{"paper":{"title":"Freezing transition of the directed polymer in a $1+d$ random medium : location of the critical temperature and unusual critical properties","arxiv_id":"cond-mat/0603041","primary_cat":"cond-mat.dis-nn","submitted_at":"2006-03-02T09:21:12+00:00","version_compiled":"v2"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13-20260917 on the question bank abstract-p1-20260923, exactly as returned. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v2","sentences":[{"i":0,"p":1.0,"id":"s0","own":false,"role":"background","text":"In dimension $d \\geq 3$, the directed polymer in a random medium undergoes a phase transition between a free phase and a disorder dominated phase.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.05,"main_claim_p":0.01,"quantitative":true,"quantitative_p":0.79,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":1,"p":1.0,"id":"s1","own":false,"role":"background","text":"For the latter, Fisher and Huse have proposed a droplet theory based on the scaling of the free energy fluctuations $\\Delta F(l) \\sim l^{\\theta}$.","hedge":{"p":0.92,"label":"asserted"},"own_p":0.02,"main_claim_p":0.01,"quantitative":false,"quantitative_p":0.21,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":2,"p":1.0,"id":"s2","own":false,"role":"background","text":"On the other hand, in related growth models belonging to the KPZ universality class, Forrest and Tang have found that the height-height correlation function is logarithmic at the transition.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.03,"main_claim_p":0.01,"quantitative":false,"quantitative_p":0.11,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":3,"p":0.71,"id":"s3","own":false,"role":"background","text":"For the directed polymer model at criticality, this translates into logarithmic free energy fluctuations $\\Delta F_{T_c}(l) \\sim (\\ln l)^{\\sigma}$ with $\\sigma=1/2$.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.09,"main_claim_p":0.02,"quantitative":true,"quantitative_p":0.97,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.17,"background":0.71,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.11}},{"i":4,"p":0.8,"id":"s4","own":true,"role":"contribution","text":"In this paper, we propose a droplet scaling analysis exactly at criticality based on this logarithmic scaling.","hedge":{"p":0.85,"label":"asserted"},"own_p":0.8,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.06,"role_probabilities":{"gap":0.0,"method":0.2,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.8,"interpretation":0.0}},{"i":5,"p":1.0,"id":"s5","own":true,"role":"result","text":"Our main conclusion is that the typical correlation length $\\xi(T)$ of the low temperature phase, diverges as $ \\ln \\xi(T) \\sim (- \\ln (T_c-T))^{1/\\sigma} \\sim (- \\ln (T_c-T))^{2} $.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.88,"main_claim_p":0.98,"quantitative":true,"quantitative_p":0.93,"role_probabilities":{"gap":0.0,"method":0.0,"result":1.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":6,"p":0.97,"id":"s6","own":true,"role":"result","text":"Furthermore, the logarithmic dependence of $\\Delta F_{T_c}(l)$ leads to the conclusion that the critical temperature $T_c$ actually coincides with the explicit upper bound $T_2$ derived by Derrida and coworkers, where $T_2$ corresponds to the temperature below which the ratio $\\bar{Z_L^2}/(\\bar{Z_L})^2$ diverges exponentially in $L$.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.66,"main_claim_p":0.04,"quantitative":true,"quantitative_p":0.89,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.97,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.03}},{"i":7,"p":0.43,"id":"s7","own":true,"role":"background","text":"Finally, since the Fisher-Huse droplet theory was initially introduced for the spin-glass phase, we briefly mention the similarities and differences with the directed polymer model.","hedge":{"p":0.6,"label":"unclear"},"own_p":0.5,"main_claim_p":0.01,"quantitative":false,"quantitative_p":0.14,"role_probabilities":{"gap":0.0,"method":0.01,"result":0.01,"background":0.43,"limitation":0.08,"future_work":0.07,"contribution":0.33,"interpretation":0.07}},{"i":8,"p":0.51,"id":"s8","own":false,"role":"interpretation","text":"If one speculates that the free energy of droplet excitations for spin-glasses is also logarithmic at $T_c$, one obtains a logarithmic decay for the mean square correlation function at criticality $\\bar{C^2(r)} \\sim 1/(\\ln r )^{\\sigma}$.","hedge":{"p":0.98,"label":"speculative"},"own_p":0.23,"main_claim_p":0.01,"quantitative":true,"quantitative_p":0.58,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.29,"background":0.03,"limitation":0.0,"future_work":0.11,"contribution":0.06,"interpretation":0.51}}]},"compiled":{"main_claim":{"p":1.0,"text":"Our main conclusion is that the typical correlation length $\\xi(T)$ of the low temperature phase, diverges as $ \\ln \\xi(T) \\sim (- \\ln (T_c-T))^{1/\\sigma} \\sim (- \\ln (T_c-T))^{2} $.","location":"abstract s5","hedge_strength":{"p":1.0,"label":"asserted"}},"provenance":{"route":{"via":"OpenRouter","model":"classifier-1.13","endpoint":"https://openrouter.ai/api/v1/systemone","provider":"TypeSafe"},"packing":"one request per 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