deletions | additions       diff --git a/We_expect_the_ISAT.tex b/We_expect_the_ISAT.tex  index cd68991..78a57ab 100644  --- a/We_expect_the_ISAT.tex  +++ b/We_expect_the_ISAT.tex 
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We expect the ISAT transition to be somewhat shifted away from $(\omega,\tau)=(1,3)$ due to considering a finite-size ensemble. If one considers the vertical line $\omega=1$, one notices that the maximal eigenvalue peaks at a value of $\tau$ somewhat greater than $3$. We remind that the phase transition for $\omega=1$ is a strong second-order phase transition where the specific heat diverges with an exponent $0.68(5)$. For $\omega<1$, there exists a line of even stronger peaks that join with the peak at $\omega=1$, from which one can infer that there exists a strong phase transition on varying $\tau$ for each $\omega<1$. This is borne out by finite-size scaling analysis. Specifically, we have studied the model when $\omega=0.5$:  for this value of $\omega$ we have simulated the model using a one-parameter flatPERM algorithm up to length $n = 1024$, running $7.9 \cdot 10^6$ iterations, and collecting $2.7 \cdot 10^{10}$ samples at the maximum length. We find that the specific heat divergence is commensurate with a first-order transition with a linear divergence. To test this assumption of a first-order transition, we consider the distribution of the number of contacts for various values of $\tau$ near the peak of the specific heat. Figure \ref{fig:double_peaks} shows a clear bimodal distribution, confirming the first-order character of the transition.  It is hence likely that there exists a line of first-order phase transitions at values of $\tau$ near $3$ for each value of $\omega\lesssim 1$. We will return to the question of whether the line of first-order transitions extends all the way to $\omega=1$ below.  Returning to our two-parameter data, let us first note that when $\tau=0$ our model is the ISAW model and hence there is a weak $\theta$-like transition at $(\omega, \tau) = (\omega_c,0)$ with $\omega_c \sim 1.94$ (see \cite{Caracciolo_2011} and references therein for recent estimates of $\omega_c$), which is reflected in the density plot by a broad peak near $\omega = 2$ on varying $\omega$ when $\tau=0$. A line of such weak peaks extends to larger values of $\tau$. When $\tau=1$, the model becomes the previously studied INNSAT \cite{Bedini:2013eg}, which also demonstrated a weak $\theta$-like transition with exponent estimates encompassing ISAW values. We thus conjecture that the entire line lies in the $\theta$-universality class.  Returning to our two-parameter data, let us first note that when $\tau=0$ our model is the ISAW model and hence there is a weak $\theta$-like transition at $(\omega, \tau) = (\omega_c,0)$ with $\omega_c \sim 1.94$ (see \cite{Caracciolo:2011iz} and references therein for recent estimates of $\omega_c$), which is reflected in the density plot by a broad peak near $\omega = 2$ on varying $\omega$ when $\tau=0$. A line of such weak peaks extends to larger values of $\tau$. When $\tau=1$, the model becomes the previously studied INNSAT \cite{Bedini:2013eg}, which also demonstrated a weak $\theta$-like transition with exponent estimates encompassing ISAW values. We thus conjecture that the entire line lies in the $\theta$-universality class. Now if the suggestion that the ISAT collapse corresponds to an infinite order multi-critical point \cite{Nahum:2013ei} \cite{Nahum_2013}  is correct, it is then natural and simplest to conjecture that the line of first-order transitions meets with the line of $\theta$-like transitions at that point. We obtain an indication of where the two lines might join by considering the ratio between the two eigenvalues of the covariance matrix $H_{ij} = \partial^2_{ij} \log Z_n$ where $i, j \in \{\omega, \tau\}$. This is based on the heuristic argument that the component $\tau\tau$ (respectively $\omega\omega$) is the specific heat associated to multiply visited sites (nearest-neighbour contacts) and will dominate the spectrum when the transition is driven by this interaction. When the two eigenvalues coincide is then argued to indicate the presence of a higher order critical point. The density plot of the eigenvalue ratio is shown on Figure~\ref{fig:eigenvalue_plot}. One clearly observes a unique point close to the ISAT collapse point, where the two eigenvalues have the same magnitude.