Place the pomegranate juice in a small pot and bring to a boil over high heat. Boil vigorously, reducing the heat to keep it from boiling over, for five to eight minutes. The juice should look syrupy and be reduced to about 1/4 cup. Set aside to cool. Alternatively, skip this step and use 1/4 cup pomegranate molasses.
Place the walnuts and garlic in a food processor and process until minced. Add the peppers and process until smooth, stopping to scrape down the sides of the bowl as needed. Add the cooled pomegranate reduction (or pomegranate molasses), lemon juice, salt, cumin, and red-pepper flakes, and process to mix. Drizzle in the olive oil with the machine still running, and process until smooth.
Photography by: Andrea D’Agosto; Prop Styling: Alicia Buszczak; Food Styling: Paul Jackman