Statistics of low energy excitations for the directed polymer in a $1+d$ random medium ($d=1,2,3$)
| dc.creator | Monthus, Cecile | |
| dc.creator | Garel, Thomas | |
| dc.date | 2006-02-08 | |
| dc.date.accessioned | 2026-07-07T07:49:27Z | |
| dc.date.available | 2026-07-07T07:49:27Z | |
| dc.description | We consider a directed polymer of length $L$ in a random medium of space dimension $d=1,2,3$. The statistics of low energy excitations as a function of their size $l$ is numerically evaluated. These excitations can be divided into bulk and boundary excitations, with respective densities $ρ^{bulk}_L(E=0,l)$ and $ρ^{boundary}_L(E=0,l)$. We find that both densities follow the scaling behavior $ρ^{bulk,boundary}_L(E=0,l) = L^{-1-θ_d} R^{bulk,boundary}(x=l/L)$, where $θ_d$ is the exponent governing the energy fluctuations at zero temperature (with the well-known exact value $θ_1=1/3$ in one dimension). In the limit $x=l/L \to 0$, both scaling functions $R^{bulk}(x)$ and $R^{boundary}(x)$ behave as $R^{bulk,boundary}(x) \sim x^{-1-θ_d}$, leading to the droplet power law $ρ^{bulk,boundary}_L(E=0,l)\sim l^{-1-θ_d} $ in the regime $1 \ll l \ll L$. Beyond their common singularity near $x \to 0$, the two scaling functions $R^{bulk,boundary}(x)$ are very different : whereas $R^{bulk}(x)$ decays monotonically for $0<x<1$, the function $R^{boundary}(x)$ first decays for $0<x<x_{min}$, then grows for $x_{min}<x<1$, and finally presents a power law singularity $R^{boundary}(x)\sim (1-x)^{-σ_d}$ near $x \to 1$. The density of excitations of length $l=L$ accordingly decays as $ρ^{boundary}_L(E=0,l=L)\sim L^{- λ_d} $ where $λ_d=1+θ_d-σ_d$. We obtain $λ_1 \simeq 0.67$, $λ_2 \simeq 0.53$ and $λ_3 \simeq 0.39$, suggesting the possible relation $λ_d= 2 θ_d$. | |
| dc.description | 15 pages, 25 figures | |
| dc.identifier | https://arxiv.org/abs/cond-mat/0602200 | |
| dc.identifier | http://arxiv.org/abs/cond-mat/0602200 | |
| dc.identifier | Phys. Rev. E 73, 056106 (2006) | |
| dc.identifier | doi:10.1103/PhysRevE.73.056106 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/124856 | |
| dc.subject | Disordered Systems and Neural Networks | |
| dc.subject | Probability | |
| dc.title | Statistics of low energy excitations for the directed polymer in a $1+d$ random medium ($d=1,2,3$) | |
| dc.type | text |