Search

We give a new derivation of mean-field percolation critical behaviour from the triangle condition that is quantitatively much better than previous proofs when the triangle diagram ∇ p c is large. In contrast to earlier methods, our approach continues to yield bounds of reasonable order when the triangle diagram ∇ p is unbounded but diverges slowly as p ↑ p c , as is expected to occur in percolation on Z d at the upper-critical dimension d = 6 . Indeed, we show in particular that if the triangle diagram diverges polylogarithmically as p ↑ p c then mean-field critical behaviour holds to within a polylogarithmic factor. We apply the methods we develop to deduce that for long-range percolation on the hierarchical lattice, mean-field critical behaviour holds to within polylogarithmic factors at the upper-critical dimension. As part of the proof, we introduce a new method for comparing diagrammatic sums on general transitive graphs that may be of independent interest. [ABSTRACT FROM AUTHOR]

The intersection numbers for p spin curves of the moduli space M ¯ g , n are considered for D l type by a matrix model. The asymptotic behavior of the large genus g limit and large p limit are derived. The remarkable features of the cases of p = 1 2 , - 1 2 , - 2 , - 3 are examined in the Laurent expansion for multiple correlation functions. The strong coupling expansions for the negative p cases are considered. [ABSTRACT FROM AUTHOR]

It is widely expected that, for a large class of models, scale invariance implies conformal invariance. A sufficient condition for this to happen is that there exists no integrated vector operator, invariant under all internal symmetries of the model, with scaling dimension - 1 . In this article, we compute the scaling dimensions of vector operators with lowest dimensions in the O(N) model. We use three different approximation schemes: ϵ expansion, large N limit and third order of the derivative expansion of Non-Perturbative Renormalization Group equations. We find that the scaling dimensions of all considered integrated vector operators are always much larger than - 1 . This strongly supports the existence of conformal invariance in this model. For the Ising model, an argument based on correlation functions inequalities was derived, which yields a lower bound for the scaling dimension of the vector perturbations. We generalize this proof to the case of the O(N) model with N ∈ 2 , 3 , 4 . [ABSTRACT FROM AUTHOR]

We study an influence network of voters subjected to correlated disordered external perturbations, and solve the dynamical equations exactly for fully connected networks. The model has a critical phase transition between disordered unimodal and ordered bimodal distribution states, characterized by an increase in the vote-share variability of the equilibrium distributions. The fluctuations (variance and correlations) in the external perturbations are shown to reduce the impact of the external influence by increasing the critical threshold needed for the bimodal distribution of opinions to appear. The external fluctuations also have the surprising effect of driving voters towards biased opinions. Furthermore, the first and second moments of the external perturbations are shown to affect the first and second moments of the vote-share distribution. This is shown analytically in the mean field limit, and confirmed numerically for fully connected networks and other network topologies. Studying the dynamic response of complex systems to disordered external perturbations could help us understand the dynamics of a wide variety of networked systems, from social networks and financial markets to amorphous magnetic spins and population genetics. [ABSTRACT FROM AUTHOR]

It has been observed that the classification into universality classes of critical behavior, as established by perturbative renormalization group in the vicinity of four or six dimensions of space by the epsilon expansion, remains valid down to three dimensions in all known cases, even when perturbative renormalization group fails in lower dimensions. In this paper we argue that this classification into universality classes remains true in lower dimensions of space, even when perturbative renormalization group fails, because of the well known phenomenon of eigenvalue repulsion. [ABSTRACT FROM AUTHOR]

We consider the n-component $$|\varphi |^4$$ lattice spin model ( $$n \ge 1$$ ) and the weakly self-avoiding walk ( $$n=0$$ ) on $$\mathbb Z^d$$ , in dimensions $$d=1,2,3$$ . We study long-range models based on the fractional Laplacian, with spin-spin interactions or walk step probabilities decaying with distance r as $$r^{-(d+\alpha )}$$ with $$\alpha \in (0,2)$$ . The upper critical dimension is $$d_c=2\alpha $$ . For $$\varepsilon >0$$ , and $$\alpha = \frac{1}{2} (d+\varepsilon )$$ , the dimension $$d=d_c-\varepsilon $$ is below the upper critical dimension. For small $$\varepsilon $$ , weak coupling, and all integers $$n \ge 0$$ , we prove that the two-point function at the critical point decays with distance as $$r^{-(d-\alpha )}$$ . This 'sticking' of the critical exponent at its mean-field value was first predicted in the physics literature in 1972. Our proof is based on a rigorous renormalisation group method. The treatment of observables differs from that used in recent work on the nearest-neighbour 4-dimensional case, via our use of a cluster expansion. [ABSTRACT FROM AUTHOR]

We propose a new way of investigating phase transitions in the context of information theory. We use an information-entropic measure of spatial complexity known as configurational entropy (CE) to quantify both the storage and exchange of information in a lattice simulation of a Ginzburg-Landau model with a scalar order parameter coupled to a heat bath. The CE is built from the Fourier spectrum of fluctuations around the mean-field and reaches a minimum at criticality. In particular, we investigate the behavior of CE near and at criticality, exploring the relation between information and the emergence of ordered domains. We show that as the temperature is increased from below, the CE displays three essential scaling regimes at different spatial scales: scale free, turbulent, and critical. Together, they offer an information-entropic characterization of critical behavior where the storage and fidelity of information processing is maximized at criticality. [ABSTRACT FROM AUTHOR]

Leo Kadanoff has worked in many fields of statistical mechanics. His contributions had an enormous impact. This holds in particular for critical phenomena, where he explained Widom's homogeneity laws by means of block-spin transformations and laid the basis for Wilson's renormalization group equation. I had the pleasure to work in his group for 1 year. A short historical account is given. [ABSTRACT FROM AUTHOR]

Branching (or tip-splitting) is a ubiquitous feature of growth processes in nature. We introduce a simple model, linear in growth probabilities, of branch competition that combines tip-splitting processes, local screening, and extinction of branches whose growth has come to an end. This model admits an exact solution; which is corroborated by numerical results. An extension of the model that depends quadratically upon growth probabilities exhibits a phase transition, with non-trivial scaling in the neighborhood of that transition. [ABSTRACT FROM AUTHOR]

I review the concept of a disorder operator, introduced originally by Kadanoff in the context of the two-dimensional Ising model. Disorder operators acquire an expectation value in the disordered phase of the classical spin system. This concept has had applications and implications to many areas of physics ranging from quantum spin chains to gauge theories to topological phases of matter. In this paper I describe the role that disorder operators play in our understanding of ordered, disordered and topological phases of matter. The role of disorder operators, and their generalizations, and their connection with dualities in different systems, as well as with majorana fermions and parafermions, is discussed in detail. Their role in recent fermion-boson and boson-boson dualities is briefly discussed. [ABSTRACT FROM AUTHOR]

We find a self-similar analytical solution for the grand-canonical-ensemble (GCE) statistics and thermodynamics in the critical region of Bose-Einstein condensation. It is valid for an arbitrary trap, loaded with an ideal gas, in the thermodynamic limit. We show that for the quantities, changing by a finite amount across the critical region, the exact GCE result differs from the corresponding canonical-ensemble result by a factor on the order of unity even in the thermodynamic limit. Thus, a widely used GCE approach does not describe correctly the critical phenomena at the phase transition for the actual systems with a fixed number of particles and yields only an asymptotics far outside the critical region. [ABSTRACT FROM AUTHOR]

This paper is the fourth in a series devoted to the development of a rigorous renormalisation group method for lattice field theories involving boson fields, fermion fields, or both. The third paper in the series presents a perturbative analysis of a supersymmetric field theory which represents the continuous-time weakly self-avoiding walk on $${{{\mathbb Z}}^d }$$ . We now present an analysis of the relevant interaction functional of the supersymmetric field theory, which permits a nonperturbative analysis to be carried out in the critical dimension $$d = 4$$ . The results in this paper include: proof of stability of the interaction, estimates which enable control of Gaussian expectations involving both boson and fermion fields, estimates which bound the errors in the perturbative analysis, and a crucial contraction estimate to handle irrelevant directions in the flow of the renormalisation group. These results are essential for the analysis of the general renormalisation group step in the fifth paper in the series. [ABSTRACT FROM AUTHOR]

This paper is the fifth in a series devoted to the development of a rigorous renormalisation group method applicable to lattice field theories containing boson and/or fermion fields, and comprises the core of the method. In the renormalisation group method, increasingly large scales are studied in a progressive manner, with an interaction parametrised by a field polynomial which evolves with the scale under the renormalisation group map. In our context, the progressive analysis is performed via a finite-range covariance decomposition. Perturbative calculations are used to track the flow of the coupling constants of the evolving polynomial, but on their own perturbative calculations are insufficient to control error terms and to obtain mathematically rigorous results. In this paper, we define an additional non-perturbative coordinate, which together with the flow of coupling constants defines the complete evolution of the renormalisation group map. We specify conditions under which the non-perturbative coordinate is contractive under a single renormalisation group step. Our framework is essentially combinatorial, but its implementation relies on analytic results developed earlier in the series of papers. The results of this paper are applied elsewhere to analyse the critical behaviour of the 4-dimensional continuous-time weakly self-avoiding walk and of the 4-dimensional $$n$$ -component $$|\varphi |^4$$ model. In particular, the existence of a logarithmic correction to mean-field scaling for the susceptibility can be proved for both models, together with other facts about critical exponents and critical behaviour. [ABSTRACT FROM AUTHOR]

In this paper, a theoretical method has been introduced for developing the crossover Peng-Robinson (CPR) equation of state (EoS) which incorporates the non-classical scaling laws asymptotically near the critical point into a classical analytic equation further away from the critical point. The CPR EoS has been adopted to describe the thermodynamic properties of some pure fluids (normal alkanes from methane to n-butane and carbon dioxide) such as density, saturated pressure, isochoric heat capacity and speed of sound. Unlike the original method for the crossover transformation made by Chen et al. (Phys Rev A 42:4470-4484, ), we have proposed a procedure which adding an additional term into the crossover transformation to obtain the thermophysical properties at the critical point more exactly. It is shown that this new crossover method yields a satisfactory representation of the thermodynamic properties close to the critical point for pure fluids relative to the original PR EoS. [ABSTRACT FROM AUTHOR]

Representing the fluid flow as a collection of coherent structures of various size, the statistical temperature of the flow state is determined as a function of the Reynolds number. It is shown that at small Reynolds numbers, associated with laminar states, the temperature is positive, while at large Reynolds numbers, associated with turbulent states, it is negative. At intermediate Reynolds numbers, the temperature changes from positive to negative as the size of the coherent structures increases, similar to what was predicted by Onsager for a system of parallel point-vortices in an inviscid fluid. It is also shown that in the range of intermediate Reynolds numbers the temperature exhibits a critical divergence. [ABSTRACT FROM AUTHOR]

We use the conformal bootstrap to perform a precision study of the operator spectrum of the critical 3d Ising model. We conjecture that the 3d Ising spectrum minimizes the central charge $$c$$ in the space of unitary solutions to crossing symmetry. Because extremal solutions to crossing symmetry are uniquely determined, we are able to precisely reconstruct the first several $$\mathbb {Z}_2$$ -even operator dimensions and their OPE coefficients. We observe that a sharp transition in the operator spectrum occurs at the 3d Ising dimension $$\Delta _\sigma = 0.518154(15)$$ , and find strong numerical evidence that operators decouple from the spectrum as one approaches the 3d Ising point. We compare this behavior to the analogous situation in 2d, where the disappearance of operators can be understood in terms of degenerate Virasoro representations. [ABSTRACT FROM AUTHOR]

We consider the $$n$$ -component $$|\varphi |^4$$ spin model on $${\mathbb {Z}}^4$$ , for all $$n \ge 1$$ , with small coupling constant. We prove that the susceptibility has a logarithmic correction to mean field scaling, with exponent $$\frac{n+2}{n+8}$$ for the logarithm. We also analyse the asymptotic behaviour of the pressure as the critical point is approached, and prove that the specific heat has fractional logarithmic scaling for $$n =1,2,3$$ ; double logarithmic scaling for $$n=4$$ ; and is bounded when $$n>4$$ . In addition, for the model defined on the $$4$$ -dimensional discrete torus, we prove that the scaling limit as the critical point is approached is a multiple of a Gaussian free field on the continuum torus, whereas, in the subcritical regime, the scaling limit is Gaussian white noise with intensity given by the susceptibility. The proofs are based on a rigorous renormalisation group method in the spirit of Wilson, developed in a companion series of papers to study the 4-dimensional weakly self-avoiding walk, and adapted here to the $$|\varphi |^4$$ model. [ABSTRACT FROM AUTHOR]

It is well know that systems with an interaction decaying as a power of the distance may have critical exponents that are different from those of short-range systems. The boundary between long-range and short-range is known, however the behavior in the crossover region is not well understood. In this paper we propose a general form for the crossover function and we compute it in a particular limit. We compare our predictions with the results of numerical simulations for two-dimensional long-range percolation. [ABSTRACT FROM AUTHOR]

The study of the statistical properties of random matrices of large size has a long history, dating back to Wigner, who suggested using Gaussian ensembles to give a statistical description of the spectrum of complex Hamiltonians and derived the famous semi-circle law, the work of Dyson and many others. Later, 't Hooft noticed that, in $$SU(N)$$ non-Abelian gauge theories, tessalated surfaces can be associated to Feynman diagrams and that the large $$N$$ expansion corresponds to an expansion in successive topologies. Following this observation, some times later, it was realized that some ensembles of random matrices in the large N expansion and the so-called double scaling limit could be used as toy models for quantum gravity: 2D quantum gravity coupled to conformal matter. This has resulted in a tremendous expansion of random matrix theory, tackled with increasingly sophisticated mathematical methods and number of matrix models have been solved exactly. However, the somewhat paradoxical situation is that either models can be solved exactly or little can be said. Since the solved models display critical points and universal properties, it is tempting to use renormalization group ideas to determine universal properties, without solving models explicitly. Initiated by Brézin and Zinn-Justin, the approach has led to encouraging results, first for matrix integrals and then quantum mechanics with matrices, but has not yet become a universal tool as initially hoped. In particular, general quantum field theories with matrix fields require more detailed investigations. To better understand some of the encountered difficulties, we first apply analogous ideas to the simpler $$O(N)$$ symmetric vector models, models that can be solved quite generally in the large $$N$$ limit. Unlike other attempts, our method is a close extension of Brézin and Zinn-Justin. Discussing vector and matrix models with similar approximation scheme, we notice that in all cases (vector and matrix integrals, vector and matrix path integrals in the local approximation), at leading order, non-trivial fixed points satisfy the same universal algebraic equation, and this is the main result of this work. However, its precise meaning and role have still to be better understood. [ABSTRACT FROM AUTHOR]

A personal and subjective recollection, concerning mainly Wilson's lectures delivered over the spring of 1972 at Princeton University (summary of a talk at Cornell University on November 16, 2013 at the occasion of the memorial Kenneth G. Wilson conference). [ABSTRACT FROM AUTHOR]

Kenneth Wilson had an enormous impact on the renormalization group and field theories in general. I had the great pleasure to work in three fields to which he contributed essentially: Critical phenomena, gauge-invariance in duality and confinement, and flow equations and similarity renormalization. [ABSTRACT FROM AUTHOR]

We study the influence of a correlated disorder on the localization phase transition in the pinning model (Random polymer models, ). When correlations are strong enough, an infinite disorder regime arises: large and frequent attractive regions appear in the environment. We present here a pinning model in random binary ( $$\{-1,1\}$$ -valued) environment. Defining infinite disorder via the requirement that the probability of the occurrence of a large attractive region is sub-exponential in its size, we prove that it coincides with the fact that the critical point is equal to its minimal possible value, namely $$h_c(\beta )=-\beta $$ . We also stress that in the infinite disorder regime, the phase transition is smoother than in the homogeneous case, whatever the critical exponent of the homogeneous model is: disorder is therefore always relevant. We illustrate these results with the example of an environment based on the sign of a Gaussian correlated sequence, in which we show that the phase transition is of infinite order in presence of infinite disorder. Our results contrast with results known in the literature, in particular in the case of an IID disorder, where the question of the influence of disorder on the critical properties is answered via the so-called Harris criterion, and where a conventional relevance/irrelevance picture holds. [ABSTRACT FROM AUTHOR]

The exponential family of random graphs is one of the most promising class of network models. Dependence between the random edges is defined through certain finite subgraphs, analogous to the use of potential energy to provide dependence between particle states in a grand canonical ensemble of statistical physics. By adjusting the specific values of these subgraph densities, one can analyze the influence of various local features on the global structure of the network. Loosely put, a phase transition occurs when a singularity arises in the limiting free energy density, as it is the generating function for the limiting expectations of all thermodynamic observables. We derive the full phase diagram for a large family of 3-parameter exponential random graph models with attraction and show that they all consist of a first order surface phase transition bordered by a second order critical curve. [ABSTRACT FROM AUTHOR]

We consider the percolation problem in the high-temperature Ising model on the two-dimensional square lattice at or near critical external fields. We show that all scaling relations, except for a single hyperscaling relation, hold under the power law assumptions for the one-arm path and the four-arm paths. [ABSTRACT FROM AUTHOR]

Random Hermitian matrices are used to model complex systems without time-reversal invariance. Adding an external source to the model can have the effect of shifting some of the matrix eigenvalues, which corresponds to shifting some of the energy levels of the physical system. We consider the case when the n× n external source matrix has two distinct real eigenvalues: a with multiplicity r and zero with multiplicity n− r. For a Gaussian potential, it was shown by Péché (Probab. Theory Relat. Fields 134:127-173, ) that when r is fixed or grows sufficiently slowly with n (a small-rank source), r eigenvalues are expected to exit the main bulk for | a| large enough. Furthermore, at the critical value of a when the outliers are at the edge of a band, the eigenvalues at the edge are described by the r-Airy kernel. We establish the universality of the r-Airy kernel for a general class of analytic potentials for $r=\mathcal{O}(n^{\gamma})$ for 0≤ γ<1/12. [ABSTRACT FROM AUTHOR]

We consider one dimensional systems of particles interacting with each other through long range interactions that are translation invariant. We seek quasi-periodic equilibrium states. Standard arguments show that if there are continuous families of quasi-periodic ground states, the system can have large scale motion, if the family of ground states is discontinuous, the system is pinned down. The transition between the two cases is called breakdown of analyticity and has been widely studied. We use recently developed fast and efficient algorithms to compute all the continuous families of ground states even close to the boundary of analyticity. We show that the boundary of analyticity can be computed by monitoring some appropriate norm of the computed solutions. We implemented these algorithms on several models. We found that there are regions where the boundary is smooth and the breakdown satisfies scaling relations. In other regions, the scalings seem to be interrupted and restart again. We present a renormalization group explanation of these phenomena. This suggest that the renormalization group may have some complicated global behavior. [ABSTRACT FROM AUTHOR]

One of the crowning achievements of mathematical, statistical physics over the past half century has been the discovery of the many aspects of structure of the critical point. It has been an exciting time and was only possible through the combined efforts of many excellent people. This article contains brief reviews of some of the parts in which I have been most interested and to which I have made some contributions. [ABSTRACT FROM AUTHOR]

We consider the 2D quenched–disordered q–state Potts ferromagnets and show that in the translation invariant measure, averaged over the disorder, at self–dual points any amalgamation of q−1 species will fail to percolate despite an overall (high) density of 1− q −1. Further, in the dilute bond version of these systems, if the system is just above threshold, then throughout the low temperature phase there is percolation of a single species despite a correspondingly small density. Finally, we demonstrate both phenomena in a single model by considering a “perturbation” of the dilute model that has a self–dual point. We also demonstrate that these phenomena occur, by a similar mechanism, in a simple coloring model invented by O. Häggström. [ABSTRACT FROM AUTHOR]

We consider bond percolation on $\{\backslash Bbb\; Z\}^d$ at the critical occupation density pc for d>6 in two different models. The first is the nearest-neighbor model in dimension d≫6. The second model is a “spread-out” model having long range parameterized by L in dimension d>6. In the spread-out case, we show that the cluster of the origin conditioned to contain the site x weakly converges to an infinite cluster as |x|→∞ when d>6 and L is sufficiently large. We also give a general criterion for this convergence to hold, which is satisfied in the case d≫6 in the nearest-neighbor model by work of Hara.(12) We further give a second construction, by taking p{\Bbb Q}^p and taking its limit as p↗p-c. The limiting object is the high-dimensional analogue of Kesten's incipient infinite cluster (IIC) in d=2. We also investigate properties of the IIC such as bounds on the growth rate of the cluster that show its four-dimensional nature. The proofs of both the existence and of the claimed properties of the IIC use the lace expansion. Finally, we give heuristics connecting the incipient infinite cluster to invasion percolation, and use this connection to support the well-known conjecture that for d>6 the probability for invasion percolation to reach a site x is asymptotic to c|x|-(d-4) as |x|→∞. [ABSTRACT FROM AUTHOR]

The article begins with some personal comments by the author on the outstanding contributions of Michael Fisher to statistical mechanics and critical phenomena. Its major aim is to trace the contributions of a number of pioneering personalities to the early history of equilibrium statistical mechanics. Four different areas are considered: (1) Classical Statistical Mechanics, (2) Quantum Statistical Mechanics, (3) Interacting Systems, and (4) The Ising Model. The article is concerned with the development and applications of statistical mechanics when certain basic assumptions are made. It does not deal with the justification of these assumptions which is a sophisticated discipline of its own. [ABSTRACT FROM AUTHOR]

The character of critical behavior in physical systems depends on the range of interactions. In the limit of infinite range of the interactions, systems will exhibit mean-field critical behavior, i.e., critical behavior not affected by fluctuations of the order parameter. If the interaction range is finite, the critical behavior asymptotically close to the critical point is determined by fluctuations and the actual critical behavior depends on the particular universality class. A variety of systems, including fluids and anisotropic ferromagnets, belongs to the three-dimensional Ising universality class. Recent numerical studies of Ising models with different interaction ranges have revealed a spectacular crossover between the asymptotic fluctuation-induced critical behavior and mean-field-type critical behavior. In this work, we compare these numerical results with a crossover Landau model based on renormalization-group matching. For this purpose we consider an application of the crossover Landau model to the three-dimensional Ising model without fitting to any adjustable parameters. The crossover behavior of the critical susceptibility and of the order parameter is analyzed over a broad range (ten orders) of the scaled distance to the critical temperature. The dependence of the coupling constant on the interaction range, governing the crossover critical behavior, is discussed. [ABSTRACT FROM AUTHOR]

Some time ago, Fisher and de Gennes pointed out that long-ranged correlations in a fluid close to its critical point Tc cause distinct effective forces between immersed colloidal particles which can even lead to flocculation [C. R. Acad. Sc. Paris B287:207 (1978)]. Here we calculate such forces between pairs of spherical particles as function of both relevant thermodynamic variables, i.e., the reduced temperature t=(T-Tc)/Tc and the field h conjugate to the order parameter. This provides the basis for specific predictions concerning the phase behavior of a suspension of colloidal particles in a near-critical solvent. [ABSTRACT FROM AUTHOR]

Typically, in order to obtain finite-size scaling laws for quantities in the microcanonical ensemble, an assumption is taken as a starting point. In this paper, consistency of such a Microcanonical Finite-Size Scaling Assumption with its commonly accepted canonical counterpart is shown, which puts Microcanonical Finite-Size Scaling on a firmer footing. [ABSTRACT FROM AUTHOR]

The partition function with boundary conditions for various two-dimensional Ising models is examined and previously unobserved properties of nonformal invariance and universality are established numerically. [ABSTRACT FROM AUTHOR]

The idea that near the critical point each block of spins behaves just like a single big spin is investigated. The case where a diamond-shaped block of spins is embedded in a (small) sea of spins is studied. Use is made of the Markov property method to make exact computations of the various spin moments needed to test this hypothesis. The residual fluctuation about the mean value of the block spin is seen to tend to a finite fraction of the length of the mean block-spin. This result is in line with previous studies which used different types of boundary conditions. [ABSTRACT FROM AUTHOR]

Within the massive field-theoretic renormalization-group approach the expressions for the β and γ functions of the anisotropic mn-vector model are obtained for general space dimension d in three-loop approximation. Resumming corresponding asymptotic series, critical exponents for the case of the weakly diluted quenched Ising model ( m = 1, n = 0), as well as estimates for the marginal order parameter component number mc of the weakly diluted quenched m-vector model, are calculated as functions of d in the region 2 ≤ d < 4. Conclusions concerning the effectiveness of different resummation techniques are drawn. [ABSTRACT FROM AUTHOR]

Linked cluster expansions are generalized from an infinite to a finite volume. They are performed to 20th order in the expansion parameter to approach the critical region from the symmetric phase. A new criterion is proposed to distinguish first- from second-order transitions within a finite-size scaling analysis. The criterion applies also to other methods for investigating the phase structure, such as Monte Carlo simulations. Our computational tools are illustrated with the example of scalar ( O(N) models with four- and six-point couplings for N=1 and N=4 in three dimensions. It is shown how to localize the tricritical line in these models. We indicate some further applications of our methods to the electroweak transition as well as to models for superconductivity. [ABSTRACT FROM AUTHOR]