4 of the correlations of the initial configuration with the evolving configurations one recovers the same result of the replica formalism. An explicit isomorphism of the two approaches can be shown to be present: the relevant computations will be presented in [23]. part that pushes the singularity in the complex plane; in the same way the coupling constant takes an imagi- nary part. Although the bare coupling is real, the fixed point may correspond to an imaginary coupling constant. One could argue that asymptotically the exponent are like those of lattice animals for the complex singularity. Let us came back to the replica approach and let us consider in more detail the stochastic differential equation (14) that is the resummation of the leading perturbative contributions. We are interested to study it in a non- perturbative way. However the previous conclusion may be to hasty. Us- ing the same arguments of [24–26] one finds that the sum of the leading diagrams is related to the solution of the stochastic differential equation governing the local fluc- tuations of the overlap φ ( x ) = q ( x ) − q (being q the space average of q ( x )) It is well know that in perturbation theory the solution of the stochastic differential equation is unique and that there an hidden supersymmetry [24–26]. The supersym- metry relates the two propagators and gives where ω ( x ) is a Gaussian short range noise, that is ω ( x ) ω ( y ) = δ ( x − y ). A and g are smooth functions of the temperature and they are chosen in such a way to implement the condition φ$_{ω}$ = 0. The two propagators are given by the relations This supersymmetric relation is at the origin of the di- mensional reduction : the critical exponents of lattice an- imals problem in dimensions D are the same of those of the Ising model near the Lee-Yang singularity in dimen- sion D − 2. G$_{1}$ ( x ) ∝ 1 x ∂G$_{0}$ ( x ) ∂x . (17) − Δ φ$_{ω}$ ( x ) + A + ϵφ$_{ω}$ ( x ) + gφ$_{ω}$ ( x ) 2 = ω ( x ) (14) G$_{0}$ ( x ) = φ$_{ω}$ ( x ) φ$_{ω}$ (0) , G$_{1}$ ( x ) = φ$_{ω}$ ( x ) ω (0) = ( 1 − Δ + ϵ + 2 gφ )$_{x,}$$_{0}$ , (15) F [ q ]$_{ω}$ = ∫ dx ( 1 2 ∂ $^{2}$q ( x ) ∂x 2 + W ( q ( x )) − ω ( x ) q ( x ) . (18) However in presence of multiple solutions (as e.g. for the Random Field Ising Model) everything becomes more complex (multiple solutions cannot be seen in perturba- tion theory so that this problem does not affect the per- turbative analysis that we have presented above). Super- symmetry and dimensional reduction are only valid if we average over all the solution with a sign depending on the parity of the corresponding Morse index: this weight is not the natural one from the physical viewpoint [24–26]. Which is the correct weight? If we stay within the replica formalism it is useful to consider the free energy where the last equality follows from integration by part (and it is correct only in case the solution to the stochastic differential equation is unique). Neglecting technicalities, the physics is quite clear. The choice of the variables σ (i.e. the initial conditions in the dynamics) induce point dependent shift of the critical temperature and the effects of these fluctuations is the dominant one. One may wonder if there is a direct role of G$_{1}$ ( x ) in the dynamics. A suggestion is the following. Let us consider a theory where the microscopic evolution equations for the particle have a stochastic nature. In this case the overlap q$_{σ,τ}$ ( x ) will be a function of both the time t and of the noise η ( x, t ) and it will be denoted by q$_{σ,η}$ ( x, t ). We can define a different dynamical susceptibility: Equation (14) can be written as δF [ q ]$_{ω}$ /δq ( x ) = 0. Now the natural choice would be to consider among the many solutions the one that minimize F [ q ]$_{ω}$. On the other hand this choice is not natural in the dynamics where we would like to take the solution q$_{M}$ ( x ) that maximize q ( x ). In- deed it can be proved [28] that there is a solution q$_{M}$ ( x ) such that q$_{M}$ ( x ) > q ( x ) for all x and for any possible solutions of the stochastic differential equation (this is a well know fact in the Random Field Ising Model [28]). In both cases dimensional reduction is no more valid and non-perturbative effects are present. A neat formulation of the problem is the following: we introduce a fictitious time s and we write the following evolution equation: and the corresponding correlation G$_{22}$ ( x, t ), where the overline denotes the average over initial conditions and angular brackets average over η [34]. One could argue that in the region of time where C ( t ) is near to (and above) the plateaux, G$_{22}$ ( x, t ) should behaves as G$_{1}$ ( x ). On the other hand in the region where C ( t ) is small, the behaviour of G$_{22}$ ( x, t ) and G$_{4}$ ( x, t ) should be similar. The whole analysis can be redone in the dynamical approach where one takes care in an explicit way the de- pendence of the correlations on the initial configuration (a fact that was neglected in [19]). The computations can be done in a neat way within the Martin-Siggia-Rose (MSR) formalism of equilibrium dynamics using a uni- versal cubic dynamical action close to the mode-coupling transition. Detailed computations shows that taking care with the boundary conditions at time s = 0 given by q ( x, 0) = 1. In the region below T$_{c}$ , where the equation ∂q ( x, s ) ∂s = − Δ q ( x, s ) + W $^{′}$( q ( x, s )) − ω ( x ) (19) V χ$_{22}$ ( t ) = 〈 Q$_{σ,η}$ ( t ) $^{2}$〉 − 〈 Q$_{σ,η}$ ( t ) 〉 2 , (16)