional optical material with a Levy distribution of scattering lengths) has motivated us to analyze its one-dimensional analogue: A linear chain of barriers with independent spacings s…

This paper introduces a new method for the efficient computation of oscillatory multidimensional lattice sums in geometries with boundaries. Such sums are ubiquitous in both pure and applied mathematics, and have immediate applications in…

In the worldline formalism, scalar Quantum Electrodynamics on a 2-dimensional lattice is related to the areas of closed loops on this lattice. We exploit this relationship in order to determine the general structure of the moments of the…