Introduction to a Renormalisation Group Method
Abstract
This is a primer on a mathematically rigorous renormalisation group theory, presenting mathematical techniques fundamental to renormalisation group analysis such as Gaussian integration, perturbative renormalisation and the stable manifold theorem. It also provides an overview of fundamental models in statistical mechanics with critical behaviour, including the Ising and φ⁴ models and the self-avoiding walk.
The book begins with critical behaviour and its basic discussion in statistical mechanics models, and subsequently explores perturbative and non-perturbative analysis in the renormalisation group. Lastly it discusses the relation of these topics to the self-avoiding walk and supersymmetry.
Including exercises in each chapter to help readers deepen their understanding, it is a valuable resource for mathematicians and mathematical physicists wanting to learn renormalisation group theory.
... It is useful to equip these algebras with a suitable norm: this will make the estimates in Sections 3 and 4 rather simple and intuitive. Previous examples of the use of norms in the context of Grassmann integration can be found in [8,30,31]. Definition 2.1 A Grassmann algebra is a complex unital algebra whose generators anticommute. ...
... Even though our assumptions are somewhat weaker than, e.g., [8,25], the proof outlined in those references carries over and is therefore here omitted. A more geometrical perspective on this statement can be found in [12,13,51]. ...
... Proof of Proposition 4. 8 We shall skip many details since the proof is analogous to the one of Proposition 3.9. The superfunction F β is well-defined thanks to (H1) and satisfies IDB for any 0 ≤ β ≤ τ γ thanks to (H2-II) τ . ...
We study discrete random Schrödinger operators via the supersymmetric formalism. We develop a cluster expansion that converges at both strong and weak disorder. We prove the exponential decay of the disorder-averaged Green’s function and the smoothness of the local density of states either at weak disorder and at energies in proximity of the unperturbed spectrum or at strong disorder and at any energy. As an application, we establish Lifshitz-tail-type estimates for the local density of states and thus localization at weak disorder.
... The lack of regularity of typical fields drawn from the measure is related to that of the noise, which is inherited by solutions to the equation and makes the renormalization of nonlinear terms necessary. Finally, in statistical mechanics nonlinear Gibbs measures of the form (1.1) are believed to describe the universal behavior of large systems close to certain phase transitions [10,34,169,170] (at least for w = δ 0 ). In this context the leading behavior close to the transition is often captured by mean-field theory whereas fluctuations around it are properly captured by the classical Gibbs measure μ. ...
... In this context the leading behavior close to the transition is often captured by mean-field theory whereas fluctuations around it are properly captured by the classical Gibbs measure μ. In the Physics literature this has been predicted to happen for Bose-Einstein condensation [7,11,12,89,93] or for Berezinskii-Kosterlitz-Thouless transitions [18,69,90,91,132,133,154]. For the classical Ising model rigorous mathematical results in this spirit for equilibrium states can be found in [10,78,79,153], whereas works about the derivation of the dynamical equation (1.5) include [16,59,64,121]. ...
... Another difficulty is to achieve this using only the real parameter ν(λ) introduced above. This is really in the spirit of renormalization in Quantum Field Theory, as ini-tiated by Dyson in [51] and further developed within statistical physics using renormalization group techniques [10,44,167,170]. ...
We provide a rigorous derivation of nonlinear Gibbs measures in two and three space dimensions, starting from many-body quantum systems in thermal equilibrium. More precisely, we prove that the grand-canonical Gibbs state of a large bosonic quantum system converges to the Gibbs measure of a nonlinear Schrödinger-type classical field theory, in terms of partition functions and reduced density matrices. The Gibbs measure thus describes the behavior of the infinite Bose gas at criticality, that is, close to the phase transition to a Bose–Einstein condensate. The Gibbs measure is concentrated on singular distributions and has to be appropriately renormalized, while the quantum system is well defined without any renormalization. By tuning a single real parameter (the chemical potential), we obtain a counter-term for the diverging repulsive interactions which provides the desired Wick renormalization of the limit classical theory. The proof relies on a new estimate on the entropy relative to quasi-free states and a novel method to control quantum variances.
... On the side of the Renormalisation Group, hierarchical models are particularly convenient since they correspond to a dynamical system operating on one-variable functions. See [GK83] or the recent book [BBS19] or Sections 4 and 5 where this will play a key role. The building brick of hierarchical RG is the hierarchical GFF which replaces the classical Gaussian Free Field (GFF) in a hierarchical setting. ...
... We present both versions as the IR form is more commonly used to run the RG flow while the UV form is naturally embedded in the continuum space R d . See for example [BBS19,Hut24] for recent references on hierarchical systems in the infrared (IR) setting. ...
The goal of this paper is to prove singularity of three natural fields in QFT with respect to their natural base measure. The fields we consider are the following ones: (1) The near-critical limit of the 2d Ising model (in the -direction) is locally singular w.r.t the critical scaling limit of 2d Ising. (N.B. In the h-direction it is not locally singular). (2) The 2d Hierarchical Sine-Gordon field is singular w.r.t the 2d hierarchical Gaussian Free Field for all . (3) The Hierarchical field is singular w.r.t the 3d hierarchical GFF. Item (1) gives the first strong indication that the energy field of critical 2d Ising model does not exist as a random Schwarz distribution on the plane. Item (2) has been proved to be singular for the non-hierarchical 2d Sine-Gordon sufficiently far from the BKT point in [GM24] while item (3) is proved to be singular for the non-hierarchical 3d field in [BG21, OOT21, HKN24]. We believe our way to detect a singular behaviour at all scales is very much down to earth and may be applicable in all settings where one has a good enough control on the so-called effective potentials.
... In d ě 4 universality is proven in the context of the strictly related φ 4 models, see e.g. [6] and references therein, where it has been rigorously shown that the value of the exponents is equal to the mean field ones, e.g. the correlation length exponent is ν " 1{2 and the specific heat exponent α " p4´dq{2. We remark that, however, while in d ě 5 the behavior is exactly the same as in mean field, in d " 4 logarithmic corrections are present; the difference is that in the first case the interaction is irrelevant in the Renormalization Group sense, while in the second is marginal (or, more precisely, marginally irrelevant). ...
... kPDα ď Cγ 2h and we get |K phq φ,φ,ω,ω 1 px, yq| ď C|λ|γ h .(5.18) Repeating the analysis of[6], one has|x´y| N |K phq φ,φ,ω,ω 1 px, yq| ď C N |λ|γ´h N |λ|γ h (5.19)which in turns imply |K phq φ,φ,ω,ω 1 px, yq| ď |λ|γ h C N 1`γ hN |x´y| N .Back to (5.3), we get at β " β c S ω,ω 1 px, yq :" I φ´ω ,x ,φ´ω1 ,y " ...
We prove that in the 2d Ising Model with a weak bidimensional quasi-periodic disorder in the interaction, the critical behavior is the same as in the non-disordered case, that is the critical exponents are identical and no logarithmic corrections are present. The result establishes the validity of the prediction based on the Harris-Luck criterion and it provides the first rigorous proof of universality in the Ising model in presence of quasi-periodic disorder. The proof combines Renormalization Group approaches with direct methods used to deal with small divisors in KAM theory.
... These two functionals are employed as generators of a commutative algebra A whose composition is the pointwise product. The main rationale at the heart of [11] and inspired by the algebraic approach [5] is the following: The stochastic behavior codified by the white noise ξ can be encoded in A by deforming its product setting for all τ 1 , τ 2 ...
... Remark 3.5. We observe that the deformation map Γ C ·Q introduced above is similar to the inverse of the one used to define Wick ordering, see, e.g., [2,Sec.2]. ...
In a recent work Dappiaggi (Commun Contemp Math 24:2150075, 2022), a novel framework aimed at studying at a perturbative level a large class of nonlinear, scalar, real, stochastic PDEs has been developed and inspired by the algebraic approach to quantum field theory. The main advantage is the possibility of computing the expectation value and the correlation functions of the underlying solutions accounting for renormalization intrinsically and without resorting to any specific regularization scheme. In this work, we prove that it is possible to extend the range of applicability of this framework to cover also the stochastic nonlinear Schrödinger equation in which randomness is codified by an additive, Gaussian, complex white noise.
... In particular, the Constructive Renormalization Group approach has given a rigorous construction of non-trivial fixed points in different cases (see for example [1,29,30], and the introductions [2,14,71,44]). All these examples show that a weakly coupled RG can be non-perturbatively implemented, and a fixed point can be found without any approximation in a Banach space of interactions. ...
... The BV differential is defined as the sum of the Koszul map and the Chevalley-Eilenberg differential, (14) s BV = δ cl + γ ce + ... . ...
In a recent paper, with Drago and Pinamonti we have introduced a Wetterich-type flow equation for scalar fields on Lorentzian manifolds, using the algebraic approach to perturbative QFT. The equation governs the flow of the average effective action, under changes of a mass parameter k. Here we introduce an analogous flow equation for gauge theories, with the aid of the Batalin-Vilkovisky (BV) formalism. We also show that the corresponding average effective action satisfies a Slavnov-Taylor identity in Zinn-Justin form. We interpret the equation as a cohomological constraint on the functional form of the average effective action, and we show that it is consistent with the flow.
... It was pioneered in the work of Wilson [Wil71a,Wil71b], building on the work of Kadanoff [Kad66], where the idea of progressive integration of scales was first introduced for studying critical phenomena. Since then, the RG techniques have witnessed a very intense and diverse development, depending on whether they are used to study, e.g., Euclidean scalar measures [BCG+80, GK85b, FMRS87, BY90, Abd07, BBS19], Grassmann measures [GK85a,FMRS86,Les87,BG90,BGM92], lattice gauge theories [Bal87,Bal88,Dim20], supersymmetric measures [BBS15b,BBS15a,AFP20], or stochastic differential equations [Kup16,Duc21,Duc22] and depending on whether the separation of scales is done in a continuum or discrete fashion. Our list of references is by no means exhaustive and should only be understood as a sample for the interested reader. ...
... Let us begin by considering the operation of localisation and renormalisation for the monomials of degree k = 1. This operation is crucial for the study of the flow equation carried out in the next section and is well-known in the study of the renormalisation group flow, see, e.g., [Mas08,BBS19]. Ultimately, this operation is necessary because we want to renormalise the interacting measure by adding a suitable local quadratic term only. ...
Building on previous work on the stochastic analysis for Grassmann random variables, we introduce a forward-backward stochastic differential equation (FBSDE) which provides a stochastic quantisation of Grassmann measures. Our method is inspired by the so-called continuous renormalisation group, but avoids the technical difficulties encountered in the direct study of the flow equation for the effective potentials. As an application, we construct a family of weakly coupled subcritical Euclidean fermionic field theories and prove exponential decay of correlations.
... (Their notation is different to ours: their θ δ is 2θ δ in our notation and their θ β is −θ β in our notation.) See [8] and references therein for related rigorous results for weakly self-avoiding walk and the ϕ 4 model on hierarchical lattices. Note that Theorem 1.3 and its strengthened form stated in (1.10) imply that these exponents must satisfy θ β ≤ θ δ ≤ θ γ if they are well-defined. ...
... Proof of Lemma A. 8 We can compute that ...
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 is large. In contrast to earlier methods, our approach continues to yield bounds of reasonable order when the triangle diagram is unbounded but diverges slowly as , as is expected to occur in percolation on at the upper-critical dimension d=6. Indeed, we show in particular that if the triangle diagram diverges polylogarithmically as 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.
... We first consider (9). Now (6) shows that on A On A ...
We study the effective radius of weakly self-avoiding star polymers in one, two, and three dimensions. Our model includes N Brownian motions up to time T, started at the origin and subject to exponential penalization based on the amount of time they spend close to each other, or close to themselves. The effective radius measures the typical distance from the origin. Our main result gives estimates for the effective radius where in two and three dimensions we impose the restriction that . One of the highlights of our results is that in two dimensions, we find that the radius is proportional to , up to logarithmic corrections.
... The connection between the Ising model and the ϕ 4 model is anticipated to be highly intricate, with both models believed to fall within the same universality class. Renormalisation group arguments, as detailed in [Gri70,Kad93] and the recent book [BBS19], suggest that many of their properties, including critical exponents, precisely coincide at their respective critical points. Griffiths and Simon [SG73] laid the groundwork for these profound connections by demonstrating that the ϕ 4 model emerges as a specific near-critical scaling limit of a collection of mean-field Ising models. ...
We study the nearest-neighbour Ising and models on with and obtain new lower bounds on their two-point functions at (and near) criticality. Together with the classical infrared bound, these bounds turn into up to constant estimates when . When d=4, we obtain an “almost” sharp lower bound corrected by a logarithmic factor. As a consequence of these results, we show that and when , where is the critical exponent associated with the decay of the model’s two-point function at criticality and is the critical exponent of the correlation length . When d=3, we improve previous results and obtain that . As a byproduct of our proofs, we also derive the blow-up at criticality of the so-called bubble diagram when d=3,4.
... In the critical phase, continuity of the phase transition in dimensions d ≥ 3 [GPPS22], triviality of the critical and near-critical scaling limits in d ≥ 4 [Aiz82,Frö82,AD21,Pan23], and bounds on the (near-)critical two-point function in dimensions d ≥ 3 [FSS76,FILS78,Sak15,DP24] have been established. Furthermore, the so-called weakly-coupled φ 4 model (which corresponds to choosing g small enough in (1.1)) has been extensively studied in the literature through the renormalization group approach [GK85,Har87,FMRS87,BBS14,ST16,BBS19]. The model's critical behaviour in dimension d = 2, weakly-coupled or not, remains a fascinating challenge. ...
In this article, we analyse the model on in the supercritical regime . We consider a random cluster representation of the model, which corresponds to an Ising random cluster model on a random environment. We prove that the supercritical phase of this percolation model on () is well behaved in the sense that for every , local uniqueness of macroscopic clusters happens with high probability, uniformly in boundary conditions. This result provides the basis for renormalisation techniques used to study several fine properties of the supercritical phase. As an application, we obtain surface order large deviation bounds for the empirical magnetisation of the model in the entire supercritical regime.
... As demonstrated in many special cases, hierarchical models exhibit phase transitions quite analogous to their relatives on the finite-dimensional lattices Z d (cf. [5,6,12] and references therein). In fact, one of the key features of hierarchical models is an effective dimension which is a tunable parameter. ...
We apply Feshbach-Krein-Schur renormalization techniques in the hierarchical Anderson model to establish a criterion on the single-site distribution which ensures exponential dynamical localization as well as positive inverse participation ratios and Poisson statistics of eigenvalues. Our criterion applies to all cases of exponentially decaying hierarchical hopping strengths and holds even for spectral dimension , which corresponds to the regime of transience of the underlying hierarchical random walk. This challenges recent numerical findings that the spectral dimension is significant as far as the Anderson transition is concerned.
... The regime d > d c forms the mean-field regime of a model. Prominent techniques such as the lace expansion [BS85] and the rigorous renormalisation group method [BBS14,BBS15a,BBS15b,BBS19] have been developed to analyse the mean-field regime. However, a drawback of these approaches is their predominantly perturbative nature, necessitating the identification of a small parameter within the model. ...
This article proposes a new way of deriving mean-field exponents for sufficiently spread-out Bernoulli percolation in dimensions . We obtain an upper bound for the full-space and half-space two-point functions in the critical and near-critical regimes. In a companion paper, we apply a similar analysis to the study of the weakly self-avoiding walk model in dimensions .
... The regime d > d c is called the mean-field regime of a model. Noteworthy methods such as the lace expansion [BS85] or the rigorous renormalisation group method [BBS14,BBS15a,BBS15b,BBS19] have emerged to carry out the analysis of the mean-field regime. However, a limitation of these methods lies in their predominantly perturbative nature, which is reflected in the necessity of exhibiting a small parameter in the model. ...
This article proposes a new way of deriving mean-field exponents for the weakly self-avoiding walk model in dimensions . Among other results, we obtain up-to-constant estimates for the full-space and half-space two-point functions in the critical and near-critical regimes. A companion paper proposes a similar analysis for spread-out Bernoulli percolation in dimensions .
... One can perturb this model with finite-ranged or higher spin interactions, or consider it on different lattices, and ask what happens to the critical behavior. In d ≥ 4, universality is proven in the context of the closely related φ 4 models (see, e.g., [8] and references therein), where it has been rigorously shown that the values of the exponents are equal to the mean-field ones, e.g., the correlation length exponent is ν = 1/2 and the specific heat exponent α = (4 − d)/2. We remark, however, that while in d ≥ 5 the behavior is exactly the same as in the mean-field theory, in d = 4 logarithmic corrections are present; the difference is that in the first case the interaction is irrelevant in the Renormalization Group sense, while in the second it is marginal (or, more precisely, marginally irrelevant). ...
We prove that in the 2D Ising model with a weak bidimensional quasi-periodic disorder in the interaction, the critical behavior is the same as in the non-disordered case; that is, the critical exponents for the specific heat and energy-energy correlations are identical, and no logarithmic corrections are present. The disorder produces a quasi-periodic modulation of the amplitude of the correlations and a renormalization of the velocities, that is, the coefficients of the rescaling of positions, and of the critical temperature. The result establishes the validity of the prediction based on the Harris–Luck criterion, and it provides the first rigorous proof of universality in the Ising model in the presence of quasi-periodic disorder in both directions and for any angle. Small divisors are controlled assuming a Diophantine condition on the frequencies, and the convergence of the series is proved by Renormalization Group analysis.
... Wilson and contemporaries addressed this by the Wilsonian regularization, i.e. considering Feynman functional integral on a smaller subspace, namely on ultraviolet (UV) damped fields. Since such a subspace is obtained via coarse-graining, i.e. local averaging of fields, physicswise it is natural to require instances with subsequent coarse-grainings to be compatible with each-other, thus the notion of Wilsonian renormalization group (RG) emerged [17][18][19][20][21][22][23][24][25][26]. A Feynman measure instance with a given UV regularization is linked to a stronger UV regularized instance by 'integrating out' high frequency modes in between, called to be the Wilsonian renormalization group equation (RGE). ...
In nonperturbative formulation of quantum field theory, the vacuum state is characterized by the Wilsonian renormalization group (RG) flow of Feynman type field correlators. Such a flow is a parametric family of ultraviolet (UV) regularized field correlators, the parameter being the strength of the UV regularization, and the instances with different strength of UV regularizations are linked by the renormalization group equation. Important RG flows are those which reach out to any UV regularization strengths. In this paper it is shown that for these flows a natural, mathematically rigorous generally covariant definition can be given, and that they form a topological vector space which is Hausdorff, locally convex, complete, nuclear, semi-Montel, Schwartz. That is, they form a generalized function space having favorable properties, similar to multivariate distributions. The other theorem proved in the paper is that for Wilsonian RG flows reaching out to all UV regularization strengths, a simple factorization formula holds in case of bosonic fields over flat (affine) spacetime: the flow always originates from a regularization-independent distributional correlator, and its running satisfies an algebraic ansatz. The conjecture is that this factorization theorem should generically hold, which is worth future investigations.
... For more details, see [18] and follow-up work. We use the notation of the Gaussian integral as a differential operator (see [38] for a detailed discussion of this notation), as a convenient bookkeeping device for the action of derivatives with respect to matrix elements of the covariance of a Gaussian measure and we denote V (σ) = ln(1−ı g/3 σ). Equation (3.23) becomes with this notation: where δ/δσ denotes the derivative with respect to σ. ...
We consider the zero-dimensional quartic O(N) vector model and present a complete study of the partition function Z(g, N) and its logarithm, the free energy W(g, N), seen as functions of the coupling g on a Riemann surface. We are, in particular, interested in the study of the transseries expansions of these quantities. The point of this paper is to recover such results using constructive field theory techniques with the aim to use them in the future for a rigorous analysis of resurgence in genuine quantum field theoretical models in higher dimensions. Using constructive field theory techniques, we prove that both Z(g, N) and W(g, N) are Borel summable functions along all the rays in the cut complex plane . We recover the transseries expansion of Z(g, N) using the intermediate field representation. We furthermore study the small-N expansions of Z(g, N) and W(g, N). For any on the sector of the Riemann surface with , the small-N expansion of Z(g, N) has infinite radius of convergence in N, while the expansion of W(g, N) has a finite radius of convergence in N for g in a subdomain of the same sector. The Taylor coefficients of these expansions, and , exhibit analytic properties similar to Z(g, N) and W(g, N) and have transseries expansions. The transseries expansion of is readily accessible: much like Z(g, N), for any n, has a zero- and a one-instanton contribution. The transseries of is obtained using Möbius inversion, and summing these transseries yields the transseries expansion of W(g, N). The transseries of and W(g, N) are markedly different: while W(g, N) displays contributions from arbitrarily many multi-instantons, exhibits contributions of only up to n-instanton sectors.
... Here the φ are fields on a D-dimensional spacetime, Dφ is a functional integration, the interaction potential V is a polynomial in the fields, K is the propagator, and Z 0 is a normalization. Such formulations allow expressing Gaussian expectation values and Wick's formula, the key ingredient of perturbative expansions, as well-defined algebraic expressions that do not suffer the potential divergences of the combinatorially equivalent integrals (see for instance [146][147][148][149]). They moreover provide a practical tool for computing Feynman diagrams. ...
This thesis investigates low-dimensional models of nonperturbative quantum gravity, with a special focus on Causal Dynamical Triangulations (CDT). We define the so-called curvature profile, a new quantum gravitational observable based on the quantum Ricci curvature. We subsequently study its coarse-graining capabilities on a class of regular, two-dimensional polygons with isolated curvature singularities, and we determine the curvature profile of (1+1)-dimensional CDT with toroidal topology. Next, we focus on CDT in 2+1 dimensions, intvestigating the behavior of the two-dimensional spatial slice geometries. We then turn our attention to matrix models, exploring a differential reformulation of the integrals over one- and two-matrix ensembles. Finally, we provide a hands-on introduction to computer simulations of CDT quantum gravity.
... The relationship between ϕ 4 and the Ising model is predicted to be very deep: they are a fundamental example of two models that are believed to belong to the same universality class. Indeed, it is predicted from renormalisation group heuristics (see [Gri70,Kad93] or the recent book [BBS19]) that at their respective critical points many properties of these two models, such as critical exponents, exactly coincide. A manifestation of this was discovered by Griffiths and Simon in [GS73], where they show that the ϕ 4 model arises as a certain near-critical scaling limit of mean-field Ising models. ...
We prove that the set of automorphism invariant Gibbs measures for the model on graphs of polynomial growth has at most two extremal measures at all values of . We also give a sufficient condition to ensure that the set of all Gibbs measures is a singleton. As an application, we show that the spontaneous magnetisation of the nearest-neighbour model on vanishes at criticality for . The analogous results were established for the Ising model in the seminal works of Aizenman, Duminil-Copin, and Sidoravicius (Comm. Math. Phys., 2015), and Raoufi (Ann. Prob., 2020) using the so-called random current representation introduced by Aizenman (Comm. Math. Phys., 1982). One of the main contributions of this paper is the development of a corresponding geometric representation for the model called the random tangled current representation.
... For more details, see [17] and followup work. We use the notation of the Gaussian integral as a differential operator (see [31] for a detailed discussion of this notation), as a convenient bookkeeping device for the action of derivatives with respect to matrix elements of the covariance of a Gaussian measure and we denote V (σ) = ln(1 − ı g/3 σ). Eq. (3.23) becomes with this notation: ...
We consider the 0-dimensional quartic O(N) vector model and present a complete study of the partition function Z(g,N) and its logarithm, the free energy W(g,N), seen as functions of the coupling g on a Riemann surface. Using constructive field theory techniques we prove that both Z(g,N) and W(g,N) are Borel summable functions along all the rays in the cut complex plane . We recover the transseries expansion of Z(g,N) using the intermediate field representation. We furthermore study the small-N expansions of Z(g,N) and W(g,N). For any on the sector of the Riemann surface with , the small-N expansion of Z(g,N) has infinite radius of convergence in N while the expansion of W(g,N) has a finite radius of convergence in N for g in a subdomain of the same sector. The Taylor coefficients of these expansions, and , exhibit analytic properties similar to Z(g,N) and W(g,N) and have transseries expansions. The transseries expansion of is readily accessible: much like Z(g,N), for any n, has a zero- and a one-instanton contribution. The transseries of is obtained using M\"oebius inversion and summing these transseries yields the transseries expansion of W(g,N). The transseries of and W(g,N) are markedly different: while W(g,N) displays contributions from arbitrarily many multi-instantons, exhibits contributions of only up to n-instanton sectors.
... In dimensions d ≥ 5, one expects that the model is trivial throughout the continuous phase transition regime -c.f. the case of Ising, where triviality was shown by Aizenman [Aiz82] and Fröhlich [Frö82]. We also refer to the review book [BBS19] for an account of renormalisation group approaches to this problem. In d = 4, there may be some distinction: for ∆ > ∆ tric , one expects a marginal triviality result as in the case for Ising, which was recently shown by Aizenman and Duminil-Copin [ADC21], whereas at ∆ tric it is unclear whether logarithmic corrections are present. ...
We prove the existence of a tricritical point for the Blume-Capel model on for every . The proof in relies on a novel combinatorial mapping to an Ising model on a larger graph, the techniques of Aizenman, Duminil-Copin, and Sidoravicious (Comm. Math. Phys, 2015), and the celebrated infrared bound. In d=2, the proof relies on a quantitative analysis of crossing probabilities of the dilute random cluster representation of the Blume-Capel. In particular, we develop a quadrichotomy result in the spirit of Duminil-Copin and Tassion (Moscow Math. J., 2020), which allows us to obtain a fine picture of the phase diagram in d=2, including asymptotic behaviour of correlations in all regions. Finally, we show that the techniques used to establish subcritical sharpness for the dilute random cluster model extend to any .
... The subject of critical phenomena and phase transitions has fascinated mathematicians for over half a century; see [1]. In the physics literatures, critical phenomena are understood via the renormalisation group(RG) method. ...
Inspired by the idea of stochastic quantization proposed by Parisi and Wu, we construct the transition probability matrix which plays a central role in the renormalization group through a stochastic differential equation. By establishing the discrete time stochastic dynamics, the renormalization procedure can be characterized from the perspective of probability. Hence, we will focus on the investigation of the infinite dimensional stochastic dynamic. From the stochastic point of view, the discrete time stochastic dynamic can induce a Markov chain. Via calculating the square field operator and the Bakry-\'Emery curvature for a class of two-points functions, the local Poincar\'e inequality is established, from which the estimate of correlation functions can also be obtained. Finally, under the condition of ergodicity, by choosing the couple relationship between the system parameter K and the system time T properly when , the two-points correlation functions for limit system are also estimated.
... In this section, we illustrate Theorem 2 on the general ϕ 4 models (see [5,Chapter 1]) for a more detailed introduction to this model from Quantum Field Theory). Let Λ be a finite set, A = (A x,y ) x,y∈Λ be a symmetric positive definite matrix, and let g > 0, ν ∈ R and h ∈ R Λ be constants. ...
We control the behavior of the Poincar{\'e} constant along the Polchinski renormalization flow using a dynamic version of -calculus. We also treat the case of higher order eigenvalues. Our method generalizes a method introduced by B. Klartag and E. Putterman to analyze the evolution of log-concave distributions along the heat flow. Furthermore, we apply it to general 4-measures and discuss the interpretation in terms of transport maps.
... He chooses their next walk with equal probability and moves forward or backward at each step. Based on the random selection, and after a few steps, the walker's path constructs a random walk called a simple random walk [126]. The random walk has a specific application in modeling the behavior of random processes. ...
Self-adaptive systems provide the ability of autonomous decision-making for handling the changes affecting the functionalities of cyber-physical systems. A self-adaptive system repeatedly monitors and analyzes the local system and the environment and makes significant decisions regarding fulfilling the system's functional optimization and safety requirements. Such a decision must be made before a deadline, and the autonomy helps the system meet the timing constraints. Suppose the model of the cyber-physical system is available. In that case, it can be used for verification against specific formal properties to reveal whether the system is committed to the properties or not. However, according to the dynamicity of such systems, the system model needs to be reconstructed and reverified at runtime. As the model of the self-adaptive systems is a composition of the local system and the environment models, the size of the composed model is relatively large. Therefore, we need efficient and scalable methods to verify the model at runtime in resource-constrained systems. Since the physical environment and the cyber part of the system usually have stochastic natures, the reflection of each behavior is modeled through probabilistic parameters, which we have some predictions about them. If the system observes or predicts some changes in the behavior of the environment or the local system, the parameter(s) are updated. This research focuses on the problem of runtime model size reduction in self-adaptive systems. As a solution, the model is partitioned into sub-models that can be verified/approximated independently. At runtime, if a change occurs, only the affected sub-models are subject to re-verification/re-approximation. Finally, with the help of an aggregation algorithm, the partial results from the sub-models are composed, and the verification result for the whole model is calculated. In some situations, updating the model may cause some delays in the decision-making. The self-adaptive system must decide about an incomplete model when a few parameters have been missed to meet the decision-making deadlines. We do this by conducting a set of behavioral simulations by random walk and matching the system's current behavior with its previous behavioral patterns. Thus, the system is equipped with a runtime parameter estimation method respecting a certain upper bound of errors. This thesis proposes a new metric for determining an upper bound of errors caused by applying the approximation technique. The metric is the basis for two proposed theorems that guarantee upper bounds of errors and accuracy of runtime verification. The evaluation results confirm that the proposed approximation framework reduces the model's size and helps decision-making within the time restrictions. The framework keeps the accuracy of the parameter estimations and verification results upper than 96.5% and 95%, respectively, while fully guaranteeing the system's safety.
... Now using Gaussian integration by parts (see [11] Exercise 2.1.3) ...
We study the Sine-Gordon model for in infinite volume. We give a variatonal characterization of it's laplace transform, and deduce from this large deviations. Along the way we obtain estimates which are strong enough to obtain a proof of the Osterwalder-Schrader axioms including exponential decay of correlations as a byproduct. Our method is based on the Boue-Dupuis formula with an emphasis on the stochastic control structure of the problem.
... Here the φ are fields on a D-dimensional spacetime, Dφ is a functional integration, the interaction potential V is a polynomial in the fields, K is the propagator, and Z 0 is a normalization. Such formulations allow expressing Gaussian expectation values and Wick's formula, the key ingredient of perturbative expansions, as well-defined algebraic expressions that do not suffer the potential divergences of the combinatorially equivalent integrals (see for instance [12][13][14][15]). They moreover provide a practical tool for computations of Feynman diagrams. ...
Differential reformulations of field theories are often used for explicit computations. We derive a one-matrix differential formulation of two-matrix models, with the help of which it is possible to diagonalize the one- and two-matrix models using a formula by Itzykson and Zuber that allows diagonalizing differential operators with respect to matrix elements of Hermitian matrices. We detail the equivalence between the expressions obtained by diagonalizing the partition function in differential or integral formulation, which is not manifest at first glance. For one-matrix models, this requires transforming certain derivatives to variables. In the case of two-matrix models, the same computation leads to a new determinant formulation of the partition function, and we discuss potential applications to new orthogonal polynomials methods.
We analyse and clarify the finite‐size scaling of the weakly‐coupled hierarchical ‐component model for all integers in all dimensions , for both free and periodic boundary conditions. For , we prove that for a volume of size with periodic boundary conditions the infinite‐volume critical point is an effective finite‐volume critical point, whereas for free boundary conditions the effective critical point is shifted smaller by an amount of order . For both boundary conditions, the average field has the same non‐Gaussian limit within a critical window of width around the effective critical point, and in that window we compute the universal scaling profile for the susceptibility. In contrast, and again for both boundary conditions, the average field has a massive Gaussian limit when above the effective critical point by an amount . In particular, at the infinite‐volume critical point the susceptibility scales as for periodic boundary conditions and as for free boundary conditions. We identify a mass generation mechanism for free boundary conditions that is responsible for this distinction and which we believe has wider validity, in particular to Euclidean (non‐hierarchical) models on in dimensions . For we prove a similar picture with logarithmic corrections. Our analysis is based on the rigorous renormalisation group method of Bauerschmidt, Brydges and Slade, which we improve and extend.
We obtain precise plateau estimates for the two-point function of the finite-volume weakly coupled hierarchical model in dimensions , for both free and periodic boundary conditions and for any number of components of the field . We prove that, within a critical window around their respective effective critical points, the two-point functions for both free and periodic boundary conditions have a plateau, in the sense that they decay as until reaching a constant plateau value of order (with a logarithmic correction for d=4), where V is the size of the finite volume. The two critical windows for free and periodic boundary conditions do not overlap. The dependence of the plateau height on the location within the critical window is governed by an explicit n-dependent universal profile which is independent of the dimension. The proof is based on a rigorous renormalisation group method and extends the method used by Michta et al. (arXiv:2306.00896) to study the finite-volume susceptibility and related quantities. Our results lead to precise conjectures concerning Euclidean (non-hierarchical) models of spin systems and self-avoiding walk in dimensions .
Inspired by the idea of stochastic quantization proposed by Parisi and Wu, we reconstruct the transition probability function that has a central role in the renormalization group using a stochastic differential equation. From a probabilistic perspective, the renormalization procedure can be characterized by a discrete-time Markov chain. Therefore, we focus on this stochastic dynamic, and establish the local Poincaré inequality by calculating the Bakry-Émery curvature for two point functions. Finally, we choose an appropriate coupling relationship between parameters K and T to obtain the Poincaré inequality of two point functions for the limiting system. Our method extends the classic Bakry-Émery criterion, and the results provide a new perspective to characterize the renormalization procedure.
In a recent paper, with Drago and Pinamonti we have introduced a Wetterich-type flow equation for scalar fields on Lorentzian manifolds, using the algebraic approach to perturbative QFT. The equation governs the flow of the effective average action, under changes of a mass parameter k. Here we introduce an analogous flow equation for gauge theories, with the aid of the Batalin–Vilkovisky (BV) formalism. We also show that the corresponding effective average action satisfies a Slavnov–Taylor identity in Zinn-Justin form. We interpret the equation as a cohomological constraint on the functional form of the effective average action, and we show that it is consistent with the flow.
We consider long‐range Bernoulli bond percolation on the dd‐dimensional hierarchical lattice in which each pair of points xx and yy are connected by an edge with probability 1−exp(−β∥x−y∥−d−α), where 0<α<d is fixed and β⩾0 is a parameter. We study the volume of clusters in this model at its critical point β=βc, proving precise estimates on the moments of all orders of the volume of the cluster of the origin inside a box. We apply these estimates to prove up‐to‐constants estimates on the tail of the volume of the cluster of the origin, denoted as KK, at criticality, namely, Pβc(|K|⩾n)≍n−(d−α)/(d+α)d<3αn−1/2(logn)1/4d=3αn−1/2d>3α.\begin{equation*} \mathbb {P}_{\beta _c}(|K|\geqslant n) \asymp {{\left\lbrace \def\eqcellsep{&}\begin{array}{ll}n^{-(d-\alpha)/(d+\alpha)} & d < 3\alpha \\[3pt] n^{-1/2}(\log n)^{1/4} & d=3\alpha \\[3pt] n^{-1/2} & d>3\alpha . \end{array} \right.}} \end{equation*}In particular, we compute the critical exponent δ to be (d+α)/(d−α) when dd is below the upper‐critical dimension dc=3α and establish the precise order of polylogarithmic corrections to scaling at the upper‐critical dimension itself. Our work also lays the foundations for the study of the scaling limit of the model: In the high‐dimensional case d⩾3α, we prove that the sized‐biased distribution of the volume of the cluster of the origin inside a box converges under suitable normalization to a chi‐squared random variable, while in the low‐dimensional case d<3α, we prove that the suitably normalized decreasing list of cluster sizes in a box is tight in ℓp∖{0} if and only if p>2d/(d+α).
The arboreal gas is the probability measure on (unrooted spanning) forests of a graph in which each forest is weighted by a factor β > 0 per edge. It arises as the q → 0 limit of the q q -state random cluster model with p = β q . We prove that in dimensions d ⩾ 3 the arboreal gas undergoes a percolation phase transition. This contrasts with the case of d=2 d = 2 where no percolation transition occurs.
The starting point for our analysis is an exact relationship between the arboreal gas and a non-linear sigma model with target space the fermionic hyperbolic plane H 0 | 2 . This latter model can be thought of as the 0-state Potts model, with the arboreal gas being its random cluster representation. Unlike the standard Potts models, the H 0 | 2 model has continuous symmetries. By combining a renormalisation group analysis with Ward identities we prove that this symmetry is spontaneously broken at low temperatures. In terms of the arboreal gas, this symmetry breaking translates into the existence of infinite trees in the thermodynamic limit. Our analysis also establishes massless free field correlations at low temperatures and the existence of a macroscopic tree on finite tori.
The goal of the present paper is to establish a framework which allows to rigorously determine the large-scale Gaussian fluctuations for a class of singular SPDEs at and above criticality, and therefore beyond the range of applicability of pathwise techniques, such as the theory of Regularity Structures. To this purpose, we focus on a d-dimensional generalization of the Stochastic Burgers equation (SBE) introduced in van Beijeren et al. (Phys Rev Lett 54(18):2026–2029, 1985. https://doi.org/10.1103/PhysRevLett.54.2026). In both the critical d=2d=2 and super-critical d≥3 cases, we show that the scaling limit of (the regularised) SBE is given by a stochastic heat equation with non-trivially renormalised coefficient, introducing a set of tools that we expect to be applicable more widely. For d≥3 the scaling adopted is the classical diffusive one, while in d=2d=2 it is the so-called weak coupling scaling which corresponds to tuning down the strength of the interaction in a scale-dependent way.
For general ferromagnetic Ising models whose coupling matrix has bounded spectral radius, we show that the log‐Sobolev constant satisfies a simple bound expressed only in terms of the susceptibility of the model. This bound implies very generally that the log‐Sobolev constant is uniform in the system size up to the critical point (including on lattices), without using any mixing conditions. Moreover, if the susceptibility satisfies the mean‐field bound as the critical point is approached, our bound implies that the log‐Sobolev constant depends polynomially on the distance to the critical point and on the volume. In particular, this applies to the Ising model on subsets of when .
The proof uses a general criterion for the log‐Sobolev inequality in terms of the Polchinski (renormalisation group) equation, a recently proved remarkable correlation inequality for Ising models with general external fields, the Perron–Frobenius theorem, and the log‐Sobolev inequality for product Bernoulli measures.
We control the behavior of the Poincaré constant along the Polchinski renormalization flow using a dynamic version of [Formula: see text]-calculus. We also treat the case of higher order eigenvalues. Our method generalizes a method introduced by Klartag and Putterman to analyze the evolution of log-concave distributions along the heat flow. Furthermore, we apply it to general [Formula: see text]-measures and discuss the interpretation in terms of transport maps.
In this study, the flow characteristics over modified semi-cylindrical weirs (MSCM) were investigated using computational fluid dynamics (CFD) simulations. The simulations included the analysis of flow velocity and pressure distribution, turbulence intensity, and streamline patterns. The numerical results were compared to laboratory observations, and a good agreement was observed. Different turbulence models, including renormalized group (RNG) k-ε, standard k-ε, k-ω two-equations, and large eddy simulation (LES), were evaluated, and all showed suitable performance in simulating the flow field and hydraulic characteristics. However, the standard k-ε model outperformed the other models. The analysis of streamline patterns from upstream to downstream of the weirs showed that the curve of the crest provided an opportunity for the flow to harmonize with the surface of the crest. The downstream ramp guided the streamlines, preventing collision with the downstream slab surface and vorticity formation. Additionally, the results indicated that adding the downstream ramp significantly reduced the turbulence intensity of the flow at the toe of the weirs. The analysis of pressure distribution showed that the flow departed from hydrostatic state when reaching the weir, and the maximum deviation occurred at the crest vertex.
Differential reformulations of field theories are often used for explicit computations. We derive a one-matrix differential formulation of two-matrix models, with the help of which it is possible to diagonalize the one- and two-matrix models using a formula by Itzykson and Zuber that allows diagonalizing differential operators with respect to matrix elements of Hermitian matrices. We detail the equivalence between the expressions obtained by diagonalizing the partition function in differential or integral formulation, which is not manifest at first glance. For one-matrix models, this requires transforming certain derivatives to variables. In the case of two-matrix models, the same computation leads to a new determinant formulation of the partition function, and we discuss potential applications to new orthogonal polynomials methods.
The flow equations of the renormalisation group permit to analyse rigorously the perturbative n-point functions of renormalisable quantum field theories including gauge theories. In this paper, we want to do a step towards a rigorous nonperturbative analysis of the flow equations (FEs). We restrict to massive scalar (one-component) fields and analyse a mean field limit where the Schwinger functions are considered to be momentum independent and thus are replaced by their zero momentum values. We analyse smooth solutions of the system of FEs for the n-point functions for different sets of boundary conditions. We will realise that allowing for nonvanishing irrelevant terms permits to construct asymptotically free nontrivial smooth solutions of the scalar field mean field FEs.
We derive a continuous-time lace expansion for a broad class of self-interacting continuous-time random walks. Our expansion applies when the self-interaction is a sufficiently nice function of the local time of a continuous-time random walk. As a special case we obtain a continuous-time lace expansion for a class of spin systems that admit continuous-time random walk representations.
We apply our lace expansion to the n-component model on when n=1,2, and prove that the critical Green's function is asymptotically a multiple of when and the coupling is weak. As another application of our method, we establish the analogous result for the lattice Edwards model at weak coupling. © 2021 Wiley Periodicals LLC.
In this review are summarized about 20 years of theoretical research with applications in the field of many-body physics for strongly correlated fermions with Rowe’s equation of motion (R-EOM) method and extended RPA equations. One major goal is to set up, via EOM, RPA equations with a correlated ground state. Since the correlations depend on the RPA amplitudes, it follows that RPA becomes a selfconsistency problem which is called Self-Consistent RPA (SCRPA). This then also improves very much the Pauli principle violated with standard RPA. The method was successfully applied to several non trivial problems, like the nuclear pairing Hamiltonian in the particle–particle channel (pp-RPA) and the Hubbard model of condensed matter. The SCRPA has several nice properites, as for instance, it can be formulated in such a way that all very appreciated qualities of standard RPA as, e.g., appearance of zero (Goldstone) modes in the case of broken symmetries, conservation laws, Ward identities, etc. are maintained. For the Goldstone mode an explicit example of a model case is presented. The formalism has its sound theoretical basis in the fact that an extension of the usual RPA operator has been found which exactly annihilates the Coupled Cluster Doubles (CCD) ground state wave function. This has been a longstanding problem for all RPA practioners from the beginning. There exists a rather simplified version of SCRPA which is the so-called renormalized RPA (r-RPA) where only the correlated occupation numbers are involved in the selfconsistent cycle. Because its numerical solution is rather similar to standard RPA, it has known quite a number of applications, like beta and double beta decays, which are reviewed in this article. In this review also an extended version of second RPA (ERPA)is described. This ERPA maintains all appreciable properties of standard RPA. Several realistic applications for, e.g., the damping of giant resonances are presented. Another important aspect of the extended RPA-theories is that it can be formulated symmetry conserving replacing and eventually improving symmetry (e.g., number) projected mean field approaches. A practical application is given in the review. The EOM formalism allows to treat alpha clustering and alpha particle (quartet) condensation in nuclear matter. This EOM approach succeeded where earlier attempts have failed. It is, for instance, shown how the critical temperature for alpha particle condensation in symmetric and asymmetric nuclear matter can efficiently be calculated.
A bstract
Much of our understanding of critical phenomena is based on the notion of Renormalization Group (RG), but the actual determination of its fixed points is usually based on approximations and truncations, and predictions of physical quantities are often of limited accuracy. The RG fixed points can be however given a fully rigorous and non- perturbative characterization, and this is what is presented here in a model of symplectic fermions with a nonlocal (“long-range”) kinetic term depending on a parameter ε and a quartic interaction. We identify the Banach space of interactions, which the fixed point belongs to, and we determine it via a convergent approximation scheme. The Banach space is not limited to relevant interactions, but it contains all possible irrelevant terms with short-ranged kernels, decaying like a stretched exponential at large distances. As the model shares a number of features in common with ϕ ⁴ or Ising models, the result can be used as a benchmark to test the validity of truncations and approximations in RG studies. The analysis is based on results coming from Constructive RG to which we provide a tutorial and self-contained introduction. In addition, we prove that the fixed point is analytic in ε , a somewhat surprising fact relying on the fermionic nature of the problem.
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