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Andrea Bedini added section_Numerical_results_We_studied__.tex
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\section{Numerical results}
We studied the Three-body ISAW model using the FlatPERM algorithm \cite{Prellberg_2004} which is based on the Pruned and Enriched Rosenbluth Method (PERM) developed in \cite{Grassberger:1997eh}.
For the PERM algorithm, at each iteration a polymer configurations is generated kinetically (which is to say that each growth step is selected at random from all possible growth steps) along with a weight factor to correct the sample bias. At each growth step, configuration with very high weight relative to other configurations of the same size are enriched (duplicated) while configuration with low weight or that cannot be grown any further are pruned (discarded). Despite introducing a correlation between each iteration, this simple mechanism greatly improves the algorithm efficiency. A single iteration is then concluded when all configurations have been pruned and the total number of samples generated during each iteration depends on the problem at hand and on the details of the enriching/pruning strategy.
FlatPERM extends this method by cleverly choosing the enrichment and pruning steps to generate for each polymer size $n$ a quasi-flat historgram in some choosen micro-canonical quantities $\mathbf{k}=(k_1,k_2,\dotsc,k_{\ell})$ and producing an estimate $W_{n,\mathbf{k}}$ of the total weight of the walks of length $n$ at fixed values of $\mathbf{k}$. From the total weight one can access physical quantities over a broad range of temperatures through a simple weighted average, e.g.
%
\begin{align}
\< \mathcal O \>_n(\tau) = \frac{\sum_{\mathbf{k}} \mathcal O_{n,\mathbf{k}}\,
\left(\prod_j \tau_j^{k_j}\right) \, W_{n,\mathbf{k}}}{\sum_\mathbf{k} \left(\prod_j \tau_j^{k_j}\right) \, W_{n,\mathbf{k}}}.
\end{align}
%
The quantities $k_j$ may be any subset of the physical parameters of
the model. To study the full two parameter phase space we set $(k_1, k_2) = (f_2, f_3)$ and $(\tau_1, \tau_2) = (\omega_2, \omega_3)$.
We have run the flatPERM algorithm limiting the configurations to a length of $n = 256$ and running $7.32 \cdot 10^6$ iterations. In this way we produced $5.72 \cdot 10^{10}$ samples at the maximum length. As common with the flatPERM algorithm, one can also count the configurations by the fraction of its steps made independently, this gives a measure of the number of ``effectively independent samples''. Our numerical study collected $2.79 \cdot 10^9$ effective samples.
To obtain a landscape of possible phase transitions, we plot the largest eigenvalue of the matrix of second derivatives of the free energy with respect to $\tau$ and $\omega$ (measuring the strength of the fluctuations and covariance in $m_t$ and $m_c$) at length $n=256$ on the left-hand side of Figure~\ref{fig:eigenvalue_plot}.
