In this paper we study the phase transition of continuum Widom–Rowlinson measures in Rd\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}^d$$\end{document} with q\documentclass[12pt]{minimal}
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\begin{document}$$q$$\end{document} types of particles and random radii. Each particle xi\documentclass[12pt]{minimal}
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\begin{document}$$x_i$$\end{document} of type i is marked by a random radius ri\documentclass[12pt]{minimal}
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\begin{document}$$r_i$$\end{document} distributed by a probability measure Qi\documentclass[12pt]{minimal}
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\begin{document}$$Q_i$$\end{document} on R+\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}^+$$\end{document}. The distributions Qi\documentclass[12pt]{minimal}
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\begin{document}$$Q_i$$\end{document} may be different for different i, this setting is called the non-symmetric case. The particles of same type do not interact with each other whereas a particle xi\documentclass[12pt]{minimal}
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\begin{document}$$x_i$$\end{document} and xj\documentclass[12pt]{minimal}
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\begin{document}$$x_j$$\end{document} with different type i≠j\documentclass[12pt]{minimal}
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\begin{document}$$i\ne j$$\end{document} interact via an exclusion hardcore interaction forcing ri+rj\documentclass[12pt]{minimal}
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\begin{document}$$r_i+r_j$$\end{document} to be smaller than |xi-xj|\documentclass[12pt]{minimal}
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\begin{document}$$|x_i-x_j|$$\end{document}. In the symmetric integrable case (i.e. ∫rdQ1(dr)<+∞\documentclass[12pt]{minimal}
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\begin{document}$$\int r^dQ_1(dr)<+\infty $$\end{document} and Qi=Q1\documentclass[12pt]{minimal}
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\begin{document}$$Q_i=Q_1$$\end{document} for every 1≤i≤q\documentclass[12pt]{minimal}
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\begin{document}$$1\le i\le q$$\end{document}), we show that the Widom–Rowlinson measures exhibit a standard phase transition providing uniqueness, when the activity is small, and co-existence of q ordered phases, when the activity is large. In the non-integrable case (i.e. ∫rdQi(dr)=+∞\documentclass[12pt]{minimal}
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\begin{document}$$\int r^dQ_i(dr)=+\infty $$\end{document}, 1≤i≤q\documentclass[12pt]{minimal}
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\begin{document}$$1\le i \le q$$\end{document}), we show another type of phase transition. We prove, when the activity is small, the existence of at least q+1\documentclass[12pt]{minimal}
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\begin{document}$$q+1$$\end{document} extremal phases and we conjecture that, when the activity is large, only the q ordered phases subsist. We prove a weak version of this conjecture in the symmetric case by showing that the Widom–Rowlinson measure with free boundary condition is a mixing of the q\documentclass[12pt]{minimal}
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\begin{document}$$q$$\end{document} ordered phases if and only if the activity is large.
