Directed polymer in a random medium - an introduction

I Directed polymers A Why bother? II Hamiltonian and Randomness A Pure case B Let’s put randomness 1 Hamiltonian 2 Partition function 3 Averages C What to look for? D Size: disorder and thermal correlation 1 Free energy fluctuation 2 Our Aim 3 Digression: Self-averaging III Q.: Is disorder relevant? A Annealed average: low temperature problem B Moments of Z 1 Hamiltonian for moments 2 Bound state: two polymer problem C RG approach 1 Expansion in potential 2 Reunion 3 Divergences 4 RG flows 5 Relevant, irrelevant and marginal 6 Fixed points: strong vs weak disorder D Ans.: Yes, it is! IV Quantitative Results on exponents A Replica 1 n → 0 and N → ∞ B Bethe ansatz 1 Flory approach V Can we handle free energy? A Free energy and the KPZ equation B Thermal correlation and an exact result 1 Exact result on response: pure like C Free energy of extension: pure like D RG of the KPZ equation 1 Scale transformation and an important relation VI Virtual reality: Intuitive picture A Rare events B Probability distribution 1 Response to a force 2 Scaling approach 3 And so the probability distribution is C Confinement energy VII Overlap VIII Summary and open problems Acknowledgments References