arXiv subjects

Sergio Caracciolo

Publications and source records attributed to Sergio Caracciolo.

At least 73 records · Page 4Linked to original sources

arXiv2006

Virial coefficients and osmotic pressure in polymer solutions in good-solvent conditions

We determine the second, third, and fourth virial coefficients appearing in the density expansion of the osmotic pressure of a monodisperse polymer solution in good-solvent conditions. Using the expected large-concentration behavior, we extrapolate the low-density expansion outside the dilute regime, obtaining the osmotic pressure for any concentration in the semidilute region. Comparison with field-theoretical predictions and experimental data shows that the obtained expression is quite accurate. The error is approximately 1-2% below the overlap concentration and rises at most to 5-10% in the limit of very large polymer concentrations.

cond-mat.stat-mech↗

arXiv2006

One-in-Two-Matching Problem is NP-complete

2-dimensional Matching Problem, which requires to find a matching of left- to right-vertices in a balanced $2n$-vertex bipartite graph, is a well-known polynomial problem, while various variants, like the 3-dimensional analogoue (3DM, with triangles on a tripartite graph), or the Hamiltonian Circuit Problem (HC, a restriction to ``unicyclic'' matchings) are among the main examples of NP-hard problems, since the first Karp reduction series of 1972. The same holds for the weighted variants of these problems, the Linear Assignment Problem being polynomial, and the Numerical 3-Dimensional Matching and Travelling Salesman Problem being NP-complete. In this paper we show that a small modification of the 2-dimensional Matching and Assignment Problems in which for each $i \leq n/2$ it is required that either $π(2i-1)=2i-1$ or $π(2i)=2i$, is a NP-complete problem. The proof is by linear reduction from SAT (or NAE-SAT), with the size $n$ of the Matching Problem being four times the number of edges in the factor graph representation of the boolean problem. As a corollary, in combination with the simple linear reduction of One-in-Two Matching to 3-Dimensional Matching, we show that SAT can be linearly reduced to 3DM, while the original Karp reduction was only cubic.

cs.CC↗

arXiv2006

Large-N_f chiral transition in the Yukawa model

We investigate the finite-temperature behavior of the Yukawa model in which $N_{f}$ fermions are coupled with a scalar field $ϕ$ in the limit $N_f \to \infty$. Close to the chiral transition the model shows a crossover between mean-field behavior (observed for $N_f = \infty$) and Ising behavior (observed for any finite $N_f$). We show that this crossover is universal and related to that observed in the weakly-coupled $ϕ^4$ theory. It corresponds to the renormalization-group flow from the unstable Gaussian fixed point to the stable Ising fixed point. This equivalence allows us to use results obtained in field theory and in medium-range spin models to compute Yukawa correlation functions in the crossover regime.

hep-lat↗

arXiv2005

Large-N_f behavior of the Yukawa model: analytic results

We investigate the Yukawa model in which $N_f$ fermions are coupled with a scalar field $ϕ$ through a Yukawa interaction. The phase diagram is rather well understood. If the fermions are massless, there is a chiral transition at $T_c$: for $T < T_c$ chiral symmetry is spontaneously broken. At $N_f=\infty$ the transition is mean-field like, while, for any finite $N_f$, standard arguments predict Ising behavior. This apparent contradiction has been explained by Kogut et al., who showed by scaling arguments and Monte Carlo simulations that in the large-$N_f$ limit the width of the Ising critical region scales as a power of $1/N_f$, so that only mean-field behavior is observed for $N_f$ strictly equal to infinity. We will show how the results of Kogut et al. can be recovered analytically in the framework of a generalized $1/N_f$ expansion. The method we use is a simple generalization of the method we have recently applied to a two-dimensional generalized Heisenberg model.

hep-lat↗

arXiv2005

Correction-to-scaling exponents for two-dimensional self-avoiding walks

We study the correction-to-scaling exponents for the two-dimensional self-avoiding walk, using a combination of series-extrapolation and Monte Carlo methods. We enumerate all self-avoiding walks up to 59 steps on the square lattice, and up to 40 steps on the triangular lattice, measuring the mean-square end-to-end distance, the mean-square radius of gyration and the mean-square distance of a monomer from the endpoints. The complete endpoint distribution is also calculated for self-avoiding walks up to 32 steps (square) and up to 22 steps (triangular). We also generate self-avoiding walks on the square lattice by Monte Carlo, using the pivot algorithm, obtaining the mean-square radii to ~0.01% accuracy up to N = 4000. We give compelling evidence that the first non-analytic correction term for two-dimensional self-avoiding walks is Delta_1 = 3/2. We compute several moments of the endpoint distribution function, finding good agreement with the field-theoretic predictions. Finally, we study a particular invariant ratio that can be shown, by conformal-field-theory arguments, to vanish asymptotically, and we find the cancellation of the leading analytic correction.

cond-mat.stat-mech↗

arXiv2005

Critical Behavior of the Two-Dimensional Randomly Driven Lattice Gas

We investigate the critical behavior of the two-dimensional randomly driven lattice gas, in which particles are driven along one of the lattice axes by an infinite external field with randomly changing sign. A finite-size scaling (FSS) analysis provides novel evidences that this model is not in the same universality class as the driven lattice gas with a constant drive (DLG), contrarily to what has been recently reported in the literature. Indeed, the FSS functions of transverse observables (i.e., related to order-parameter fluctuations with wave vector perpendicular to the direction of the field) differ from the mean-field behavior predicted and observed within the DLG universality class. At variance with the DLG case, FSS is attained on lattices with fixed aspect ratio and anisotropy exponent equal to 1 and the transverse Binder cumulant does not vanish at the critical point.

cond-mat.stat-mech↗

arXiv2005

O(n) vector model at n=-1, -2 on random planar lattices: a direct combinatorial derivation

The O(n) vector model with logarithmic action on a lattice of coordination 3 is related to a gas of self-avoiding loops on the lattice. This formulation allows for analytical continuation in n: critical behaviour is found in the real interval [-2,2]. The solution of the model on random planar lattices, recovered by random matrices, also involves an analytic continuation in the number n of auxiliary matrices. Here we show that, in the two cases n=-1, -2, a combinatorial reformulation of the loop gas problem allows to achieve the random matrix solution with no need of this analytical continuation.

cond-mat.stat-mech↗

arXiv2004

Two-Dimensional Heisenberg Model with Nonlinear Interactions: 1/N Corrections

We investigate a two-dimensional classical $-vector model with a generic nearest-neighbor interaction $W(\bsigma_i\cdot \bsigma_j)$ in the large-N limit, focusing on the finite-temperature transition point at which energy-energy correlations become critical. We show that this transition belongs to the Ising universality class. However, the width of the region in which Ising behavior is observed scales as $1/N^{3/2}$ along the magnetic direction and as 1/N in the thermal direction; outside a crossover to mean-field behavior occurs. This explains why only mean-field behavior is observed for $N=\infty$

cond-mat.stat-mech↗

arXiv2004

Fermionic field theory for trees and forests

We prove a generalization of Kirchhoff's matrix-tree theorem in which a large class of combinatorial objects are represented by non-Gaussian Grassmann integrals. As a special case, we show that unrooted spanning forests, which arise as a q \to 0 limit of the Potts model, can be represented by a Grassmann theory involving a Gaussian term and a particular bilocal four-fermion term. We show that this latter model can be mapped, to all orders in perturbation theory, onto the N-vector model at N=-1 or, equivalently, onto the sigma-model taking values in the unit supersphere in R^{1|2}. It follows that, in two dimensions, this fermionic model is perturbatively asymptotically free.

cond-mat.stat-mech↗

arXiv2003

Finite-Size Scaling in the Driven Lattice Gas

We present a Monte Carlo study of the high-temperature phase of the two-dimensional driven lattice gas at infinite driving field. We define a finite-volume correlation length, verify that this definition has a good infinite-volume limit independent of the lattice geometry, and study its finite-size-scaling behavior. The results for the correlation length are in good agreement with the predictions based on the field theory proposed by Janssen, Schmittmann, Leung, and Cardy. The theoretical predictions for the susceptibility and the magnetization are also well verified. We show that the transverse Binder parameter vanishes at the critical point in all dimensions $d\ge 2$ and discuss how such result should be expected in the theory of Janssen et al. in spite of the existence of a dangerously irrelevant operator. Our results confirm the Gaussian nature of the transverse excitations.

cond-mat.stat-mech↗

arXiv2003

General duality for abelian-group-valued statistical-mechanics models

We introduce a general class of statistical-mechanics models, taking values in an abelian group, which includes examples of both spin and gauge models, both ordered and disordered. The model is described by a set of ``variables'' and a set of ``interactions''. A Gibbs factor is associated to each variable and to each interaction. We introduce a duality transformation for systems in this class. The duality exchanges the abelian group with its dual, the Gibbs factors with their Fourier transforms, and the interactions with the variables. High (low) couplings in the interaction terms are mapped into low (high) couplings in the one-body terms. The idea is that our class of systems extends the one for which the classical procedure 'a la Kramers and Wannier holds, up to include randomness into the pattern of interaction. We introduce and study some physical examples: a random Gaussian Model, a random Potts-like model, and a random variant of discrete scalar QED. We shortly describe the consequence of duality for each example.

cond-mat.stat-mech↗

arXiv2003

Comment on ``Dynamic behavior of anisotropic non-equilibrium driving lattice gases''

In a recent Letter Albano and Saracco study the dynamic critical behavior of some anisotropic driven lattice gases by Monte Carlo (MC) simulations. In this Comment we point out that the Ansätze they use to relate the measured scaling exponents with the critical exponents analytically computed within different field-theoretical approaches do not take properly into account the strongly anisotropic nature of the phase transition, by implicitly assuming $z = z_{\bot} = z_{\parallel}$. As a consequence, at variance with the claims by the authors, their MC data are not conclusive to determine which one of the field theories proposed in the literature correctly describes the universal properties of the phase transition in these lattice gases.

cond-mat↗

arXiv2003

Shape dependence of the finite-size scaling limit in a strongly anisotropic $O(\infty)$ model

We discuss the shape dependence of the finite-size scaling limit in a strongly anisotropic O(N) model in the large-N limit. We show that scaling is observed even if an incorrect value for the anisotropy exponent is considered. However, the related exponents may only be effective ones, differing from the correct critical exponents of the model. We discuss the implications of our results for numerical finite-size scaling studies of strongly anisotropic systems.

cond-mat.stat-mech↗

arXiv2002

Transverse Fluctuations in the Driven Lattice Gas

We define a transverse correlation length suitable to discuss the finite-size-scaling behavior of an out-of-equilibrium lattice gas, whose correlation functions decay algebraically with the distance. By numerical simulations we verify that this definition has a good infinite-volume limit independent of the lattice geometry. We study the transverse fluctuations as they can select the correct field-theoretical description. By means of a careful finite-size scaling analysis, without tunable parameters, we show that they are Gaussian, in agreement with the predictions of the model proposed by Janssen, Schmittmann, Leung, and Cardy.

cond-mat.stat-mech↗

arXiv2002

An exactly solvable random satisfiability problem

We introduce a new model for the generation of random satisfiability problems. It is an extension of the hyper-SAT model of Ricci-Tersenghi, Weigt and Zecchina, which is a variant of the famous K-SAT model: it is extended to q-state variables and relates to a different choice of the statistical ensemble. The model has an exactly solvable statistic: the critical exponents and scaling functions of the SAT/UNSAT transition are calculable at zero temperature, with no need of replicas, also with exact finite-size corrections. We also introduce an exact duality of the model, and show an analogy of thermodynamic properties with the Random Energy Model of disordered spin systems theory. Relations with Error-Correcting Codes are also discussed.

cond-mat.stat-mech↗

arXiv2002

Two-Dimensional Heisenberg Model with Nonlinear Interactions

We investigate a two-dimensional classical $N$-vector model with a nonlinear interaction $(1 + \bsigma_i\cdot \bsigma_j)^p$ in the large-N limit. As observed for N=3 by Blöte {\em et al.} [Phys. Rev. Lett. {\bf 88}, 047203 (2002)], we find a first-order transition for $p>p_c$ and no finite-temperature phase transitions for $p < p_c$. For $p>p_c$, both phases have short-range order, the correlation length showing a finite discontinuity at the transition. For $p=p_c$, there is a peculiar transition, where the spin-spin correlation length is finite while the energy-energy correlation length diverges.

cond-mat.stat-mech↗

arXiv2001

Discrete non-Abelian groups and asymptotically free models

We consider a two-dimensional $σ$-model with discrete icosahedral/dodecahedral symmetry. Using the perturbative renormalization group, we argue that this model has a different continuum limit with respect to the O(3) $σ$ model. Such an argument is confirmed by a high-precision numerical simulation.

hep-lat↗

arXiv2001

Dynamic Critical Behavior of an Extended Reptation Dynamics for Self-Avoiding Walks

We consider lattice self-avoiding walks and discuss the dynamic critical behavior of two dynamics that use local and bilocal moves and generalize the usual reptation dynamics. We determine the integrated and exponential autocorrelation times for several observables, perform a dynamic finite-size scaling study of the autocorrelation functions, and compute the associated dynamic critical exponents $z$. For the variables that describe the size of the walks, in the absence of interactions we find $z \approx 2.2$ in two dimensions and $z\approx 2.1$ in three dimensions. At the $θ$-point in two dimensions we have $z\approx 2.3$.

cond-mat.stat-mech↗