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The Continuous‐Time Lace Expansion
Abstract
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.
... Another way to prove an infrared bound is the lace expansion. It has been successful in various models, such as self-avoiding walk [10,18], oriented/unoriented percolation [16,29], lattice trees and lattice animals [17], the contact process [30], the Ising model [31], the φ 4 model [9,32], the randomconnection model [19] and self-repellent Brownian bridges [8]. Since the lace expansion yields a renewal equation for the two-point function, the infrared asymptotics at the critical point can be derived by deconvolution [25], without assuming reflection positivity. ...
... We use the bra-ket notation commonly used in physics to denote the eigenvectors of S (3) by 9) and denote its transpose by ⟨ #» σ |. ...
... The lace expansion for q > 0 is far more involved, due to pivotal intervals (cf., pivotal bridges in Definition 4.2(iii) below), and will be reported separately in the sequel [21]. The lace expansion is one of the few methods to rigorously prove critical behavior in high dimensions for various models, such as self-avoiding walk [10,18], lattice trees and lattice animals [17], percolation [16], the classical Ising model [31,33] and the lattice φ 4 model [9,32]. David Brydges, who established the methodology of the lace expansion for the first time in 1985 with Thomas Spencer, was awarded the Henri Poincaré Prize in 2024. ...
The transverse-field Ising model is widely studied as one of the simplest quantum spin systems. It is known that this model exhibits a phase transition at the critical inverse temperature , where q is the strength of the transverse field. Bj\"ornberg [Commun. Math. Phys., 232 (2013)] investigated the divergence rate of the susceptibility for the nearest-neighbor model as the critical point is approached by simultaneously changing q and the spin-spin coupling J in a proper manner, with fixed temperature. In this paper, we prove that the susceptibility diverges as as for assuming an infrared bound on the space-time two-point function. One of the key elements is a stochastic-geometric representation in Bj\"ornberg & Grimmett [J. Stat. Phys., 136 (2009)] and Crawford & Ioffe [Commun. Math. Phys., 296 (2010)]. As a byproduct, we derive a new lace expansion for the classical Ising model (i.e., q=0).
... The lace expansion has been successful in showing mean-field critical behaviour in high dimensions for various models, including lattice trees and lattice animals for d > 8 (e.g., [6,8,10]). The other models are self-avoiding walk for d > 4 (e.g., [2,6,9]), percolation for d > 6 (e.g., [6,7]), oriented percolation and the contact process for the spatial dimension d > 4 (e.g., [22,24]), and the Ising and ϕ 4 models for d > 4 (e.g., [1,[25][26][27]). For the nearest-neighbor lattice trees and lattice animals, in particular, Hara and Slade [6,8,10] show mean-field behaviour for both models in dimensions higher than an unspecified number bigger than 8. ...
... For the spread-out models of self-avoiding walk, percolation, oriented percolation and the contact process, van der Hofstad and Sakai [12] use the base point p 1 = 1, because of the unity of the 1-point function for those models. Since the analysis in terms of the underlying random walks is very simple, we do not have to know in detail the lace expansion; the exception is in Lemma 2.1 below, where we investigate the first lace-expansion coefficientπ (1) p to prove p c − p 1 = O(L −2d ). However, the basic facts (summarized in Proposition 2.2 below) and a minimum definition about the lace-expansion coefficients should be enough to read the proof, which we hope makes this paper more accessible to wider audience. ...
... Since pg p is increasing in p with p 1 g p 1 = 1, it suffices to show p c g p c > 1. By (2.6) and (2.14), it then suffices to show thatπ (1) p c −ĥ p c is bounded from below by β times a positive constant for large L. Here, and only here, we use the actual definition of the lace-expansion coefficientπ (1) p (see, e.g., [8]). We can easily check thatπ (1) p for both models is larger than the sum of triangles consisting only of three distinct edges: 3 , which is enough for lattice trees becauseĥ p ≡ 0. For lattice animals, we show belowĥ p ≤ 1 4π (1) p + O(β 2 ) for p ≤ p c in high dimensions d > 8. ...
A spread-out lattice animal is a finite connected set of edges in . A lattice tree is a lattice animal with no loops. The best estimate on the critical point so far was achieved by Penrose (J. Stat. Phys. 77, 3–15, 1994) : for both models for all . In this paper, we show that for all , where the model-dependent constant C has the random-walk representation\begin{align*} C_{\textrm{LT}}=\sum _{n=2}^\infty \frac{n+1}{2e}U^{*n}(o),&& C_{\textrm{LA}}=C_{\textrm{LT}}-\frac 1{2e^2}\sum _{n=3}^\infty U^{*n}(o), \end{align*}
where is the n-fold convolution of the uniform distribution on the d-dimensional ball . The proof is based on a novel use of the lace expansion for the 2-point function and detailed analysis of the 1-point function at a certain value of p that is designed to make the analysis extremely simple.
... The lace expansion has been successful in showing mean-field critical behavior in high dimensions for various models, including lattice trees and lattice animals for d > 8 (e.g., [6,8,10]). The other models are self-avoiding walk for d > 4 (e.g., [2,6,9]), percolation for d > 6 (e.g., [6,7]), oriented percolation and the contact process for the spatial dimension d > 4 (e.g., [22,24]), and the Ising and φ 4 models for d > 4 (e.g., [1,25,26,27]). For the nearest-neighbor lattice trees and lattice animals, in particular, Hara and Slade showed the mean-field critical behavior for both models in sufficiently high dimensions d > 8 in [6,8,10], where they did not mention about the specific dimension above which their results can be applied. ...
... For the spread-out models of self-avoiding walk, percolation, oriented percolation and the contact process, van der Hofstad and Sakai [12] simply used the base point p 1 = 1, because of the unity of the 1-point function for those models. Since the analysis in terms of the underlying random walks is very simple, we do not have to know in detail the lace expansion; the exception is in Lemma 2.1 below, where we investigate the first lace-expansion coefficientπ (1) p to prove p c − p 1 = O(L −2d ). However, the basic facts (summarized in Proposition 2.2 below) and a minimum definition about the lace expansion coefficients should be enough to read the proof, which we hope makes this paper more accessible to wider audience. ...
... Since pg p is increasing in p with p 1 g p 1 = 1, it suffices to show p c g pc > 1. By (2.6) and (2.14), it then suffices to show thatπ (1) pc −ĥ pc is bounded from below by β times a positive constant for large L. Here, and only here, we use the actual definition of the lace-expansion coefficientπ (1) p (see, e.g., [8]). We can easily check thatπ (1) p for both models is larger than the sum of triangles consisting only of three distinct edges:π (1) p ≥ |Λ|(|Λ|−1)(p/|Λ|) 3 , which is enough for lattice trees becauseĥ p ≡ 0. For lattice animals, we show belowĥ p ≤ 1 ...
A spread-out lattice animal is a finite connected set of edges in {{x, y} ⊂ Z d : 0 < ∥x − y∥ ≤ L}. A lattice tree is a lattice animal with no loops. The best estimate on the critical point p c so far was achieved by Penrose [23]: p c = 1/e + O(L −2d/7 log L) for both models for all d ≥ 1. In this paper, we show that p c = 1/e + CL −d + O(L −d−1) for all d > 8, where the model-dependent constant C has the random-walk representation C LT = ∞ n=2 n + 1 2e U * n (o), C LA = C LT − 1 2e 2 ∞ n=3
... (1.13) This is proved for strictly self-avoiding walk in [16,17], for weakly self-avoiding walk in [3,37], and for the continuous-time weakly self-avoiding walk (also known as the lattice Edwards model) in [6]. All these proofs use the lace expansion. ...
... We apply (4.6) with = min{2, d − 4}, so 2 + ≤ d − 2, and conclude from Proposition 4.1 and the fact that z (x) = z (−x) that the last term on the right-hand 5 Briefly, if χ(m(z)) were finite then G (m(z)) z would decay exponentially and this contradicts the fact that m(z) is by (1.9) the exponential decay rate of G z . 6 We expect that the proposition in fact remains true for all s ∈ [0, 2d −6) but its restriction to s ∈ [0, d −2] is sufficient for our needs. side of (4.4) iŝ ...
... This yields the required analogue of (5.25) for derivatives ofˆ [6,32,33], which raises the possibility that our results could be extended to these spin models. However, for percolation the bound on (x) is |x| −2(d−2) and for lattice trees and lattice animals it is |x| −(2d−6) [17], so for neither does have finite (d − 2) nd moment in any dimension. ...
We use the lace expansion to study the long-distance decay of the two-point function of weakly self-avoiding walk on the integer lattice Zd in dimensions d>4, in the vicinity of the critical point, and prove an upper bound |x|-(d-2)exp[-c|x|/ξ], where the correlation length ξ has a square root divergence at the critical point. As an application, we prove that the two-point function for weakly self-avoiding walk on a discrete torus in dimensions d>4 has a “plateau.” We also discuss the significance and consequences of the plateau for the analysis of critical behaviour on the torus.
... The lace expansion has been used to prove |x| −(d−2) decay for the long-distance behaviour of critical twopoint functions in a variety of statistical mechanical lattice models above their upper critical dimensions, including self-avoiding walk for d > 4 [1,2,6,7], percolation for d > 6 [6,7], lattice trees and lattice animals for d > 8 [6,7], the Ising model for d > 4 [11], and the ϕ 4 model for d > 4 [2,12]. Related results for long-range models are proved in [4]. ...
... The lace expansion has been used to prove |x| −(d−2) decay for the long-distance behaviour of critical twopoint functions in a variety of statistical mechanical lattice models above their upper critical dimensions, including self-avoiding walk for d > 4 [1,2,6,7], percolation for d > 6 [6,7], lattice trees and lattice animals for d > 8 [6,7], the Ising model for d > 4 [11], and the ϕ 4 model for d > 4 [2,12]. Related results for long-range models are proved in [4]. ...
... Our proof is adapted from Kotani's proof of [10, Theorem 1.6.1] in his unpublished 1991 handwritten notes in Japanese [8], which were translated into English at that time by Takashi Hara. In Section 3, we show that the sufficient condition in the extension of Kotani's Theorem is very convenient to use in conjunction with the lace expansion for weakly self-avoiding walk in dimensions d > 4, and provides an alternative approach in that context to the methods used in [1,2,6,7]. Although the realm of application of Kotani's Theorem to the lace expansion appears to be less general than other methods, its application to weakly self-avoiding walk is strikingly simple and provides a new tool for problems of this genre. ...
In 1991, Shinichi Kotani proved a theorem giving a sufficient condition to conclude that a function f(x) on decays like for large x, assuming that its Fourier transform is such that is well behaved for k near zero. We prove an extension of Kotani's Theorem and combine it with the lace expansion to give a simple proof that the critical two-point function for weakly self-avoiding walk has decay in dimensions .
... Lace expansion methods [Sak07,Sak15,Sak22] were applied in very large dimensions to obtain, not only that for both models η = 0, but also exact asymptotics for the critical two-point function, showing that it is equivalent (at large scales) to A/|x| d−2 2 (where A > 0 is a model-dependent constant). For the case of the weakly-coupled ϕ 4 model (i.e. with small coupling g), lace expansion was successfully implemented for d ≥ 5 in [BHH21]. The renormalisation group method was applied up to dimension d = 4 to obtain exact asymptotics in this setup. ...
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.
... The lace expansion was originally developed to study the self-avoiding walk in dimensions d > 4 [1,4,13,26], and the method has been extended to many models on Z d , including percolation in d > 6 [7,11,14], Ising and ϕ 4 models in d > 4 [3,24,25], and lattice trees and lattice animals in d > 8 [8,12]. More recently, lace expansions have also been derived for statistical mechanical models on R d . ...
We consider the convolution equation on , , where is the Dirac delta function and J,g are given functions. We provide conditions on that ensure the deconvolution G(x) to decay as for large , where is a positive-definite diagonal matrix. This extends a recent deconvolution theorem on proved by the author and Slade to the possibly anisotropic, continuum setting while maintaining its simplicity. Our motivation comes from studies of statistical mechanical models on based on the lace expansion. As an example, we apply our theorem to a self-repellent Brownian motion in dimensions , proving its critical two-point function to decay as , like the Green function of the Laplace operator .
... The long-distance asymptotic behaviour of the two-point function is an essential feature in the analysis of critical phenomena in lattice statistical mechanical models such as percolation, the Ising model, or the self-avoiding walk. In high dimensions, x −(d−2) 2 decay of the critical two-point function has been proved in several cases, including Hara et al. (2003); Hara (2008); Brydges et al. (2021);Sakai (2007). However, the near-critical behaviour, which merges the subcritical exponential decay and the power-law critical decay, has received scant attention despite the fact that it has the potential to reveal important and hitherto unstudied aspects of the critical behaviour, particularly for models defined on a torus. ...
... The first lace expansion was invented by Brydges and Spencer [8] for weakly self-avoiding walk. Since then, it has been extended to strictly self-avoiding walk [19], lattice trees and lattice animals [18], oriented percolation [23], the contact process [24], the Ising model [25], the |ϕ| 4 model [7,26] and the random-connection model [20]; see also [27] for the development of the subject until mid 2000s. In general, the lace expansion gives rise to a recursion equation for the two-point function, which is almost identical to that for the Green function of the underlying random walk. ...
The lace expansion for the Ising two-point function was successfully derived in (Sakai in Commun Math Phys 272:283–344, 2007, Proposition 1.1). It is an identity that involves an alternating series of lace-expansion coefficients. In the same paper, we claimed that the expansion coefficients obey certain diagrammatic bounds which imply faster x-space decay (as the two-point function cubed) above the critical dimension (=4 for finite-variance models) if the spin-spin coupling is ferromagnetic, translation-invariant, summable and symmetric with respect to the underlying lattice symmetries. However, we recently found a flaw in the proof of (Sakai in Commun Math Phys 272:283–344, 2007, Lemma 4.2), a key lemma to the aforementioned diagrammatic bounds. In this paper, we no longer use the problematic (Sakai 2007, Lemma 4.2), and prove new diagrammatic bounds on the expansion coefficients that are slightly more complicated than those in (Sakai 2007, Proposition 4.1) but nonetheless obey the same fast decay above the critical dimension . Consequently, the lace-expansion results for the Ising and models in the literature are all saved. The proof is based on the random-current representation and its source-switching technique of Griffiths, Hurst and Sherman, combined with a double expansion: a lace expansion for the lace-expansion coefficients.
... There exist a few results about phase transitions in systems with continuous symmetry that were proved with different methods, see [8,19]. But reflection positivity and infrared bound remains the most prolific method and the only one that has been applied to quantum systems. ...
The method of reflection positivity and infrared bounds allows to prove the occurrence of phase transitions in systems with continuous symmetries. We review the method in the context of quantum spin systems. The novel aspect is a proof for long-range interactions that involve the Euclidean distance between sites.
... has been proved [36] (see also [5,17,18] for closely related results). In [34], Theorem 3.1 is applied to prove that the torus two-point function has a "plateau" in the sense of the following theorem. ...
How long does a self-avoiding walk on a discrete d-dimensional torus have to be before it begins to behave differently from a self-avoiding walk on ? We consider a version of this question for weakly self-avoiding walk on a torus in dimensions . On for , the partition function for n-step weakly self-avoiding walk is known to be asymptotically purely exponential, of the form , where is the growth constant for weakly self-avoiding walk on . We prove the identical asymptotic behaviour on the torus (with the same A and as on ) until n reaches order , where V is the number of vertices in the torus. This shows that the walk must have length of order at least before it ``feels'' the torus in its leading asymptotics. Our results support the conjecture that the behaviour of the partition function does change once n reaches , and we relate this to a conjectural critical scaling window which separates the dilute phase from the dense phase . To prove the conjecture and to establish the existence of the scaling window remains a challenging open problem. The proof uses a novel lace expansion analysis based on the ``plateau'' for the torus two-point function obtained in previous work.
... Proof of Theorem 1.1. Since b(z) ≤ b(Ω −1 ) ≤ 1 for z ≤ Ω −1 by definition, it follows from Proposition 3.2 and the continuity of the function b that the interval (2,3] is forbidden for values of b(z), so b(z) ≤ 2 for all z ∈ [0, z c ). By monotone convergence, also b(z c ) ≤ 2, and λ z approaches a limit λ zc = 1 ...
We use the lace expansion to give a simple proof that the critical two-point function for weakly self-avoiding walk on has decay in dimensions .
... The first lace expansion was invented by Brydges and Spencer [8] for weakly self-avoiding walk. Since then, it has been extended to strictly self-avoiding walk [19], lattice trees and lattice animals [18], oriented percolation [23], the contact process [24], the Ising model [25], the |ϕ| 4 model [7,26] and the random-connection model [20]; see also [27] for the development of the subject until mid 2000s. In general, the lace expansion gives rise to a recursion equation for the two-point function, which is almost identical to that for the Green function of the underlying random walk. ...
The lace expansion for the Ising two-point function was successfully derived in Sakai (Commun. Math. Phys., 272 (2007): 283--344). It is an identity that involves an alternating series of the lace-expansion coefficients. In the same paper, we claimed that the expansion coefficients obey certain diagrammatic bounds which imply faster x-space decay (as the two-point function cubed) above the critical dimension (=4 for finite-variance models), if the spin-spin coupling is ferromagnetic, translation-invariant, summable and symmetric with respect to the underlying lattice symmetries. However, we recently found a flaw in the proof of Lemma 4.2 in Sakai (2007), a key lemma to the aforementioned diagrammatic bounds. In this paper, we no longer use the problematic Lemma 4.2 of Sakai (2007), and prove new diagrammatic bounds on the expansion coefficients that are slightly more complicated than those in Proposition 4.1 of Sakai (2007) but nonetheless obey the same fast decay above the critical dimension . Consequently, the lace-expansion results for the Ising and models so far are all saved. The proof is based on the random-current representation and its source-switching technique of Griffiths, Hurst and Sherman, combined with a double expansion: a lace expansion for the lace-expansion coefficients.
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 give conditions on a real-valued function F on , for , which ensure that the solution G to the convolution equation has Gaussian decay for large |x|. Precursors of our results were obtained in the 2000s, using intricate Fourier analysis. In 2022, a very simple deconvolution theorem was proved, but its applicability was limited. We extend the 2022 theorem to remove its limitations while maintaining its simplicity—our main tools are Hölder’s inequality, weak derivatives, and basic Fourier theory in space. Our motivation comes from critical phenomena in equilibrium statistical mechanics, where the convolution equation is provided by the lace expansion and G is a critical two-point function. Our results significantly simplify existing proofs of critical decay in high dimensions for self-avoiding walk, Ising and models, percolation, and lattice trees and lattice animals. We also improve previous error estimates.
This article is concerned with self-avoiding walks (SAW) on that are subject to a self-attraction. The attraction, which rewards instances of adjacent parallel edges, introduces difficulties that are not present in ordinary SAW. Ueltschi has shown how to overcome these difficulties for sufficiently regular infinite-range step distributions and weak self-attractions. This article considers the case of bounded step distributions. For weak self-attractions we show that the connective constant exists, and, in , carry out a lace expansion analysis to prove the mean-field behaviour of the critical two-point function, hereby addressing a problem posed by den Hollander.
We prove that for critical (spread-out) lattice trees in dimensions d > 8, the unique paths from the origin to vertices of large tree distance converge to Brownian motion. This provides an important ingredient for proving weak convergence of the corresponding historical processes.
We show Green's function asymptotics for the two-point function of weakly
self-avoiding walk in dimension bigger than 4, reproving a classic result by
Brydges and Spencer in 1985. Our proof relies on Banach algebras to analyze the
lace-expansion fixed point equation and is considerably simpler than previous
approaches.
Using the Griffiths-Simon construction of the model and the lace
expansion for the Ising model, we prove that, if the strength of
nonlinearity is sufficiently small for a large class of short-range models in
dimensions , then the critical two-point function
is asymptotically times
a model-dependent constant, and the critical point is estimated as
, where
is the massless point for the Gaussian model.
Green's function G(x) of a zero mean random walk on the N-dimensional integer lattice N≥2 is expanded in powers of 1/∣x∣ under suitable moment conditions. In particular, we find minimal moment conditions for G(x) to behave like a constant times the Newtonian potential (or logarithmic potential in two dimensions) for large values of
∣x∣. Asymptotic estimates of G(x) in dimensions N≥4, which are valid even when these moment conditions are violated, are computed. Such estimates are applied to determine
the Martin boundary of the random walk. If N= 3 or 4 and the random walk has zero mean and finite second moment, the Martin boundary consists of one point, whereas if
N≥ 5, this is not the case, because non-harmonic functions arise as Martin boundary points for a large class of such random
walks. A criterion for when this happens is provided. 1991 Mathematics Subject Classification: 60J15, 60J45, 31C20.
We approximate a (spatially cutoff) (f4)2 Euclidean field theory by an ensemble of spin 1/2 Ising spins with ferromagnetic pair interactions. This approximation is used to prove a Lee-Yang theorem and GHS type correlation inequalities for the (f4)2 theory. Application of these results are presented.
We derive a perturbation expansion for general self-interacting random walks, where steps are made on the basis of the history of the path. Examples of models where this expansion applies are reinforced random walk, excited random walk, the true (weakly) self-avoiding walk, loop-erased random walk, and annealed random walk in random environment. In this paper we show that the expansion gives rise to useful formulae for the speed and variance of the random walk, when these quantities are known to exist. The results and formulae of this paper have been used elsewhere by the authors to prove monotonicity properties for the speed (in high dimensions) of excited random walk and related models, and certain models of random walk in random environment. We also derive a law of large numbers and central limit theorem (with explicit error terms) directly from this expansion, under strong assumptions on the expansion coefficients. The assumptions are shown to be satisfied by excited random walk in high dimensions with small excitation parameter, a model of reinforced random walk with underlying drift and small reinforcement parameter, and certain models of random walk in random environment under strong ellipticity conditions. This is the extended version of the paper [20], where we provide all proofs.
We construct the thermodynamic limit of the critical (massless) φ4 model in 4 dimensions with an ultraviolet cutoff by means of a “partly renormalized” phase space expansion. This expansion requires in a natural way the introduction of effective or “running” constants, and the infrared asymptotic freedom of the model, i.e. the decay of the running coupling constant, plays a crucial rôle. We prove also that the correlation functions of the model are the Borel sums of their perturbation expansion.
The triangle condition for percolation states that is finite at the critical point, where τ(x, y) is the probability that the sitesx andy are connected. We use an expansion related to the lace expansion for a self-avoiding walk to prove that the triangle condition is satisfied in two situations: (i) for nearest-neighbour independent bond percolation on thed-dimensional hypercubic lattice, ifd is sufficiently large, and (ii) in more than six dimensions for a class of “spread-out” models of independent bond percolation which are believed to be in the same universality class as the nearest-neighbour model. The class of models in (ii) includes the case where the bond occupation probability is constant for bonds of length less than some large number, and is zero otherwise. In the course of the proof an infrared bound is obtained. The triangle condition is known to imply that various critical exponents take their mean-field (Bethe lattice) values(\gamma = \beta = 1,\delta = \Delta _t = 2, t\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \geqslant } 2) and that the percolation density is continuous at the critical point. We also prove thatv
2 in (i) and (ii), wherev
2 is the critical exponent for the correlation length.
We prove a conjecture by E. Lieb, which leads to the Lieb inequality for plane rotors. As in the Ising model case, this inequality implies the existence of an algorithm to compute the transition temperature of this model.
We present a new method for rigorously proving the existence of phase transitions. In particular, we prove that phase transitions occur in (φ·φ)
32 quantum field theories and classical, isotropic Heisenberg models in 3 or more dimensions. The central element of the proof is that for fixed ferromagnetic nearest neighbor coupling, the absolutely continuous part of the two point function ink space is bounded by 0(k
−2). When applicable, our results can be fairly accurate numerically. For example, our lower bounds on the critical temperature in the three dimensional Ising (resp. classical Heisenberg) model agrees with that obtained by high temperature expansions to within 14% (resp. a factor of 9%).
We use a transformation due to Percus to give a simple derivation of the Griffiths, Hurst, and Sherman, and some other new inequalities, for Ising ferromagnets with pair interactions. The proof makes use of the Griffiths, Kelly, and Sherman and the Fortuin, Kasteleyn, and Ginibre inequalities.
We consider spread-out models of self-avoiding walk, bond percolation, lattice trees and bond lattice animals on Z d , having long finite-range connections, above their upper critical dimensions d = 4 (self-avoiding walk), d = 6 (percolation) and d = 8 (trees and animals). The two-point functions for these models are respectively the generating function for selfavoiding walks from the origin to x 2 Z d , the probability of a connection from 0 to x, and the generating function for lattice trees or lattice animals containing 0 and x. We use the lace expansion to prove that for sufficiently spread-out models above the upper critical dimension, the two-point function of each model decays, at the critical point, as a multiple of jxj 2Gammad as x ! 1. We use a new unified method to prove convergence of the lace expansion. The method is based on x-space methods rather than the Fourier transform. Our results also yield unified and simplified proofs of the bubble condition for self-avoi...
We consider the critical spread-out contact process in \Zd with , whose infection range is denoted by . The two-point function is the probability that x\in\Zd is infected at time t by the infected individual located at the origin o\in\Zd at time 0. We prove Gaussian behavior for the two-point function with for some finite for . When , we also perform a local mean-field limit to obtain Gaussian behaviour for with fixed and when the infection range depends on T such that for any . The proof is based on the lace expansion and an adaptation of the inductive approach applied to the discretized contact process. We prove the existence of several critical exponents and show that they take on mean-field values. The results in this paper provide crucial ingredients to prove convergence of the finite-dimensional distributions for the contact process towards the canonical measure of super-Brownian motion, which we defer to a sequel of this paper.
The lace expansion has been a powerful tool for investigating mean-field behavior for various stochastic-geometrical models, such as self-avoiding walk and percolation, above their respective upper-critical dimension. In this paper, we prove the lace expansion for the Ising model that is valid for any spin-spin coupling. For the ferromagnetic case, we also prove that the expansion coefficients obey certain diagrammatic bounds that are similar to the diagrammatic bounds on the lace-expansion coefficients for self-avoiding walk. As a result, we obtain Gaussian asymptotics of the critical two-point function for the nearest-neighbor model with d>>4 and for the spread-out model with d>4 and L>>1, without assuming reflection positivity.
We prove that the scaling limits of spin fluctuations in four-dimensional Ising-type models with nearest-neighbor ferromagnetic interaction at or near the critical point are Gaussian. A similar statement is proven for the λ𝜑4 fields over R4 with a lattice ultraviolet cutoff, in the limit of infinite volume and vanishing lattice spacing. The proofs are enabled by the models' random current representation, in which the correlation functions' deviation from Wick's law is expressed in terms of intersection probabilities of random currents with sources at distances that are large on the model's lattice scale. Guided by the analogy with random walk intersection amplitudes, the analysis focuses on the improvement of the so-called tree diagram bound by a logarithmic correction term, which is derived here through multi-scale analysis.
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.
We give a sufficient condition for tightness for convergence of rescaled critical spatial structures to the canonical measure of super-Brownian motion. This condition is formulated in terms of the r-point functions for r = 2, . . ., 5. The r-point functions describe the expected number of particles at given times and spatial locations, and have been investigated in the literature for many high-dimensional statistical physics models, such as oriented percolation and the contact process above 4 dimensions and lattice trees above 8 dimensions. In these settings, convergence of the finite-dimensional distributions is known through an analysis of the r-point functions, but the lack of tightness has been an obstruction to proving convergence on path space. We apply our tightness condition first to critical branching random walk to illustrate the method as tightness here is well known. We then use it to prove tightness for sufficiently spread-out lattice trees above 8 dimensions, thus proving that the measure-valued process describing the distribution of mass as a function of time converges in distribution to the canonical measure of super-Brownian motion. We conjecture that the criterion will also apply to other statistical physics models.
Ferromagnetic lattice spin systems can be expressed as gases of random walks interacting via a soft core repulsion. By using a mixed spinrandom walk representation we present a unified approach to many recently established correlation inequalities. As an application of these inequalities we obtain a simple proof of the mass gap for the λ(φ 4) 2 quantum field model. We also establish new upper bounds on critical temperatures.
We extend and apply a rigorous renormalisation group method to study critical
correlation functions, on the 4-dimensional lattice , for the
weakly coupled n-component spin model for all , and
for the continuous-time weakly self-avoiding walk. For the model,
we prove that the critical two-point function has (Gaussian) decay
asymptotically, for . We also determine the asymptotic decay of the
critical correlations of the squares of components of , including the
logarithmic corrections to Gaussian scaling, for . The above extends
previously known results for to all , and also observes new
phenomena for , all with a new method of proof. For the continuous-time
weakly self-avoiding walk, we determine the decay of the critical generating
function for the "watermelon" network consisting of p weakly mutually- and
self-avoiding walks, for all , including the logarithmic corrections.
This extends a previously known result for , for which there is no
logarithmic correction, to a much more general setting. In addition, for both
models, we study the approach to the critical point and prove existence of
logarithmic corrections to scaling for certain correlation functions. Our
method gives a rigorous analysis of the weakly self-avoiding walk as the case of the model, and provides a unified treatment of both
models, and of all the above results.
Loop-weighted walk with parameter is a non-Markovian model of
random walks that is related to the loop O(N) model of statistical mechanics.
A walk receives weight if it contains k loops; whether this is
a reward or punishment for containing loops depends on the value of .
A challenging feature of loop-weighted walk is that it is not purely repulsive,
meaning the weight of the future of a walk may either increase or decrease if
the past is forgotten. Repulsion is typically an essential property for lace
expansion arguments. This article circumvents the lack of repulsion and proves,
via the lace expansion, that for any loop-weighted walk is
diffusive in high dimensions.
Random walks are stochastic processes formed by successive summation of independent, identically distributed random variables and are one of the most studied topics in probability theory. This contemporary introduction evolved from courses taught at Cornell University and the University of Chicago by the first author, who is one of the most highly regarded researchers in the field of stochastic processes. This text meets the need for a modern reference to the detailed properties of an important class of random walks on the integer lattice. It is suitable for probabilists, mathematicians working in related fields, and for researchers in other disciplines who use random walks in modeling.
The subject of this book is equilibrium statistical mechanics, in particular the theory of critical phenomena, and quantum field theory. The central theme is the use of random-walk representations as a tool to derive correlation inequalities. The consequences of these inequalities for critical-exponent theory are expounded in detail. The book contains some previously unpublished results. It addresses both the researcher and the graduate student in modern statistical mechanics and quantum field theory.
I Foundations of the renormalization group.- Phase Transitions and the Scaling Hypothesis.- Mean-Field Theory and the Gaussian Approximation.- Wilsonian Renormalization Group.- Critical Behavior of the Ising Model Close to Four Dimensions.- Field-Theoretical Renormalization Group.- II Introduction to the functional renormalization group.- Functional Methods.- Exact FRG Flow Equations.- Vertex Expansion.- Derivative Expansion.- III Functional renormalization group approach to fermions.- Fermionic Functional Renormalization Group.- Normal Fermions: Partial Bosonization in the Forward Scattering Channel.- Superfluid Fermions: Partial Bosonization in the Particle-Particle Channel.
We use Schwinger-Dyson equations combined with rigorous ``perturbation-theoretic'' correlation inequalities to give a new and extremely simple proof of the existence and nontriviality of the weakly-coupled continuum varphi{2/4} and varphi{3/4} quantum field theories, constructed as subsequence limits of lattice theories. We prove an asymptotic expansion to order lambda or lambda2 for the correlation functions and for the mass gap. All Osterwalder-Schrader axioms are satisfied except perhaps Euclidean (rotation) invariance.
The covariant quantum electrodynamics of Tomonaga, Schwinger, and Feynman is used as the basis for a general treatment of scattering problems involving electrons, positrons, and photons. Scattering processes, including the creation and annihilation of particles, are completely described by the S matrix of Heisenberg. It is shown that the elements of this matrix can be calculated, by a consistent use of perturbation theory, to any desired order in the fine-structure constant. Detailed rules are given for carrying out such calculations, and it is shown that divergences arising from higher order radiative corrections can be removed from the S matrix by a consistent use of the ideas of mass and charge renormalization.
We provide here the details of the proof, announced in [1], that in d>4 dimensions the (even) phi4 Euclidean field theory, with a lattice cut-off, is inevitably free in the continuum limit (in the single phase regime). The analysis is nonperturbative, and is based on a representation of the field variables (or spins in Ising systems) as source/sink creation operators in a system of random currents --- which may be viewed as the mediators of correlations. In this dual representation, the onset of long-range-order is attributed to percolation in an ensemble of sourceless currents, and the physical interaction in the phi4 field --- and other aspects of the critical behavior in Ising models --- are directly related to the intersection properties of long current clusters. An insight into the criticality of the dimension d=4 is derived from an analogy (foreseen by K. Symanzik) with the intersection properties of paths of Brownian motion. Other results include the proof that in certain respect, the critical behavior in Ising models is in exact agreement with the mean-field approximation in high dimensions d>4, but not in the low dimension d=2 --- for which we establish the ``universality'' of hyperscaling.
In a recent paper Brydges, Fröhlich, and Spencer have successfully applied Markov chains to classical spin systems. Using probabilistic methods in a more systematic way, we improve some of their results. The methods can be extended to fields with continuous parameter such as P(??) fields.
Some new correlation inequalities are described which bound large-distance behavior of correlations in ferromagnets from above by correlations at intermediate distances. Among applications are (1) an inequality, \eta, on the decay of correlations at the critical point; (2) an inequality \chi\geqq relating the zero-field susceptibility and the mass gap in a nearest-neighbor ferromagnet; (3) a finite algorithm for rigorously computing a sequence of upper bounds guaranteed to converge to the true transition temperature in Ising ferromagnets.
It is rigorously proved that the continuum limits of Euclidean varphi4d lattice fields are free fields in d>4. An exact geometric characterization of criticality in Ising models is introduced, and used to prove other mean-field features for d>4 and hyperscaling in d=2.
It is shown that one- and two-component λϕ4
theories and non-linear σ-models in five or more dimensions
approach free or generalized free fields in the continuum (scaling)
limit, and that in four dimensions the same result holds, provided there
is infinite field strength renormalization, as expected. Some critical
exponents for the lattice theories in five or more dimensions are shown
to be mean field. The main tools are Symanzik's polymer representation
of scalar field theories and correlation inequalities.
Using an expansion based on the renormalization group philosophy we prove that for aT step weakly self-avoiding random walk in five or more dimensions the variance of the endpoint is of orderT and the scaling limit is gaussian, asT.
The Gaussian inequality is proven for multicomponent rotators with negative correlations between two spin components. In the case of one-component systems, the Gaussian inequality is shown to be a consequence of Lebowitz' inequality. For multicomponent models, the Gaussian inequality implies that the decay rate of the truncated correlation (or Schwinger) functions is dominated by that of the two-point function. Applied to field theory, these inequalities give information on the absence of bound states in the (1
2 + 1
2)2 model.
We give a rigorous proof of mean-field critical behavior for the susceptibility (γ=1/2) and the correlation length (v=1/4) for models of lattice trees and lattice animals in two cases: (i) for the usual model with trees or animals constructed
from nearest-neighbor bonds, in sufficiently high dimensions, and (ii) for a class of “spread-out” or long-range models in
which trees and animals are constructed from bonds of various lengths, above eight dimensions. This provides further evidence
that for these models the upper critical dimension is equal to eight. The proof involves obtaining an infrared bound and showing
that a certain “square diagram” is finite at the critical point, and uses an expansion related to the lace expansion for the
self-avoiding walk.
Key wordsLattice animals–branched polymers–upper critical dimension–lace expansion–critical exponents–mean-field behavior
The aim of this paper is to stu the behavior asm tends to ∞ of a family of measures exp[-Φ
(m)(x)]dx
(m) on ℝm
, whereΦ
(m) is a potential on ℝm
which is a perturbation “in a suitable sense” of the harmonic potential Σj
x
j2.
We prove that above eight dimensions the scaling limit of sufficiently spread-out lattice trees is the variant of super-Brownian
motion known as integrated super-Brownian excursion (ISE), as conjectured by Aldous. The same is true for nearest-neighbour
lattice trees in sufficiently high dimensions. The proof uses the lace expansion.
We consider oriented bond percolation on , at the critical occupation density pc, for d>4. The model is a “spread-out” model having long range parameterised by L. We consider configurations in which the cluster of the origin survives to time n, and scale space by n1/2. We prove that for L sufficiently large all the moment measures converge, as n→∞, to those of the canonical measure of super-Brownian motion. This extends a previous result of Nguyen and Yang, who proved Gaussian behaviour for the critical two-point function, to all r-point functions (r⩾2). We use lace expansion methods for the two-point function, and prove convergence of the expansion using a general inductive method that we developed in a previous paper. For the r-point functions with r⩾3, we use a new expansion method.RésuméOn considére un modèle de percolation orientée sur les liens de à la densité d'occupation critique pc, pour d>4. Le modèle comporte un effet de dispersion à longue portée paramétré par une longueur L. On considère les configurations dans lesquelles l'amas comprenant l'origine survit jusqu'au temps n, et on rééchelonne l'espace par un facteur n1/2. On montre, pour L assez grand, la convergence de tous les moments à valeurs mesures vers ceux du super mouvement brownien. On étend ainsi un résultat préalablement obtenu par Nguyen et Yang, qui ont montré le comportement gaussien de la fonction à deux points critique, au cas des fonctions à r points. La convergence du développement de la fonction à deux points est établie à l'aide d'une méthode générale d'induction développée dans un article précédent. Une nouvelle méthode de développement est employée pour r supérieur à trois.
We prove a law of large numbers for random walks in certain kinds of i.i.d.
random environments in Z^d that is an extension of a result of Bolthausen,
Sznitman and Zeitouni (2003). We use this result, along with the lace expansion
for self-interacting random walks, to prove a monotonicity result for the
?first coordinate of the speed of the random walk under some strong assumptions
on the distribution of the environment.
The Brydges-Spencer lace expansion is used to prove that the scaling limit of the finite-dimensional distributions of self-avoiding random walk in the d-dimensional cubic lattice is Gaussian, if d is sufficiently large. It is also shown that the critical exponent for the number of self-avoiding walks is equal to 1, if d is sufficiently large.
In this paper, we apply the Brydges-Spencer lace expansion and the Hara-Slade analysis to obtain the triangle condition for the nearest-neighbor oriented bond percolation in high dimensions and for the spread-out oriented bond percolation in . Furthermore, we also establish the infrared bound in the subcritical region and the mean-field behavior for these models.
We use the Brydges-Spencer lace expansion to prove that the mean square displacement of aT step strictly self-avoiding random walk in thed dimensional square lattice is asymptotically of the formDT asT approaches infinity, ifd is sufficiently large. The diffusion constantD is greater than one.
A general formulation is given of Simon's Ising model inequality:
whereB is any set of spins separating α from γ. We show that 〈σbσα〉 can be replaced by 〈σbσα〉A whereA is the spin system “inside”B containing α. An advantage of this is that a finite algorithm can be given to compute the transition temperature to any desired accuracy. The analogous inequality for plane rotors is shown to hold if a certain conjecture can be proved. This conjecture is indeed verified in the simplest case, and leads to an upper bound on the critical temperature. (The conjecture has been proved in general by Rivasseau. See notes added in proof.)
We study excited random walks in i.i.d. random cookie environments in high dimensions, where the kth cookie at a site determines the transition probabilities (to the left and right) for the kth departure from that site. We show that in high dimensions, when the expected right drift of the first cookie is sufficiently large, the velocity is strictly positive, regardless of the strengths and signs of subsequent cookies. Under additional conditions on the cookie environment, we show that the limiting velocity of the random walk is continuous in various parameters of the model and is monotone in the expected strength of the first cookie at the origin. We also give non-trivial examples where the first cookie drift is in the opposite direction to all subsequent cookie drifts and the velocity is zero. The proofs are based on a cut-times result of Bolthausen, Sznitman and Zeitouni, the lace expansion for self-interacting random walks of van der Hofstad and Holmes, and a coupling argument. © Association des Publications de l'Institut Henri Poincaré, 2012.
A self-avoiding walk with small attractive interactions is described here. The existence of the connective constant is established, and the diffusive behavior is proved using the method of the lace expansion.