The quenched limiting distributions of a charged-polymer model

The limit distributions of the charged-polymer Hamiltonian of Kantor and Kardar [Bernoulli case] and Derrida, Griffiths and Higgs [Gaussian case] are considered. Two sources of randomness enter in the definition: a random field $q= (q_i){i\geq 1}$ of i.i.d.\ random variables, which is called the random \emph{charges}, and a random walk $S = (S_n){n \in \NN}$ evolving in $\ZZ^d$, independent of the charges. The energy or Hamiltonian $K = (K_n){n \geq 2}$ is then defined as $$K_n := \sum{1\leq i < j\leq n} q_i q_j {\bf 1}_{{S_i=S_j}}.$$ The law of $K$ under the joint law of $q$ and $S$ is called ''annealed'', and the conditional law given $q$ is called ''quenched''. Recently, strong approximations under the annealed law were proved for $K$. In this paper we consider the limit distributions of $K$ under the quenched law.