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We explore mathematical relationship between holographic Wilsonian renormalization group(HWRG) and stochastic quantization(SQ) motivated by the similarity of the monotonicity in RG flow with Langevin dynamics of non-equilibrium thermodynamics. We look at scalar field theory in AdS space with its generic mass, self-interaction, and boundary deformation in the momentum space. Identifying the stochastic time $t$ with radial coordinate $r$ in AdS, we establish maps between the fictitious time evolution of stochastic multi point correlation function and the radial evolution of multi-trace deformation, which respectively, express the relaxation process of Langevin dynamics and holographic RG flow. We especially consider marginal multi-trace deformation on the AdS boundary which is successfully captured by a Langevin dynamics of SQ., Comment: 55+1 pages, 3 figures, typos are corrected, conclusion section is added

We explore the mathematical relationship between holographic Wilsonian renormalization group(HWRG) and stochastic quantization(SQ) of scalar field theory with its generic mass, self-interaction and $n$-multiple-trace deformation on the $d$-dimensional conformal boundary defined in AdS$_{d+1}$ spacetime. We understand that once we define our Euclidean action, $S_E$ as $S_E\equiv -2S_B$, then the stochastic process will reconstruct the holographic Wilsonian renormalization group data via solving Langevin equation and computing stochastic correlation functions. The $S_B$ is given by $S_B=S_{\rm ct}+S_{\rm def}$, where $S_{\rm ct}$ is the boundary counter term and $S_{\rm def}$ is the boundary deformation which gives a boundary condition. In our study, we choose the boundary condition adding (marginal)$n$-multiple trace deformation to the holographic dual field theory. In this theory, we establish maps bewteen ficticious time, $t$ evolution of stochastic $n$-point, ($2n-2$)-point correlation functions and the (AdS)radial, $r$ evolution of $n$-multiple-trace and ($2n-2$)-multiple-trace deformations respectively once we take identifications of $r=t$ and between some of constants appearing in both sides., Comment: 41 pages and 3 figures, typos are corrected

We study holographic entanglement entropy in 5-dimensional charged black brane geometry obtained from Einstein-SU(2)Yang-Mills theory defined in asymptotically AdS space. This gravity system undergoes second order phase transition near its critical point affected by a spatial component of the Yang-Mills fields, which is normalizable mode of the solution. This is known as phase transition between isotropic and anisotropic phases. We get analytic solutions of holographic entanglement entropies by utilizing the solution of bulk spacetime geometry given in arXiv:1109.4592, where we consider subsystems defined on AdS boundary of which shapes are wide and thin slabs and a cylinder. It turns out that the entanglement entropies near the critical point shows scaling behavior such that for both of the slabs and cylinder, $\Delta_\varepsilon S\sim\left(1-\frac{T}{T_c}\right)^\beta$ and the critical exponent $\beta=1$, where $\Delta_\varepsilon S\equiv S^{iso}-S^{aniso}$, and $S^{iso}$ denotes the entanglement entropy in isotropic phase whereas $S^{aniso}$ denotes that in anisotropic phase. We suggest a quantity $O_{12}\equiv S_1-S_2$ as a new order parameter near the critical point, where $S_1$ is entanglement entropy when the slab is perpendicular to the direction of the vector order whereas $S_2$ is that when the slab is parallel to the vector order. $O_{12}=0$ in isotropic phase but in anisotropic phase, the order parameter becomes non-zero showing the same scaling behavior. Finally, we show that even near the critical point, the first law of entanglement entropy is hold. Especially, we find that the entanglement temperature for the cylinder is $\mathcal T_{cy}=\frac{c_{ent}}{a}$, where $c_{ent}=0.163004\pm0.000001$ and $a$ is the radius of the cylinder., Comment: 1+29 pages, 4 figures, typos are corrected, a few paragraphs are added in introduction and discussion section is added