> <\body> <\hide-preamble> > > >>> ]>>> \; \ > model>|Duch>|<\author-affiliation> Institute of Mathematics Polish Academy of Sciences Poland |>>|Gubinelli>|<\author-affiliation> Mathematical Institute University of Oxford United Kingdom |<\author-email> gubinelli@maths.ox.ac.uk >>> <\abstract> We present a new, simple and self-contained technique for proving a mass gap for Euclidean quantum field theories in the high-temperature regime using the stochastic quantisation approach. As an application, we establish exponential decay of spatial correlations for the fractional > model on the plane in the Wick-ordering regime. The key idea is to analyse the propagation of perturbations using the linearised stochastic quantisation equation. Spatial propagation is controlled by an auxiliary random walk, while the potentially destabilising effect of the renormalised nonlinearity is controlled by combining dissipative energy estimates for the nonlinear dynamic with the Boué\UDupuis variational formula. <\table-of-contents|toc> Introduction and main results> |.>>>>|> Spatial propagation> |.>>>>|> Block decomposition and random walk representation |.>>>>|> > An abstract propagation criterion |.>>>>|> > Stochastic enhancement and remainder dynamic> |.>>>>|> Pathwise energy and Rayleigh estimates> |.>>>>|> Weighted energy inequality |.>>>>|> > Coming down from infinity and dissipation |.>>>>|> > Local Rayleigh-quotient bound |.>>>>|> > Path estimates and Boué\UDupuis> |.>>>>|> Bounds along block paths |.>>>>|> > Exponential path bound |.>>>>|> > Proof of the mass gap> |.>>>>|> Random-walk estimates> |.>>>>|> Weighted lattice Besov estimates> |.>>>>|> Extension to the >-invariant model> |.>>>>|> Synchronisation by noise> |.>>>>|> |.>>>>|> Set >=\*\> and ,\>=\>/\*\>, where the ultraviolet and volume cutoffs are <\equation> \=2,\=M*L,n\\,M\\. Here \> is a fixed constant chosen below. For \>>, we define the nearest neighbour Laplacian <\equation*> \>*f\\**e|)>+f**e|)>-2*f|)>. For \>, the operator >\>|)>/2>> is defined on >> as the Fourier multiplier. Write ,\>> for the Gaussian measure on ,\>>> with covariance >>. The interacting fractional > measure on ,\>>> is <\equation> \,\>\|)>\,\>>*exp|4>*\*\,\>>-6*C,\>*\+3*C,\>|)>|)>*\,\>\|)>. We study the above measure in the regime in which the so-called Wick renormalisation is sufficient to control the continuum limit \0>. Hence, the counterterm coincides with <\equation> C,\>\\>G>|)>, where >> is the kernel of >> on >>. Let \,\>>. An observable ,\>\C> is called -supported if \F|)>\A>, where F> is the gradient relative to |\,\|\>>. We denote by >> the >> distance on the torus >=\/*\|)>> and write >> for the distance between the >-periodic sets \>>. <\theorem> Let \>. There exists \0> such that the following holds. For every \> there is \\> such that, for every \|]>>, every pair ,\|)>> as in, and all bounded observables with bounded gradients and support sets of diameter at most , <\equation> ,\>>|\|>\C*e>/>*sup\L>|\F|)>|sup\L>|\<\|\|\>>*\G|)>|\<\|\|\>>. The assumption that \0> is small enough corresponds to the high-temperature regime. The case =2> corresponds to the standard > model. The exponential decay relies on our choice of the fractional operator >|)>/2>>. For \2>, the alternative natural choice >|)>/2>> gives a Gaussian measure whose covariance decays only polynomially.\ The proof technique does not use the scalar nature of the model and extends to the vectorial model as briefly discussed in Appendix. By a straightforward adaptation of the technique developed in, after a suitable extension to |)>>, the family of measures ,\>|)>,\>> is tight in the entire subcritical regime \>, for a suitable choice of counterterm that coincides with when \>. The bound holds for any accumulation point for\>. The extension for \>, that is beyond the Wick-ordering regime, requires additional technical arguments, which we leave for future work. The proof of the above theorem is presented in Section. It uses the Helffer\USjöstrand covariance representation combined with the analysis of the parabolic stochastic quantisation equation of the model, <\equation> \=->*\+\-3*C,\>*\|)>|]>t+W,\=\, where is standard Brownian motion in =L*,\>,\|)>> extended periodically to >>. In particular, <\equation*> \W|]>=\*\|)>|}>>*min. At fixed cutoffs is a globally well-posed finite-dimensional SDE. Its unique invariant measure is, and the Markov semigroup > is reversible in ,\>|)>>. Write \>> for the solution of started from > at time. For an arbitrary initial time , we write \>> for the solution satisfying >=\>. Its derivative with respect to the initial condition is <\equation*> U>=D>*\>. Note that U>f> solves <\equation> \v+>+3*\*>|)>-C,\>|)>|]>v=0,v=f. We divide the lattice ,\>> into blocks of size and denote by >> a bump function supported in a block centred at a lattice point >; see Section. Our main technical result is the following contractive estimate for the linearised dynamic. <\theorem> Let \>. There exist ,C\0>, uniform in the cutoffs and block indices, such that for every \|]>>, every pair ,\|)>> as in, every 0> and all block indices ,\>, we have <\equation> \*sup\\>sup>,\>|)>>|\>*U>*>*f|)>|\<\|\|\>>,\>|)>>\C*e>,\|)>/L>, where >,\>|)>> is the unit ball in ,\>|)>> centred at zero and \>:\\L,\>|)>|}>> is the random set of fields that can be reached at time one from arbitrary initial conditions. <\proof> Proposition proves the hypothesis of Proposition, which then implies the stated bound. The bound shows that a perturbation of the initial condition decays in time and has exponentially small influence on distant spatial regions uniformly in the cutoffs. This estimate extends to the infinite-volume continuum dynamic and yields exponential synchronisation of solutions driven by the same noise. Consequently, the dynamic is ergodic and admits a unique invariant measure, with every solution converging exponentially fast to the unique stationary solution. The invariant measure is invariant under all Euclidean transformations and satisfies reflection positivity. We give a self-contained proof of the results stated as theorems. The aforementioned conclusions about the limiting dynamic and its invariant measure are not proved here, but follow easily along the lines of. The proof of synchronisation by noise is outlined in Appendix. We first divide the lattice into blocks and dominate the propagation of the linearised equation by a conservative random walk > on the block lattice, whose jump rates have uniformly bounded exponential moments (see Proposition). The resulting estimate for the localised energy contains a deterministic dissipative term and a random contribution that may weaken the dissipation, measured by the local Rayleigh quotient |^>>> introduced in. Note that |^>>> involves the Hessian of the potential of the model, which is not bounded from below because of the presence of the counterterm. This is the principal reason why estimating |^>>> and the corresponding accumulated loss of dissipation is highly nontrivial. The Feynman\UKac representation reduces the problem to estimating exponential moments of the accumulated loss of dissipation along paths of the walk (see Proposition). The key tool is the Boué\UDupuis variational formula, which allows us to estimate this exponential moment by studying how the loss of dissipation changes when an arbitrary adapted drift is added to the driving Wiener process (see Proposition). More precisely, the formula expresses the logarithm of the exponential moment as the supremum of the expected loss of dissipation for the equation with additional drift , minus a cost functional equal to the squared >-norm of . Thus, it suffices to bound the expected loss of dissipation uniformly over all such drifts, allowing an additional term at most quadratic in , with a sufficiently small coefficient to be absorbed by the cost functional (see Lemma). To establish this estimate, we write the solution of the stochastic quantisation equation with the additional drift as =Z+\>, separating the explicitly constructed random field independent of the control from the more regular remainder >. The stochastic bounds of Section and the dissipative energy estimates of Section control the loss of dissipation uniformly in the initial condition after a positive time. The coercive bound from Proposition depends quadratically on the drift , which, as explained above, is crucial for the application in the Boué\UDupuis representation. The mass gap for scalar Euclidean field theories with polynomial interactions in two dimensions was studied in the classical works. For weakly coupled >, the existence of the infinite-volume theory and exponential decay of correlations were established in. The mass gap for two-dimensional Euclidean fields with exponential interactions was proved in, while more recently a proof based on stochastic quantisation was given in. Uniqueness of ultraregular Gibbs states for weakly coupled |)>> models was proved in. Logarithmic Sobolev inequalities for the > and > measures throughout the entire high-temperature phase were established in. Building on this result, proved uniqueness of the invariant measure for the infinite-volume > stochastic quantisation dynamic in the same regime. Ergodicity of the infinite-volume > stochastic quantisation dynamic at high temperature, together with exponential decay of spatial correlations of its invariant measure, was established in. For the construction of the fractional > model and its stochastic dynamic, we refer to. .>Our approach differs substantially from that of, where the linearised evolution is estimated by a time-chopping argument. This technique was originally introduced in to study integrability and tail estimates for rough differential equations, and was subsequently applied to the dynamic > model on a torus in. The main idea is to divide the time interval into sufficiently many random subintervals such that on every subinterval the stochastic terms are small enough to obtain a pathwise contractive estimate by fixed point arguments. The number of subintervals needed to reach a deterministic time is controlled with the use of the strong Markov property and probabilistic bounds for the stopping times. Note that in this approach the stochastic objects have to be constructed with zero initial condition on every subinterval to ensure smallness. This excludes the possibility of working with stationary stochastic data. In contrast, our approach avoids completely the stopping-time argument and instead controls the possible loss of dissipation for the dynamic described by the linearised equation directly. The key ingredients are the Boué\UDupuis variational representation combined with elementary energy estimates for the nonlinear dynamic driven by the noise shifted by the Boué\UDupuis control. Moreover, an auxiliary random walk representation is used to analyse the spatial propagation of perturbations. We identify functions ,\>\\> with periodic functions on >>. Kernels of operators are defined with respect to > times counting measure. The convolution is defined by <\equation*> >\f|)>\\*\>>K>**f. The > inner product and > norms are over ,\>>. In particular, <\equation*> |f,g|\>\\*\,\>>f*g,|f|\<\|\|\>>=|f|\<\|\|\>>,\>|)>>\|f,f|\>. We stress that sums in inner products, norms and quadratic forms always run over ,\>>, whereas the sum in the definition of the convolution is over >>. M.G. acknowledges support from the UKRI Frontier Research / EPSRC grant \PStochastic Analysis of Quantum fields\Q (EP/Z534328/1) and of the Simons Collaboration on \PProbabilistic QFT\Q. This paper has been written in TEXMACS (). An AI model helped with the derivation of the random-walk estimates stated in Lemmas and. It was also used to review the entire manuscript after its completion, correct some mathematical arguments, improve presentation as well as prepare the bibliographic entries. The authors take full responsibility for the content of the paper. We begin by studying how a perturbation of the initial data propagates under the linearised dynamic. We divide the spatial lattice into overlapping blocks of size , indexed by their centres>. For each block, we define the local > energy >> of the linearised solution inside a block >. The kinetic operator >> transfers energy between blocks, and controlling this transfer is essential for establishing decay of correlations. The main result shows that the transfer of energy between blocks is dominated by an auxiliary random walk with exponentially decaying jump rates. This reduces the problem of studying spatial propagation to estimating the auxiliary evolution on the lattice>> of block centres. Fix a nonnegative function \C>|)>>, supported in > and strictly positive on ,|]>>. The parameter 16> introduced below is fixed sufficiently big, depending only on > and >, such that the bounds, , and below hold true. Subsequently, the constant \0> is fixed sufficiently small. In what follows, constants |)>> may change from line to line, and are always independent of the cutoffs, the coupling constant \|]>>, and block indices. For torus size =M L>, let <\equation*> >\L*\/\*\\\,\> be the lattice of block centres. We reserve the symbols ,\,|\>,|\>\>> for block labels. For \,\>>, set <\equation*> |~>>\\>\+\*n|L>|)>,\>\|~>>|\>>|~>>|)>>. By our assumptions on >, the denominator is strictly positive. Thus, >> is smooth, and <\equation*> \>>\>=1,\>\B>\>|)>\L|}>. The support assumption gives <\equation*> \>\\>\\\d>,\|)>\2*L. Put <\equation*> \>>\\L:\\B>|}>,w>\>|)>|L>|)>. In two dimensions, we have <\equation> \*w>\C*L,sup >w>\C,\>\C*w>. Let >> be the kernel introduced in Lemma such that <\equation*> >|)>v|)>=\*q>*-v|)>,q>=0. For any lattice field define the local energy and the localisation error <\equation> e>\|\>*v|\<\|\|\>>,\>\\*q>*>-\>|)>*v*v. A direct computation yields <\equation> |\>*v,\>*v|\>=|\>*v,\>*>*v|)>|\>-*\>. For each deterministic initial condition >, define <\equation> \>>\inf\\>>>>||\|\<\|\|\>>=1>>>>>>|\,\>*\|\>+3*\*|>|)>-C,\>,\|\>|}>,|^>>>=>>|)>. Note that the local Rayleigh quotient >>> describes the smallest value of the quadratic form of the random Schrödinger operator <\equation*> \>+3\>|)>-C,\>|)>, appearing in the linearised equation, on functions supported in a given block >. <\proposition> There exists a conservative Markov generator > on >>, with nonnegative rates *\>>, \\>, such that, for every real lattice field , <\equation> \>\*e|)>>+e>,*e|)>>=\\>j*\>*>-e>|)>, and <\equation> sup,\> \\>j*\>*e>,\|)>/L>\. Moreover, if solves the linearised equation along >>, then <\equation> \*e>\*e|)>>-e>+2*|^>>>*e>. <\proof> Set <\equation*> k>=\*q>*>-\>|)>. By symmetry and Cauchy\USchwarz, <\equation> \>\\*k>*v. The bound >-\>|\|>\C*L*d>> and imply >\C*L>. If >|)>\L>, then >=0> and >\0> can only hold if >\d>|)>-L>. Thus, <\equation*> k>\>*e>|)>-L|)>/4>\>*e>|)>/L>, where the last inequality holds for 8>, after increasing . If >\0>, then >,\|)>\d>|)>+L> by the triangle inequality. Using >\>=1>, we obtain <\equation*> k>\>a*\>*\>,a*\>\>*e>,\|)>/L>. It follows from that <\equation> \>\>a*\>*e>. We have <\equation> sup> >a*\>+sup> >a*\>*e>,\|)>/L>\>. Let > be a conservative Markov generator with rates *\>=a*\>> for \\>. Then <\equation*> \>\>a*\>*e>=*e|)>>+>a*\>|)>*e>. Choosing sufficiently large and using we obtain and. Finally, differentiating >> in time and using, the localisation identity, the definition of the Rayleigh quotient, and gives. The local energy inequality contains two competing terms: a dissipative term >>, which produces exponential decay, and a random potential |^>>>>, which may produce a local instability. The Feynman\UKac representation allows us to express the resulting energy bound as an average over paths of an auxiliary random walk > on spatial blocks. For 0>, 1>, and a piecewise-constant path :\>>, define <\equation*> I\\,|)>\:r\I|}>, and <\equation> \|)>\1>|)>|\|>,\|)>\1>e|)>|\|>. Along each path, the random potential in produces an exponential factor determined by the accumulated loss of dissipation. To preserve exponential decay, we formulate the criterion, which allows a small cost |)>/8> for each block visited by the walk during a unit time interval. It also permits an additional, potentially big, contribution |)>> arising from the coming-down-from-infinity estimate for the nonlinear dynamic (see the first term on the RHS of). The distinction between these two contributions matters. Since the jump rates of the block walk are uniformly bounded, the exponential moments of |)>> grow at most exponentially in . Thus, their growth must be sufficiently small to be dominated by the deterministic dissipation. In contrast, the exponentially decaying weights in |)>> imply that its exponential moments are bounded uniformly in , even when multiplied by an arbitrarily large fixed constant. Thus the contribution involving |)>> changes only the multiplicative constant in the final estimate, whereas the contribution involving |)>> reduces the exponential decay rate. Note also that the naive bounds\ <\equation*> \>>\1-3\C,\>,|^>>>\3\C,\>, cannot be used directly to verify our criterion as ,\>\\> as \0>. To show one has to exhibit stochastic cancellations in the local Rayleigh quotient >>>. These cancellations will be analysed in subsequent sections. We also introduce the path functional <\equation> \>|)>\|^>>>r,\>|)>=sup\L> \>|)>, measuring the accumulated loss of dissipation, where the supremum in the definition of >|)>> runs over deterministic initial conditions >. Since >|)>> depends continuously on > and the finite dimensional state space is separable, >|)>> is measurable. The terminal time is determined by the domain of >. Let > be the continuous-time random walk on >> generated by >, independent of the noise, and denote by >>> the expectation conditioned on =\>. Define the uniform exponential jump cost <\equation> \\sup,\> \\>j*\>*>,\|)>/L+b>-1|)>, where \>> is the jump kernel of > from. <\proposition> Suppose that, for every 0>, and every deterministic piecewise-constant path :\>>, one has <\equation> \*exp>|)>|)>\exp|)>+\|)>|)> with a deterministic independent of > and the cutoffs. Then for every 0>, holds with deterministic independent of ,\,T,\,\>. <\proof> Let >> solve the linearised equation along >> on >, with >=\>*f>, and put >>=|\>v>|\<\|\|\>>>. The positive-semigroup comparison applied to and backward Feynman\UKac formula yield <\equation> e>>\e*\>>|^>>>r|)>*e>>|]>. The inequality holds pathwise for every initial condition. Since >=\>*f>, we have <\equation*> e>>=|\>*\>*f|\<\|\|\>>\e>,\|)>/L>|f|\<\|\|\>>. Define the random backward path =\>. Taking the supremum over initial conditions in gives <\equation*> <\aligned> \L> e>>*>|e*\>>>|)>|)>*|\>*\>*f|\<\|\|\>>|]>>>||e|f|\<\|\|\>>*\>>*>|)>|)>*e>,\|)>/L>|]>.>>>> Taking the supremum over |f|\<\|\|\>>\1> therefore yields <\equation*> sup\L>|f\ \>*U>>*>f|)>|\<\|\|\>>\L>\e*\>>*>|)>|)>*e>,\|)>/L>|]>. Hence by, we obtain <\equation*> \\L>|f\ \>*U>>*>f|)>|\<\|\|\>>\L>|]>\e*\>>*|)>+\|)>|)>*e>,\|)>/L>|]>. Lemma, applied with and , now gives <\equation*> \\L>|f\\>*U>>*>f|)>|\<\|\|\>>\L>|]>\C*e>,\|)>/L>*e*|]>*T>. Finally, by, <\equation> +\*\+*e\. This proves. The energy estimates discussed in Section will be applied to a decomposition of the stochastic quantisation solution into an explicit singular stochastic part and a more regular remainder. We first introduce the stationary Gaussian field and its Wick powers. For \3/2>, the Wick cube > is sufficiently regular to be treated directly as a forcing term in the energy estimates, and one may simply use the decomposition =X+|\>>. To cover the entire range \4/3>, we introduce the second order correction to remove > from the remainder equation. Let be the two sided Wiener process. Define stationary stochastic convolutions <\equation*> +\>|)>X=*>,X=X-C,\>,X=X-3*C,\>*X,+\>|)>Y=X, where ,\>> is the counterterm introduced in. Note that ,\>=\|]>>. Let >> be the infinite-lattice convolution kernel of >>>. We have <\equation*> X=*>>\W|)>,Y=>>\X|)>r. Put X-\*Y> and <\equation*> V\X-2*\*X*Y+\*Y=Z-C,\>,H\-3*\*X*Y+3*\*X*Y-\*Y. The fields and play the role of the random potential and the forcing term in the equation for the remainder \\-Z>. For a deterministic initial condition > and a control , let ,u>> denote the solution of the stochastic quantisation equation with the additional drift u> and the initial condition ,u>=\>. Define ,u>:=\,u>-Z>. We suppress the dependence on the initial condition in the notation when > is fixed, and write =\,u>> and =\,u>>. <\lemma> The remainder > satisfies <\equation> +\>|)>\+\|)>=-3*\*Z|)>-3*\*V*\-\*H+*u. <\proof> Expand |)>-3*C,\>*|)>> using *Y>, and cancel the cubic Wick term using the equation for . Fix \\-4|)>/12> and set <\equation> \=1-|2>+\. Then -4\\0> and \\/2>. Choose <\equation*> p\\,p\max |3\-4-6\>|}>,\= and define <\equation*> \>=>|)>|L>|)>>. The corresponding weighted Besov norms are defined in Appendix. The constant plays the role of the exponent of the size of the enhancement >> introduced below, and is chosen sufficiently large to absorb the powers of >> arising later in the energy and Rayleigh-quotient estimates in Section. The choice of > ensures \2>, which guarantees the spatial integrability of >*\>> used in Lemma. <\definition> For \>> and \>, define <\equation*> >\>||X|\<\|\|\>>>>|)>>+|X|\<\|\|\>>>>|)>>>>|||Y|\<\|\|\>>-3*\>>|)>>+|X*Y|\<\|\|\>>>>|)>>.>>>>> Note that >> does not depend on >. By Lemma, there is a deterministic constant such that for all \>>, \>, and \\1>, <\equation> |Z|\<\|\|\>>>>|)>>\|V|\<\|\|\>>>>|)>>\|H|\<\|\|\>>>>|)>>\C*\>. <\proposition> For every 1>, there is \\> such that <\equation> sup,\> sup\>> sup\> \*\>\C. <\proof> The result can be proved by adapting the proof technique from Section6 of, where analogous stochastic estimates are proved in the more complicated setting of the > model. To establish the exponential-moment bound required by Proposition, we shall apply in Section the Boué\UDupuis variational formula. In this section, we derive suitable pathwise energy estimates for solutions of the stochastic quantisation equation with an additional drift , which will play the role of the control in the Boué\UDupuis representation. For 1> define <\equation*> |f|\<\|\|\>>=\*|\|>,|f|\<\|\|\>>>=|>|)>*f|\<\|\|\>>,a\0. Let L*;L|)>> be a control. Recall that the controlled dynamic satisfies with an additional drift *u> and the initial condition =\>. Define <\equation> \>\|w>*\|\<\|\|\>>,\>\|w>*\|\<\|\|\>>/2>>+\*|w>*\|\<\|\|\>>, where =\-Z>. <\proposition> For \|]>>, there is 0> uniform in ,\,\>, the deterministic initial condition >, and the control , such that <\equation> |t>*\>+*\>+|2*L>>|)>+*\>\C\<\|\|\>w>*u\<\|\|\>+C*\*\>. <\proof> Write <\equation*> E=\>,K=|w>*\|\<\|\|\>>/2>>,P=|w>*\|\<\|\|\>>. Testing against >*\> and applying Lemmas and gives <\equation*> |t>*E+*K+*\*P\*E+2*\||w>*\,w>*u|\>\|+C*\*\>. Since E> and |w>*\,w>*u|\>|\|>\*E+C*|w>*u|\<\|\|\>>>, we obtain <\equation*> |t>*E+*K+*\*P\C\<\|\|\>w>*u\<\|\|\>+C*\*\>. Finally, gives *E\2*P>. Hence <\equation*> *E+*K\*K,|2*L>*E+|2>*P\*\*P. This proves. The energy inequality obtained above depends on the initial condition, whereas the estimate stated in Theorem is uniform in the initial data. To obtain the desired bound, we use the coming-down-from-infinity estimate below to bound the initial energy in terms of the forcing at earlier times, with a weight decaying exponentially in time. <\lemma> Let 0>, 1>, and recall the intervals > introduced in Section. Define <\equation> K\e/2>*r\k|}>>+I|}>>. Then <\equation> \*>\>r\C*L*e+C*\*K*|w>*u|\<\|\|\>>+\*\>|)>r. Moreover, <\equation> sup> 1>K\C,sup1> Kr\C. The constants above are deterministic and independent of ,\,\>, the initial condition >, the control , and \|]>>. <\proof> Let be as in and put <\equation*> h=C*e/2>*|w>*u|\<\|\|\>>+\*\>|)>s. Since *h+h/2> coincides with the forcing in and 0>, setting >-h|)>>, we arrive at <\equation> \*g+*g+|2L>*g\0, where we used >\g>. If 0>, the inverse satisfies <\equation*> \\+|2L>. Integrating this inequality gives <\equation*> >\|g>+|L>*-1|)>\|L>*-1|)>. If vanishes at some point in time, it must be identically zero by. Consequently, <\equation*> >-h|)>\|\*-1|)>>. Thus, independently of the initial energy, <\equation> \>\|\*-1|)>>+C*e/2>*|w>*u|\<\|\|\>>+\*\>|)>s,r\0. The bound follows by integrating over >, discarding the nonnegative terms, and bounding the energy at by, using -1|)>\C*e> for 1>. The kernel bounds follow by elementary estimates. For a reference solution =Z+\> starting from the initial condition > and driven by the noise shifted by , let >,u>> be defined by with >> replaced by ,u>>, and set |^>>,u>=>,u>|)>>. The main difficulty in estimating the local Rayleigh quotient is the absence of a uniform lower bound on the random potential due to the presence of the counterterm. The decomposition =Z+\> of the nonlinear solution separates this potential into a non-negative term quadratic in the remainder and two potentially negative terms involving the stochastic coefficients, <\equation> |)>-C,\>=|)>+V+2*Z*\. We estimate the last two terms using the weighted Besov estimates, which allow us to bound their contributions by the kinetic term and >>. As a result, we can bound |^>>,u>> entirely in terms of the dissipation >> and the stochastic enhancement. <\proposition> For \|]>> and any \\>>> such that |\|\<\|\|\>>=1>, one has <\equation> 3*\*|V+2*Z*\,\|\>\-|\,>-I|)>*\|\>-C**\>+\*\>|)>. In particular, <\equation> |^>>,u>\C*\*>+\>|)>. The constants are uniform in ,\,\,\,u>. <\proof> By and Lemma, we have <\equation*> <\aligned> |V,\|\>|\|>>|C*\>|\|\<\|\|\>>|\|\<\|\|\>>+2*\>>,>>||Z*\,\|\>|\|>>|C*\>*|w>\|\<\|\|\>>/2>>|\|\<\|\|\>>+\/2>>.>>>> Since |\|\<\|\|\>>=1>, interpolation between Sobolev spaces gives <\equation*> |\|\<\|\|\>>+2*\>>\|\|\<\|\|\>>/2>>>,|\|\<\|\|\>>+\/2>>\|\|\<\|\|\>>/2>>>, with <\equation*> \=+4*\|\>\1,\=+2*\|\>\1. Hence, by Young's inequality <\align*> |V,\|\>|\|>>|C*\*\>|\|\<\|\|\>>/2>>>\\\<\|\|\>\\<\|\|\>/2>>+C*\*\>|)>>,>>||Z*\,\|\>|\|>>|C*\*\>*|w>*\|\<\|\|\>>/2>>|\|\<\|\|\>>/2>>>\\\<\|\|\>\*|/2>>+\|\<\|\|\>>*w>*\\<\|\|\>/2>>+C*\*\>|)>>.>>>> By the assumptions on > and from Section, <\equation*> >\4\p,>=|3*\-4-6*\>\p. Since >\1>, |\|\<\|\|\>>/2>>\1+|\,>-I|)>*\|\>> and |w>*\|\<\|\|\>>/2>>\\>>, the above estimates imply. Finally, using and discarding the nonnegative first term in, we obtain. For a path :\>> and a control , define ,u>|)>> and >|)>> by replacing |^>>>> with |^>>,u>> in. Explicitly, >|)>\sup\L> \,u>|)>>. In this section, we estimate the accumulated loss of dissipation >|)>> along a fixed path> by combining the local Rayleigh-quotient bound with the energy estimates for the nonlinear dynamic. The estimate is uniform in the initial condition and the cutoffs and provides the main input for the application of the Boué\UDupuis formula in the next subsection. <\lemma> For every \|]>>, every 0>, every piecewise-constant path :\>>, and every control , <\equation> \>|)>\C*\|)>+C*\*|u|\<\|\|\>>r+C*\*\|)>,\*\|)>\C*\|)>. The constant is deterministic and uniform in ,\,\,T,\,u>. The nonnegative random variable |)>> depends only on the enhancement and the path, not on or >. <\proof> For I>, \|)>>. Hence yields <\equation*> \,u>|)>\C*\*1>\|)>>>>+\>|)>r. The contribution from the first term on the RHS of sums to *\|)>>. As \>> and \1>, the enhancement terms are bounded by *\|)>>, where <\equation> \|)>=*1>\|)>>K*\>r. By Proposition and, *\|)>\C*\|)>>. The contribution from the term on the RHS of involving the control is bounded independently of the number of blocks visited by the path. Indeed, by, we have <\equation*> \|)>>|w>*u|\<\|\|\>>\C|u|\<\|\|\>>. Summing over and using, we bound the total control contribution by *|u|\<\|\|\>>r>. Since the estimates above hold pathwise with a right-hand side independent of the initial condition, taking the supremum over all initial conditions gives. We finally establish the exponential-moment estimate in the propagation criterion formulated in Proposition. This finishes the proof of the decay estimate for the linearised evolution stated in Theorem. <\proposition> There exists a sufficiently small \0> such that, for every \|]>>, every 0>, and every piecewise-constant path :\>>, <\equation> \*exp >|)>|)>\exp|)>+\|)>|)>. The expectation is over the noise, and the constant is independent of ,T,\> and the cutoffs. <\proof> Since >\I> and -C,\>|)>\-3*\*C,\>>, we have \,u>|)>\3*\*C,\>*T>. For deterministic initial data the functional >|)>> depends only on the Wiener process after time zero. The same properties are true for >|)>>. Thus the Boué\UDupuis formula applies to this bounded functional and yields <\equation> log \*e>|)>>=sup \*>|)>-*|u|\<\|\|\>>r|]>. The supremum is over controls with finite expected > energy progressively measurable with respect to the filtration generated by the Wiener process after time zero. To bound the shifted functional, we use the decomposition ,u>=Z+\,u>> and use the enhancement from Section, which depends on the original, unshifted noise also before time zero. Note that the use of the noise before time zero plays an auxiliary role and does not affect the variational formulation. To finish the proof insert and fix > sufficiently small that \1/8>. To prove Theorem we combine the estimate for the linearised dynamic from Theorem with the Helffer\USjöstrand representation of the covariance. The estimate for the linearised evolution uses the coming-down-from-infinity property and becomes uniform in the initial condition only after a positive time. To exploit this estimate in the covariance representation below, we use stationarity to shift the tangent evolution from > to >. <\lemma> Let > be a stationary solution of with initial law ,\>>. For bounded > observables with bounded gradients, one has <\equation> ,\>>=>\,\>>|U>*\F|)>,\G|)>|\>|]> T. <\proof> Let > be the Markov semigroup associated to and write =\,\>> for the invariant measure. Reversibility and integration by parts give <\equation*> |T>*F*P*G\=-|\*F,\*P*G|\>\. Since the finite-dimensional dynamic is ergodic, *G\G\> in |)>> as \>. Integrating the preceding identity yields <\equation*> >=>|\*F,\*P*G|\>\T. By the chain rule, <\equation*> |\P*G|)>,f|\>=\|\G>|)>,U>f|\>|]>. Combining the last two identities and using stationarity to shift the time proves. <\render-proof|Proof of Theorem> Let > be stationary with fixed-time law ,\>>, and set <\equation*> f=\F|)>,g=\G|)>. By the partition of unity, <\equation*> |U>*f,g|\>=,\>|\>*U>*>*f|)>,\>*g|\>. Set <\equation*> F>=sup\L>|\>*\F|)>|\<\|\|\>>,G>=sup\L>|\>*\G|)>|\<\|\|\>>. Cauchy\USchwarz and Theorem, applied conditionally on > with initial data >*f> for the linearised dynamic, give <\equation*> <\aligned> >|U>*f,g|\>|\|>>|,\>G>*>*|\>*U>>*f|)>|\<\|\|\>>|)>>>||C*e*,\>F>*G>*e>,\|)>/>.>>>> As and have diameter at most , there are only finitely many > and > such that \>\ A\\> and \>\ B\\>. Moreover, for such indices, <\equation*> d>,\|)>\d>-2L. Consequently, <\equation*> >|U>*f,g|\>|\|>\C*e>/>*sup\L>|\*F|)>|sup\L>|\<\|\|\>>*\*G|)>|\<\|\|\>>. Integrating over 0> and using proves. In this appendix we collect estimates for the discrete random walk. We first represent the nonlocal operator >> as the generator of a symmetric random walk and establish exponential moment bounds for its jump kernel. These bounds are then used to control errors in the spatial localisation. Finally, we estimate exponential moments of the number of blocks visited by the random walk. The latter estimate is the key ingredient in the proof of the abstract propagation criterion. <\lemma> The operator >> is a symmetric conservative Markov generator. There is a nonnegative symmetric periodic kernel >>, with >=0>, such that <\equation> >|)>*f|)>=\*q>*-f|)>. Moreover, <\equation> sup,\,z> \*q>*d>*e>/4>\C>. <\proof> For =2>, the kernel is given, for z>, by <\equation*> q>=\*\|)>|}>>+\|)>|}>>|)>, where => and =>. The claimed properties follow directly. Suppose \2>. Let > be the continuous-time nearest-neighbour walk on >>, started at , with generator >>, and let >> denote its projection onto ,\>>. The identity <\equation*> I-\>=/2|\*/2|)>>*>e*s/2>*>>-I|)>s gives the desired generator representation with >=0> and, for z>, <\equation*> q>=/2|\*\*/2|)>>*>e*s/2>*\>=y-z|)>s. In particular, >> is nonnegative and symmetric. Let \>. Since e*\\z>> is an eigenfunction of >> with eigenvalue **\|)>-1|)>>, we obtain <\equation*> \*e*\\S>=exp>**\|)>-1|)>|)>. Differentiating twice in >, summing over >, and using \\1>, we obtain <\equation*> \|\|>*e|\|>/4>|]>\C*|)>*e,c=4*-1|)>\1. Since >>|)>\|\|>>, it follows that <\equation*> <\aligned> |*q>*d>*e>/4>>>||\/2|\*/2|)>>*>e*s/2>*\*|\|>*e|\|>/4>|]>s>>||\C>*>|)>*s/2>*e*s>s\C>.>>>> The integral is finite because \2> and 1>. This proves. <\lemma> For the fixed sufficiently large , all \>>, and all real lattice fields >, <\equation> 2*|w>*\,\>*\|\>\2\<\|\|\>w>*\\<\|\|\>/2>>-*|w>*\|\<\|\|\>> and <\equation> L*|w>*\|\<\|\|\>>\2*|w>*\|\<\|\|\>>. <\proof> Using the fact that \log /L|)>> is Lipschitz on |)>> with Lipschitz constant >, as well as the triangle inequality for >>, we obtain >-log w>|\|>\2*d>/L>. Consequently, for 8>, we have <\equation*> >-w>|)>|w>*w>>=2*>-log w>|)>-1|)>\>*d>*e>/4>. By we have <\equation*> 2*|w>*\,\>*\|\>=2\<\|\|\>w>*\\<\|\|\>/2>>-\*q>*>-w>|)>*\*\. Therefore, together with \a+b> give <\equation*> 2*|w>*\,\>*\|\>\2*|w>*\|\<\|\|\>>/2>>-C*L*|w>*\|\<\|\|\>>. Taking sufficiently large proves. For the second inequality, we note that <\equation*> \*w>\\*\>>/L|)>\/L|)>*>/L|)>x\2*L. Consequently, Cauchy\USchwarz gives <\equation*> |w>*\|\<\|\|\>>\*w>|)>*|w>*\|\<\|\|\>>\2*L*|w>*\|\<\|\|\>>, which proves. <\lemma> For every fixed 0> there is \\>, independent of , the cutoffs, and the block indices ,\>, such that <\equation> \>>|)>+b*\|)>>*e>,\|)>/L>|]>\C*e>,\|)>/L>*e|]>*T>. Here =\> and > is the exponential jump cost. <\proof> Set >=1>e\\>. For almost every >, let > be fixed by I>>. In the following, sums over r\T> run over the times when the walk > jumps. We have <\equation*> \|)>\T+1+N>,\|)>\c>+r\T>e/2>, where >> is the number of jumps of > during >. Moreover, the triangle inequality and =\> give <\equation*> d>,\|)>\d>,\|)>+r\T>d>,\|)>. Combining the preceding estimates yields <\equation> e|)>+b*\|)>>*e>,\|)>/L>\e>+b*-d>,\|)>/L>*exp r\T>F*,\|)>|)>, where <\equation*> F*|\>,|\>|)>\>|\>,|\>|)>|L>+b+a*e/2>. For s\T>, set <\equation*> h|\>|)>\\|\>>>*exp r\T>F*,\|)>|)>, where |\>>>> denotes the expectation for the walk started at |\>> at time . By conditioning on the first jump, satisfies, for almost every >, <\equation*> -\*h|\>|)>=|\>\|\>>j|\>|\>>**|\>,|\>|)>>*h|\>|)>-h|\>|)>|)>,h|\>|)>=1. Set <\equation*> m=sup|\>> |\>\|\>>j|\>|\>>**|\>,|\>|)>>-1|)>. The spatially constant function =expmr|)>> is a supersolution of the above backward equation, with the same terminal condition. Since the off-diagonal coefficients are nonnegative, comparison gives |)>\v>, proving <\equation> \>>*expr\T>F*,\|)>|)>\expsup|\>> |\>\|\>>j|\>|\>>**|\>,|\>|)>>-1|)>r|)>. By, , and >-1\a*e*\> for \\1>, <\equation*> <\aligned> |\>\|\>>j|\>|\>>**|\>,|\>|)>>-1|)>>||\>\|\>>j|\>|\>>*>|\>,|\>|)>/L+b>-1|)>+e*/2>>-1|)>*|\>\|\>>j|\>|\>>*e>|\>,|\>|)>/L>>>||\+a*e*e/2>.>>>> We have <\equation*> e/2>r\1>=k|}>|\|> e\1>e\c>. Consequently, by we arrive at <\equation*> \>>*exp r\T>F*,\|)>|)>\exp*T+a*e*c>|)>. Combining this with, we obtain with =exp >+a*e*c>|)>>. In this appendix we collect deterministic estimates in weighted Besov spaces on the lattice >> used to control products involving the singular stochastic terms and the nonlinear remainder. <\definition> Choose a smooth dyadic partition of unity |)>-1>> on > as in2.4>. Given the lattice spacing \0>, let >> be the smallest index such that the support of >>> intersects the boundary of the Brillouin zone and set <\equation*> \>=\,j\J>,\>>>=1-J>>\. Let >> be the inverse Fourier transform of >> on the infinite lattice. For periodic : >\\>, define <\equation*> \>*f=K>\f,-1\j\J>. For \\>, 0>, 1>, and periodic : >\\>, define <\equation> |F|\<\|\|\>>>>|)>>=supj\J>> 2>*|\>*\>*F|\<\|\|\>>> and <\equation> |f|\<\|\|\>>>>|)>>=>>2>*|\>\>*f|\<\|\|\>>,|f|\<\|\|\>>>>>|)>>=supj\J>> 2>*|\>*\>*f|\<\|\|\>> We omit the weight in the notation if it is identically equal to . Note that by Minkowski's inequality and /\*\C*/L|)>>>, we have <\equation> |\*f|)>|\<\|\|\>>\C**\>>|\|>*/L|)>>|)>*|\*f|\<\|\|\>>. The Fourier symbol of >> satisfies <\equation*> >|\|>\>*sin*\|2>|)>\|\|> on the Brillouin zone. Therefore, for \\>, <\equation> |f|\<\|\|\>>>>\>>2>*|\>*f|\<\|\|\>>. <\lemma> For \\1>, m\3>, and periodic >\\>, <\equation> |F,f|\>|\|>\C|F|\<\|\|\>>>>|)>>*|\>*f|\<\|\|\>>>>. <\proof> The result follows by Besov duality fromA.1--A.2>. Note that the periodic weights >> may be replaced with smooth periodic weights, without changing the norms. <\lemma> Let \> with 0>, and 0> with 3>. For periodic >\\>, <\equation> |f*g|\<\|\|\>>>>|)>>\C|f|\<\|\|\>>>|)>>|g|\<\|\|\>>>|)>>. <\proof> Apply the Bony decomposition of the product and use4.2>. <\lemma> For every \>>, every \\>>>, and periodic :\>\\>, <\align> |F,\|\>|\|>>|C|F|\<\|\|\>>>>|)>>|\|\<\|\|\>>|\|\<\|\|\>>+2*\>>,>>||G*\,\|\>|\|>>|C|G|\<\|\|\>>>>|)>>*|w>*\|\<\|\|\>>/2>>|\|\<\|\|\>>+\/2>>,>>>> where > is defined by. <\proof> Write >> and =\>>. Since \\B>>, the weights and their inverses are bounded and Lipschitz on a fixed neighbourhood of \>. Hence, the Sobolev multiplier estimate yields <\equation> |\*\|\<\|\|\>>>+||)>*\|\<\|\|\>>>\C|\|\<\|\|\>>>,0\b\\/2. By andA.7>, <\equation*> |F,\|\>|\|>\C|F|\<\|\|\>>>|)>>|*\|)>|\<\|\|\>>>>\C|F|\<\|\|\>>>|)>>|\|\<\|\|\>>|\|\<\|\|\>>+2*\>>. This proves. For the second estimate, first note that the paraproduct bounds of4.2>, together with the Sobolev embeddings, yield <\equation*> |f*f*f|\<\|\|\>>>>\C*|f|\<\|\|\>>>>,\\s\1,s\1+\. Apply this with =w*\>, =f=|)>*\> and =\/2>, =s=\+\/2>. The bound follows then by and. <\lemma> For every \>, \>>, and periodic :\>\\>, <\equation> |Z,w>*\|\>|\|>+6*|V,w>*\|\>|\|>+2*|H,w>*\|\>|\|>>>|\*|w>*\|\<\|\|\>>/2>>+*|w>*\|\<\|\|\>>+C*\>.>>>>> The constant is independent of ,\,\,t,\>, and \>. <\proof> Write >> and =\>>. Since \2>, we have <\equation*> \*\,\>>w*\\C. Therefore, <\equation*> |w*\|\<\|\|\>>+|w*\|\<\|\|\>>+|w*\|\<\|\|\>>\C. We use the weighted Besov estimates of, replacing our periodic weight by comparable smooth periodic weights. Observe also that by and the choice of >, <\equation*> \+2\\|4>,2\+2\\|2>. By, we obtain <\equation*> |w*\*\|\<\|\|\>>+2*\>>\C*|w*\|\<\|\|\>> |\|\<\|\|\>>/4>>|)>>\C*|w*\|\<\|\|\>>/2>>*|w*\|\<\|\|\>>. Hence, by and, we have <\equation*> <\aligned> |Z,w*\|\>|\|>>|C*|Z|>|)>>|\<\|\|\>>*w*\|\<\|\|\>>*|w*\*\|\<\|\|\>>+2*\>>>>||C*|Z|>|)>>|\<\|\|\>>*w*\|/2>>|\<\|\|\>>*w*\|\<\|\|\>>.>>>> Similarly, the quadratic estimate of gives <\equation*> <\aligned> |V,w*\|\>|\|>>|C*|V|>|)>>|\<\|\|\>>*w*\*\|\<\|\|\>>*|w*\|\<\|\|\>>+2*\>>>>||C*|V|>|)>>|\<\|\|\>>*w*\|/2>>|\<\|\|\>>*w*\|\<\|\|\>>.>>>> For the linear term, weighted Besov embedding and Hölder's inequality give <\equation*> <\aligned> |H,w*\|\>|\|>>|C*|H|>|)>>|\<\|\|\>>*w*\|\<\|\|\>>*|w*\|\<\|\|\>>+\>>>>||C*|H|>|)>>|\<\|\|\>>*w*\|\<\|\|\>>/2>>.>>>> Combining these estimates and applying Young's inequality, we get <\equation*> <\aligned> ||Z,w*\|\>*|V,w*\|\>*|H,w*\|\>|\|>>>||\\<\|\|\>w*\\<\|\|\>/2>>+*|w*\|\<\|\|\>>+C*|Z|\<\|\|\>>>|)>>+|V|\<\|\|\>>>|)>>+|H|\<\|\|\>>>|)>>|)>.>>>> The result follows now from, since 8>. >-invariant model> We briefly discuss the extension to the > model with fields valued in >, for fixed \>. The interaction potential is <\equation*> |4> :|\|>\|4>*|\|>-2**C,\>*|\|>+N**C,\>|)>,|\|>\|\|>+\+|\|>. The corresponding stochastic quantisation equation is <\equation*> \=->*\+\*|\|>-*C,\>|)>*\|]>t+W,\=,\,\|)>, where the components of ,\,W|)>> are independent. Let us explain how to adapt the construction of the stochastic objects from Section. We define to be the stationary vector-valued Gaussian convolution and set <\equation*> +\>|)>*Y=:*X:=-*C,\>|)>*X,Z=X-\*Y. The scalar Wick square > is replaced by the matrix =X\X-C,\>*I>. Summing up, the required enhancement consists of <\equation*> X,|)>,Y,|)>*Y,1\i,j,k\N, with the same weighted Besov regularities as , >, , and *Y> in the scalar case. To see why the deterministic estimates have the same structure, put =Z+\> and =Z\Z-C,\>*I>, where > is the identity matrix. Expansion of the original cubic nonlinearity produces the coercive term |\|>*\>, quadratic in > term involving , a linear term involving >, and a forcing coinciding with a linear combination of terms of the types *Y>, >, and >. In order to obtain a desired energy bound we can apply componentwise the product estimates in Lemma. The main point in the local Rayleigh estimate is that the Hessian of the interaction is matrix-valued but has the decomposition <\equation*> <\aligned> ||\|>-*C,\>|)>*I+2*\\\>>||=|\|>*I+2*\\\+B|)>*I+2*B+2*\|)>*I+2*\+\\Z|)>,>>>> where the first two terms have quadratic form |\|>+2*\v|)>\0>. The remaining terms are controlled componentwise by the quadratic and trilinear bounds of Lemma. This gives the analogue of Proposition, with the vector-valued local Rayleigh quotient defined using the symmetric quadratic form above. Finally, the localisation and block-transfer argument acts on the scalar energies >=|\>*v|\<\|\|\>>> and requires no positivity of a matrix-valued propagator. The block walk, Boué\UDupuis argument (with vector-valued controls), and covariance representation do not require any modifications. Thus, the proofs of Theorems and require only very minor modifications. Let us explain how to conclude almost sure exponential synchronisation of solutions of the infinite volume continuum equation driven by the same noise. This requires a minor extension of Theorem. Recall that =>:\\L|}>>. We claim that the estimate is also valid when the supremum over \\> is replaced with the supremum over its closed convex hull |\>|)>>.\ Let us first show that holds uniformly over \|\>|)>>. By convexity of the squared > norm, <\equation*> sup\|\>|)>>|w>-Z|)>|\<\|\|\>>\sup\L>|w>,u>-Z|)>|\<\|\|\>>, where \,u>:\\L|}>> and is the process introduced in Section. The RHS is bounded by the coming-down estimate. Consequently, applying the energy inequality to solutions restarted at time one with the initial condition \|\>|)>> yields. Thus, the proof of Lemma applies uniformly over |\>|)>>. Under a Cameron\UMartin shift, > is transformed into >, so Proposition and the argument of Proposition prove with the supremum over \|\>|)>>. This proves the convex hull extension of Theorem. To establish synchronisation, we study the random diameter of the solution set at time , <\equation*> \>\sup,\>|\> >-\>|)>|\<\|\|\>>>, where the supremum runs over all initial conditions. For arbitrary initial conditions ,\>, write =\>>. Differentiability of the solution map gives <\equation*> \>-\>=U*\+a*\>-\|)>*a. Since *\+a*\\|\>|)>>, we obtain <\equation*> \>\\>>\*\>*\>,\*\>\sup\|\>|)>>|\>*U>*\>|\<\|\|\>>\L>. By, , and Proposition, <\equation*> sup,\,\> \*\>\C>. Consequently, by the convex-hull extension of, discussed above, and the Cauchy\USchwarz inequality, we obtain <\equation*> \*\>\C>*e*\>>e>*,\|)>/>\C>*e. This implies almost sure exponential contraction of the diameter with a deterministic decay rate independent of the cutoffs. Unlike, this argument does not use order preservation and is therefore applicable to the vector-valued model. <\bibliography|bib|tm-alpha|fractional_phi42_mass.bib> <\bib-list|22> S.AlbeverioR.Høegh-Krohn. The Wightman axioms and the mass gap for strong interactions of exponential type in two-dimensional space-time. , 16:39\U82, 1974. S.Albeverio, R.Høegh-KrohnB.Zegarlinski. Uniqueness of Gibbs states for general |)>> weak-coupling models by cluster expansion. , 121(4):683\U697, 1989. M.BouéP.Dupuis. A variational representation for certain functionals of Brownian motion. , 26:1641\U1659, 1998. R.BauerschmidtB.Dagallier. Log-Sobolev inequality for the > and > measures. , 77(5):2579\U2612, 2024. R.Bauerschmidt, B.DagallierH.Weber. Holley\UStroock uniqueness method for the > dynamics. 2025. arXiv:2504.08606. D.C.Brydges, J.FröhlichA.D.Sokal. A new proof of the existence and nontriviality of the continuum > and > quantum field theories. , 91:141\U186, 1983. T.Cass, C.LittererT.Lyons. Integrability and tail estimates for Gaussian rough differential equations. , 41(4):3026\U3050, 2013. P.Duch, M.GubinelliP.Rinaldi. Parabolic stochastic quantisation of the fractional > model in the full subcritical regime. , 2026. Published online first. P.Duch, M.Hairer, J.YiW.Zhao. Ergodicity of infinite volume > at high temperature. 2025. arXiv:2508.07776. P.Duch. Flow equation approach to singular stochastic PDEs. , 6(2):327\U437, 2025. J.S.FeldmanK.Osterwalder. The Wightman axioms and the mass gap for weakly coupled |)>> quantum field theories. , 97(1):80\U135, 1976. M.GubinelliM.Hofmanová. A PDE construction of the Euclidean > quantum field theory. , 384:1\U75, 2021. M.Gubinelli, M.HofmanováN.Rana. Decay of correlations in stochastic quantization: the exponential Euclidean field in two dimensions. , 13:107\U145, 2025. J.Glimm, A.JaffeT.Spencer. The Wightman axioms and particle structure in the |)>> quantum field model. , 100(3):585\U632, 1974. F.Guerra, L.RosenB.Simon. Correlation inequalities and the mass gap in |)>>. III. Mass gap for a class of strongly coupled theories with nonzero external field. , 41(1):19\U32, 1975. B.GessP.Tsatsoulis. Synchronization by noise for the stochastic quantization equation in dimensions 2 and 3. , 20(6):2040006, 2020. B.HelfferJ.Sjöstrand. On the correlation for Kac-like models in the convex case. , 74(1\U2):349\U409, 1994. F.KunickP.Tsatsoulis. Gradient-type estimates for the dynamic >-model. , 181:104548, 2025. J.MartinN.Perkowski. Paracontrolled distributions on Bravais lattices and weak universality of the 2d parabolic Anderson model. , 55:2058\U2110, 2019. J.MagnenR.Sénéor. The infinite volume limit of the > model. , 24(2):95\U159, 1976. B.Simon. Correlation inequalities and the mass gap in |)>>. I. Domination by the two-point function. , 31:127\U136, 1973. R.ZhuX.Zhu. Lattice approximation to the dynamical > model. , 46(1):397\U455, 2018. > <\attachments> <\collection> <\associate|bib-bibliography> <\db-entry|+2DlbeszOKxZiZQ1|article|DuchGubinelliRinaldi2026> <|db-entry> M. P. > > model in the full subcritical regime> <\db-entry|+28mAsTlb18FmxCgh|article|HelfferSjostrand1994> <|db-entry> J. > <\db-entry|+28mAsTlb18FmxCgM|misc|DuchHairerYiZhao2025> <|db-entry> M. J. W. > > at high temperature> > <\db-entry|+2DlbeszOKxZiZQ2|article|GubinelliHofmanova2021> <|db-entry> M. > > quantum field theory> <\db-entry|+2DlbeszOKxZiZQ0|article|BoueDupuis1998> <|db-entry> P. > <\db-entry|+28mAsTlb18FmxCgQ|article|Simon1973> <|db-entry> > |)>>. I. Domination by the two-point function> <\db-entry|+28mAsTlb18FmxCgP|article|GlimmJaffeSpencer1974> <|db-entry> A. T. > |)>> quantum field model> <\db-entry|+28mAsTlb18FmxCgR|article|GuerraRosenSimon1975> <|db-entry> L. B. > |)>>. III. Mass gap for a class of strongly coupled theories with nonzero external field> <\db-entry|+28mAsTlb18FmxCgi|article|FeldmanOsterwalder1976> <|db-entry> K. > |)>> quantum field theories> <\db-entry|+ln6khbG2Tyo5VC|article|MagnenSeneor1976> <|db-entry> R. > > model> > <\db-entry|+28mAsTlb18FmxCgj|article|BrydgesFrohlichSokal1983> <|db-entry> J. A. D. > > and > quantum field theories> <\db-entry|+2EKYjfWH1XrHH8ne|article|AlbeverioHoeghKrohn1974> <|db-entry> R. > <\db-entry|+28mAsTlb18FmxCgL|article|GubinelliHofmanovaRana2025> <|db-entry> M. N. > <\db-entry|+2EKYjfWH1XrHH8nd|article|AlbeverioHoeghKrohnZegarlinski1989> <|db-entry> R. B. > |)>> weak-coupling models by cluster expansion> <\db-entry|+28mAsTlb18FmxCgT|article|BauerschmidtDagallier2024> <|db-entry> B. > > and > measures> <\db-entry|+28mAsTlb18FmxCgk|misc|BauerschmidtDagallierWeber2025> <|db-entry> B. H. > > dynamics> <\db-entry|+2EKYjfWH1XrHH8nf|article|Duch2025Flow> <|db-entry> > <\db-entry|+28mAsTlb18FmxCgn|article|CassLittererLyons2013> <|db-entry> C. T. > <\db-entry|+28mAsTlb18FmxCgo|article|KunickTsatsoulis2025> <|db-entry> P. > >-model> <\db-entry|+2DlbeszOKxZiZQ4|article|ZhuZhu2018> <|db-entry> X. > > model> <\db-entry|+2DlbeszOKxZiZQ3|article|MartinPerkowski2019> <|db-entry> N. > <\db-entry|+28mAsTlb18FmxCgp|article|GessTsatsoulis2020> <|db-entry> P. > <\references> <\collection> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > <\auxiliary> <\collection> <\associate|bib> DuchGubinelliRinaldi2026 HelfferSjostrand1994 DuchHairerYiZhao2025 GubinelliHofmanova2021 BoueDupuis1998 Simon1973 GlimmJaffeSpencer1974 GuerraRosenSimon1975 FeldmanOsterwalder1976 MagnenSeneor1976 BrydgesFrohlichSokal1983 AlbeverioHoeghKrohn1974 GubinelliHofmanovaRana2025 AlbeverioHoeghKrohnZegarlinski1989 BauerschmidtDagallier2024 BauerschmidtDagallierWeber2025 DuchHairerYiZhao2025 Duch2025Flow DuchGubinelliRinaldi2026 DuchHairerYiZhao2025 DuchHairerYiZhao2025 CassLittererLyons2013 KunickTsatsoulis2025 ZhuZhu2018 BoueDupuis1998 HelfferSjostrand1994 MartinPerkowski2019 GubinelliHofmanova2021 MartinPerkowski2019 GubinelliHofmanova2021 MartinPerkowski2019 GubinelliHofmanova2021 GubinelliHofmanova2021 GubinelliHofmanova2021 GubinelliHofmanova2021 GessTsatsoulis2020 DuchHairerYiZhao2025 <\associate|toc> |math-font-series||1Introduction and main results> |.>>>>|> |math-font-series||2Spatial propagation> |.>>>>|> |2.1Block decomposition and random walk representation |.>>>>|> > |2.2An abstract propagation criterion |.>>>>|> > |math-font-series||3Stochastic enhancement and remainder dynamic> |.>>>>|> |math-font-series||4Pathwise energy and Rayleigh estimates> |.>>>>|> |4.1Weighted energy inequality |.>>>>|> > |4.2Coming down from infinity and dissipation |.>>>>|> > |4.3Local Rayleigh-quotient bound |.>>>>|> > |math-font-series||5Path estimates and Boué\UDupuis> |.>>>>|> |5.1Bounds along block paths |.>>>>|> > |5.2Exponential path bound |.>>>>|> > |math-font-series||6Proof of the mass gap> |.>>>>|> |math-font-series||Appendix ARandom-walk estimates> |.>>>>|> |math-font-series||Appendix BWeighted lattice Besov estimates> |.>>>>|> |math-font-series||Appendix CExtension to the |O>-invariant model> |.>>>>|> |math-font-series||Appendix DSynchronisation by noise> |.>>>>|> |math-font-series||Bibliography> |.>>>>|>