# Self-Organization When Pedestrians Move in Opposite Directions. Multi-Lane Circular Track Model
## 8 Self-Organization Occurs with Probability 1  
Since \(\Omega\) has only a finite number of elements, \(L^{2N}\) to be precise, and \(f_{n+N^{2}}({\bf v})=f_{n}({\bf v})\) for all non-negative integers \(n\) and all \({\bf v}\) in \(\Omega\), the set \(\{f_{n}({\bf v})\) such that \(n\) is a non-negative integer and \({\bf v}\) is in \(\Omega\}\) contains, at most, \(N^{2}L^{2N}\) different numbers. Thus, since it is finite, this set has a minimum. For future reference, we summarize this statement in the next observation. **Observation 1**. : _There exists a non-negative integer \(n\), that can be taken to be no greater than \(N^{2}\), and \({\bf v}\) in \(\Omega\) such that \(f_{n}({\bf v})\leq f_{k}({\bf u})\) for any non-negative integer \(k\) and \({\bf u}\) in \(\Omega\). For future reference, we call this minimum value \(\lambda\), i.e.,_  
\[\lambda=\min_{k,{\bf u}}f_{k}({\bf u}), \tag{15}\]  
_where the minimum is taken over all non-negative integers \(k\) and all \({\bf u}\) in \(\Omega\)._  
Let \(x,x_{1},\ldots,x_{r}\) be all real numbers. We say that \(x\) is a convex combination of \(x_{1},\ldots,x_{r}\) if there exits non-negative numbers \(a_{1},\ldots,a_{r}\) such that \(a_{1}+\ldots+a_{r}=1\) and \(x=a_{1}x_{1}+\ldots+a_{r}x_{r}\). There is a very simple and well known fact that will be useful to us. Namely, if \(x\) is a convex combination of \(x_{1},\ldots,x_{r}\) with positive coefficients (i.e., \(a_{i}>0\) for all \(i\)) and \(x\leq x_{i}\) for all \(i\), then \(x_{i}=x\) for all \(1\leq i\leq r\)Note that Equation (14) implies that \(f_{n}(\mathbf{u})\) is a convex combination of \(f_{n+1}(\mathbf{v})\) with \(\mathbf{v}\) in \(\Omega\). By restricting the sum on the right hand side of Equation (14) to only those \(\mathbf{v}\) such that \(P\left(\ell^{(n+1)}=\mathbf{v}\left|\ell^{(n)}=\mathbf{u}\right.\right)>0\), we can apply the fact stated in the previous paragraph to prove the validity of the following observation. **Observation 2**. : _Let \(n\) be a non-negative integer and \(\mathbf{u}\) in \(\Omega\). Assume \(f_{n}(\mathbf{u})=\lambda\) (as defined in Equation (15)). Let \(\mathbf{v}\) be also in \(\Omega\). If \(P\left(\ell^{(n+1)}=\mathbf{v}\left|\ell^{(n)}=\mathbf{u}\right.\right)>0\), then \(f_{n+1}(\mathbf{v})=\lambda\)_  
Repeated applications of Observation 2 leads to the next Observation. **Observation 3**. : _Let \(m\) be a non-negative integer and \(\mathbf{u}^{(0)}\) in \(\Omega\). Assume \(f_{m}\left(\mathbf{u}^{(0)}\right)=\lambda\). Let \(\mathbf{u}^{(1)},\ldots,\mathbf{u}^{(s)}\) be elements in \(\Omega\). Assume that \(P\left(\ell^{(m+k+1)}=\mathbf{u}^{(k+1)}\left|\ell^{(m+k)}=\mathbf{u}^{(k)} \right.\right)>0\) for all \(0\leq k<s\), then \(f_{m+s}\left(\mathbf{u}^{(s)}\right)=\lambda\)._  
For each positive integer \(n\), we define a set of functions \(\mathbf{F}_{n}\) defined on \(\Omega\) that also take values on \(\Omega\). Setting \(\mathbf{v}=\mathbf{F}_{n}(\mathbf{u})\), these functions are determined by the following rules:  
1. If \(u_{I(n)}\
eq u_{N+I(n)}\) then \(\mathbf{v}=\mathbf{u}\). 2. If \(u_{I(n)}=u_{N+I(n)}=1\) then \(v_{N+I(n)}=2\) and \(v_{i}=u_{i}\) for all \(i\
eq N+J(n)\). 3. If \(u_{I(n)}=u_{N+I(n)}\
eq 1\) then \(v_{I(n)}=u_{I(n)}-1\) and \(v_{i}=u_{i}\) for all \(i\
eq I(n)\). Note that, if \(\ell^{(n-1)}=\mathbf{u}\) and \(\ell^{(n)}=\mathbf{F}_{n}(\mathbf{u})\), each individual stayed in its lane if there was no collision at time \(t_{n}\); the \(J(n)\) individual moving clockwise moved to lane 2 if there was a collision at time \(t_{n}\) and that collision occurred in lane 1; the \(I(n)\) individual moving counterclockwise moved to lane \(\ell-1\) if there was a collision at time \(t_{n}\) and that collision occurred in lane \(\ell\) with \(\ell>1\).