We prove an almost sure central limit theorem (CLT) for spatial extension of stretched (meaning subject to a non-zero pulling force) polymers at very weak disorder in all dimensions $d+1\geq 4$.
"An almost sure CLT for stretched polymers." Electron. J. Probab. 18 1 - 20, 2013. https://doi.org/10.1214/EJP.v18-2231