arXiv subjects

Pete Rigas

Publications and source records attributed to Pete Rigas.

29 records · Page 2Linked to original sources

arXiv2025

Phase transition of the long range Ising model in lower dimensions, for $d < α\leq d + 1$, with a Peierls argument

We extend previous results due to Ding and Zhuang in order to prove that a phase transition occurs for the long range Ising model in lower dimensions. By making use of a recent argument due to Affonso, Bissacot and Maia from 2022 which establishes that a phase transition occurs for the long range, random-field Ising model, from a suggestion of the authors we demonstrate that a phase transition also occurs for the long range Ising model, from a set of appropriately defined contours for the long range system, and a Peierls' argument.

math.PR↗

arXiv2025

The emptiness formation probability, and representations for nonlocal correlation functions, of the 20-vertex model

We study the emptiness formation probability, along with various representations for nonlocal correlation functions, of the 20-vertex model. In doing so, we leverage previous arguments for representations of nonlocal correlation functions for the 6-vertex model, under domain-wall boundary conditions, due to Colomo, Di Giulio, and Pronko, in addition to the inhomogeneous, and homogeneous, determinantal representations for the 20-vertex partition function due to Di Francesco, also under domain-wall boundary conditions. By taking a product of row configuration probabilities, we obtain a desired contour integral representation for nonlocal correlations from a determinantal representation. Finally, a counterpart of the emptiness formation probability is introduced for the 20-vertex model.

math-ph↗

arXiv2025

Poisson structure of the 4-vertex model, and the higher-spin XXX chain, and Yang-Baxter algebras

We implement the quantum inverse scattering method for the 4-vertex model. In comparison to previous works of the author which examined the 6-vertex, and 20-vertex, models, the 4-vertex model exhibits different characteristics, ranging from L-operators expressed in terms of projectors and Pauli matrices to algebraic and combinatorial properties, including Poisson structure and boxed plane partitions. With far fewer computations with an L-operator provided for the 4-vertex model by Bogoliubov in 2007, in comparison to those for L-operators of the 6, and 20, vertex models, from lower order expansions of the transfer matrix we derive a system of relations from the structure of operators that can be leveraged for studying characteristics of the higher-spin XXX chain in the weak finite volume limit. In comparison to quantum inverse scattering methods for the 6, and 20, vertex models which can be used to further study integrability, and exact solvability, an adaptation of such an approach for the 4-vertex model can be used to approximate, asymptotically in the weak finite volume limit, sixteen brackets which generate the Poisson structure. From explicit relations for operators of the 4-vertex transfer matrix, we conclude by discussing corresponding aspects of the Yang-Baxter algebra, which is closely related to the operators obtained from products of L-operators for approximating the transfer, and quantum monodromy, matrices. The structure of computations from L-operators of the 4-vertex model directly transfers to L-operators of the higher-spin XXX chain, revealing a similar structure of another Yang-Baxter algebra of interest.

math-ph↗

arXiv2025

Intertwining vectors, and Boltzmann weight matrices, of a Solid-on-Solid model from the 20-vertex model

We initiate a new study on the correspondence between the 20-vertex model and a SOS (Solid-on-Solid) model. In comparison to two previous works of the author in 2024 which characterized properties of the transfer, and quantum monodromy, matrices of the 20-vertex model from the perspective of the quantum inverse scattering method, in addition to the structure of nonlocal correlations, the forthcoming approach is an adaptation of a study on the rational 7-vertex model. For the rational 7-vertex model, Antoenneko and Valinevich demonstrated that the intertwining vectors can be used to transform the R-matrix of a vertex model into the Boltzman weight matrix of an SOS model. To further develop perspectives on classes of higher dimensional SOS models, by leveraging previous computations with L-operators due to the author, and by also manipulating q-exponentials, and other related factors from a factorization of the universal R-matrix due to Boos et al, analogs of intertwining vectors for the 20- vertex model can be obtained. In comparison to the system of equations that have been obtained for the rational 7-vertex model, the system that one obtains for the 20-vertex model consists of 9 equations for each entry of the R-matrix. Furthermore, from the fact that the 20-vertex model contains more possible configurations in the sample space than the rational 7-vertex model does, the Boltzman weight matrix of the corresponding SOS model asymptotically depends upon several contributions, a few of which include tensor products of basis elements, and spectral parameters over the triangular lattice.

math-ph↗

arXiv2025

Quantum strategies, error bounds, optimality, and duality gaps for multiplayer XOR, $\mathrm{XOR}^{}$, compiled XOR, $\mathrm{XOR}^{}$, and strong parallel repetiton of XOR, $\mathrm{XOR}^{*}$, and FFL games

We characterize exact, and approximate, optimality of games that players can interact with using quantum strategies. In comparison to a previous work of the author, arXiv: 2311.12887, which applied a 2016 framework due to Ostrev for constructing error bounds beyond CHSH and XOR games, in addition to the existence of well-posed semidefinite programs for determining primal feasible solutions, along with quantum-classical duality gaps, it continues to remain of interest to further develop the construction of error bounds, and related objects, to game-theoretic settings with several participants. In such settings, one encounters a rich information theoretic landscape, not only from the fact that there exists a significantly larger combinatorial space of possible strategies for each player, but also several opportunities for pronounced quantum advantage. We conclude this effort by describing other variants of other possible strategies, as proposed sources for quantum advantage, in $\mathrm{XOR}^{*}$, compiled $\mathrm{XOR}^{*}$, and strong parallel repetition variants of $\mathrm{XOR}^{*}$ games.

quant-ph↗

arXiv2024

Operator formalism for discretely holomorphic parafermions of the two-color Ashkin-Teller, loop $\mathrm{O} ( 1 )$, staggered eight-vertex, odd eight-vertex, and abelian sandpile models

We extend an operator formalism, developed by Hongler, Kytölä, and Zahabi in 2012 for the Ising model, to the Ashkin-Teller and loop $\mathrm{O} ( n ) $ models. The formalism is primarily dependent upon notions of massive, and massless, s-holomorphicity of the Ising model, which are respectively satisfied at, and below, the critical temperature, hence allowing for rigorous analyses of the transfer matrix, correlation functions, and lattice fermion operators. From results of another paper, by Tanhayi-Ahari and Rouhani in 2012, which demonstrates that parafermionic observables for the staggered eight-vertex model coincide with those of the Ashkin-Teller model, as well as a 2003 paper, by Wu and Kunz, which establishes a correspondence between the staggered eight-vertex and odd eight-vertex models, we establish further associations.

math.PR↗

arXiv2024

From logarithmic delocalization of the six-vertex height function under sloped boundary conditions to weakened crossing probability estimates for the Ashkin-Teller, generalized random-cluster, and $(q_σ,q_τ)$-cubic models

To obtain Russo-Seymour-Welsh estimates for the height function of the six-vertex model under sloped boundary conditions, which can be leveraged to demonstrate that the height function logarithmically delocalizes under a broader class of boundary conditions, we formulate crossing probability estimates in strips of the square lattice and the cylinder, for parameters satisfying $a\equiv b$, $c \in [1,2]$, and $\mathrm{max} \{ a , b \} \leq c$, in which each of the first two conditions respectively relate to invariance under vertical and diagonal reflections enforced through the symmetry $σξ\geq -ξ$ for domains in strips of the square lattice, and satisfaction of FKG, for the height function and for its absolute value. To determine whether arguments for estimating crossing probabilities of the height function for flat boundary conditions from a recent work due to Duminil-Copin, Karila, Manolescu, and Oulamara remain applicable for sloped boundary conditions, from the set of possible slopes given by the interior of the set of rational points from $[-1,1] \times [-1,1]$, we analyze sloped Gibbs states, which do not have infinitely many disjointly oriented circuits. In comparison to Russo-Seymour-Welsh arguments for flat boundary conditions, arguments for sloped boundary conditions present additional complications for both planar and cylindrical settings, in which crossing events that are considered in the strip, and then extended to the annulus and cylinder, must be maintained across rectangles of large aspect ratio, in spite of the fact that some proportion of faces within the strip freeze with positive probability.

math.PR↗

arXiv2024

The phase transition for the Gaussian free field is sharp

We prove that the phase transition for the Gaussian free field (GFF) is sharp. In comparison to a previous argument due to Rodriguez in 2017 which characterized a $0-1$ law for the Massive Gaussian Free Field by analyzing crossing probabilities below a threshold $h_{**}$, we implement a strategy due to Duminil-Copin and Manolescu in 2016, which establishes that two parameters are equal, one of which encapsulates the probability of obtaining an infinite connected component under free boundary conditions, while the other encapsulates the natural logarithm of the probability of obtaining a connected component from the origin to the box of length $n$ which is also taken under free boundary conditions. We quantify the probability of obtaining crossings in easy and hard directions, without imposing conditions that the graph is invariant with respect to reflections, in addition to making use of a differential inequality adapted for the GFF. The sharpness of the phase transition is characterized by the fact that below a certain height parameter of the GFF, the probability of obtaining an infinite cluster a.s. decays exponentially fast, while above the parameter, the probability of obtaining an infinite cluster occurs a.s. with good probability.

math.PR↗

arXiv2024

Eigenvectors of Toeplitz matrices from Fisher-Hartwig symbols with greater than, or equal to, one singularity

Asymptotically, we analytically derive the form of eigenvectors for two Fisher-Hartwig symbols besides those which were previously investigated in a $2016$ work due to Movassagh and Kadanoff, in which the authors characterized the eigenpairs of Toeplitz matrices generated by Fisher-Hartwig symbols with one singularity. To perform such computations, we extend their methods which consists of formulating an eigenvalue problem, obtained by a Wiener-Hopf method, from which a suitable winding number is defined for passing to Fourier space, and introducing a factorization dependent upon the winding number, for other Fisher-Hartwig symbols which have previously been defined in the literature. Following the computations required for the proof after obtaining the asymptotic approximation of the eigenvectors, we provide a table from which eigenvalues of one Fisher-Hartwig symbol for different complex valued functions $b$ can be inferred.

math-ph↗

arXiv2024

Eigenvalue attraction in open quantum systems, biophysical systems, and Parity-Time symmetric materials

We investigate eigenvalue attraction for open quantum systems, biophysical systems, and for Parity-Time symmetric materials. To determine whether an eigenvalue and its complex conjugate of a real matrix attract, we derive expressions for the second derivative of eigenvalues, which is dependent upon contributions from inertial forces, attraction between an eigenvalue and its complex conjugate, as well as the force of the remaining eigenvalues in the spectrum.

quant-ph↗

arXiv2022

Correlation length lower bound for the random-field Potts model with the greedy lattice animal

Motivated by recent developments over the past few years in the study of the correlation length of the random-field Ising model due to Ding and Wirth in a paper first available in 2020, we pursue one natural direction of research that the authors propose is of interest, namely in confirming that the same scaling for the correlation length for the random-field Ising model equals that of the random-field Potts model. To demonstrate that the $\frac{4}{3}$ emergence in the correlation length scaling for the random-field Potts model coincides with that correlation length scaling for the random-field Ising model, we refer to arguments due to Talagrand, in which an upper bound on the greedy lattice animal readily provides a corresponding lower bound on the correlation length. Contributions from the $\frac{4}{3}$ exponent appearing in the correlation length scaling for the random-field Ising, and random-field Potts models alike are reminiscent of $\frac{4}{3}$ exponents in upper bounds taken under the Liouville quantum gravity metric in high temperature.

math.PR↗